Murray Wiki - User contributions [en]

CDS 140b, Spring 2014

2014-06-01T03:46:25Z

Macmardg:

{| width=100%
|-
| colspan=2 align=center |
Differential Equations and Dynamical Systems__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* Richard Murray (CDS/BE), murray@cds.caltech.edu
* Doug MacMartin (CDS), macmardg@cds.caltech.edu
* Lectures: MWF, 1-2 pm, 314 ANB
* Office hours: TBD
| width=50% |
'''Teaching Assistant'''
* Katie Broersma (CDS)
* Contact: cds140-tas@cds.caltech.edu
* Office hours: Wed, 3:30-4:30 pm, 101 STH
|}

=== Course Description ===
'''CDS 140b is a continuation of CDS 140a'''. A large part of the course will focus on tools from nonlinear dynamics, such as perturbation theory and averaging, advanced stability analysis, the existence of periodic orbits, bifurcation theory, chaos, etc. In addition, guest lecturers will give an introduction to current research topics in dynamical systems theory. There will be 8 homeworks throughout the term but no exams.


=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Topic'''
| '''Suggested Reading/Lecture Notes'''
| '''Homework'''
|- valign=top
| 0
| 31 Mar RMM
| Course overview
|
|
|- valign=top
| 1
| 2 Apr 4 Apr* DGM
| Lagrangian and Hamiltonian systems I
* Definition and properties
* Newtonian Systems
|
*Perko 2.14
* [http://www.cds.caltech.edu/~macmardg/courses/cds140a/MarsdenMechSystems.pdf Marsden, Mechanical systems]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_03_32.pdf Lecture notes]
| [[CDS 140b Spring 2014 Homework 1|HW 1]] Due: 10 Apr (Thu)
|- valign=top
| 2
| 9 Apr* 11 Apr DGM
| Lagrangian and Hamiltonian systems II
* Hamiltonian systems with 2 degrees of freedom
* Mechanical systems
|
* Perko 3.6
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_05_21.pdf Lecture notes]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/hetero_halos_4-12-00.pdf Notes on Lagrange points]
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw2-wi14.pdf HW 2] Due: 17 Apr (Thu)
|- valign=top
| 3
| 18 Apr 21 Apr RMM
| Advanced stability theory
* Non-autonomous (time-varying) systems
* Input-to-state stability, input-output stability, small gain theorem
|
* Khalil 4.5, 4.6-4.7 (skim), 5.1-5.4
| [[CDS 140b Spring 2014 Homework 3|HW 3]] 
Due 1 May (Thu)
|- valign=top
| 4
| 28 Apr '''(12-1 pm)''' 30 Apr RMM
| Stability of perturbed systems
* Vanishing perturbations
* Non-vanishing perturbations
* Slowiy varying systems
|
* Khalil, 9.1, 9.2, 9.6
| [[CDS 140b Spring 2014 Homework 4|HW 4]] 
Due 8 May (Thu)
|- valign=top
| 5
| 5 May 7 May RMM
| Perturbation theory
* Periodic perturbations
* Averaging
* Singular Perturbations
|
* Khalil, 10.3-10.4, 11.1-11.3, 11.5
| [[CDS 140b Spring 2014 Homework 5|HW 5]] 
Due 15 May (Thu)
|- valign=top
| 6
| 12 May 14 May RMM
| Interconnected systems
* Absolute stability
* Describing functions
|
* Khalil, 7.1-7.2
| [[CDS 140b Spring 2014 Homework 6|HW 6]] 
Due 22 May (Thu)
|- valign=top
| 7
| 19 May 21 May DGM
| Nonlinear control I
* Overview of techniques
* Controllability and Lie brackets
* Gain scheduling
* Feedback linearization
|
* Khalil 12.2,12.5 (gain scheduling), 13.1-13.3 (feedback linearization)
* Isidori (chapter 2) or Nijmeijer and van der Schaft, 3.1 (for more on controllability)
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/NonlinearControl1.pdf Lecture notes]
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw7-sp14.pdf HW 7] Due: 29 May (Thu)
|- valign=top
| 8
| 23 May* 30 May DGM
| Nonlinear control II
* Backstepping
* Sliding mode control
|
* Khalil 14.3, 14.1
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/Backstepping.pdf Lecture notes]
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw8-sp14.pdf HW 8] Due: 6 June (Thu)
|}

=== References: ===

'''Course Textbooks'''

* H. Khalil, Nonlinear Systems, Prentice Hall; 3rd edition, 2001. ISBN: 978-0130673893
* S. Strogatz, Nonlinear Dynamics And Chaos, Westview Press, 1994. ISBN: 978-0738204536
* F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer; 2ed Edition, 1996. ISBN: 978-3540609346

'''Additional Sources:'''
* L. Perko, Differential Equations and Dynamical Systems (3rd), Springer, 2001. ISBN: 978-0387951164
* S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer; 2nd edition, 2003. ISBN: 978-0387001777

=== Policies ===

'''Collaboration Policy'''

Homeworks are to be done and handed in individually. To improve the learning process, students are encouraged to discuss the problems with, provide guidance to and get help from other students, the TAs and instructors. However, to make sure each student understands the concepts, solutions must be written independently and should reﬂect your understanding of the subject matter at the time of writing. Copying solutions, using solutions from previous years, having someone else type or dictate any part of the solution manual or using publicly available solutions (from the Internet) are not allowed.

'''Grading Policy'''

The final grades will be evaluated based on homework assignments, with the lowest score dropped in computing the final grade.


[[Category:Courses]]

CDS 140b, Spring 2014

2014-05-23T21:29:22Z

Macmardg:

{| width=100%
|-
| colspan=2 align=center |
Differential Equations and Dynamical Systems__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* Richard Murray (CDS/BE), murray@cds.caltech.edu
* Doug MacMartin (CDS), macmardg@cds.caltech.edu
* Lectures: MWF, 1-2 pm, 314 ANB
* Office hours: TBD
| width=50% |
'''Teaching Assistant'''
* Katie Broersma (CDS)
* Contact: cds140-tas@cds.caltech.edu
* Office hours: Wed, 3:30-4:30 pm, 101 STH
|}

=== Course Description ===
'''CDS 140b is a continuation of CDS 140a'''. A large part of the course will focus on tools from nonlinear dynamics, such as perturbation theory and averaging, advanced stability analysis, the existence of periodic orbits, bifurcation theory, chaos, etc. In addition, guest lecturers will give an introduction to current research topics in dynamical systems theory. There will be 5 homeworks throughout the term but no exams. Instead, the students are required to select a research topic and a journal paper related to CDS140b and present a brief review of the paper. The details of the projects will be discussed in the class.

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Topic'''
| '''Suggested Reading/Lecture Notes'''
| '''Homework'''
|- valign=top
| 0
| 31 Mar RMM
| Course overview
|
|
|- valign=top
| 1
| 2 Apr 4 Apr* DGM
| Lagrangian and Hamiltonian systems I
* Definition and properties
* Newtonian Systems
|
*Perko 2.14
* [http://www.cds.caltech.edu/~macmardg/courses/cds140a/MarsdenMechSystems.pdf Marsden, Mechanical systems]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_03_32.pdf Lecture notes]
| [[CDS 140b Spring 2014 Homework 1|HW 1]] Due: 10 Apr (Thu)
|- valign=top
| 2
| 9 Apr* 11 Apr DGM
| Lagrangian and Hamiltonian systems II
* Hamiltonian systems with 2 degrees of freedom
* Mechanical systems
|
* Perko 3.6
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_05_21.pdf Lecture notes]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/hetero_halos_4-12-00.pdf Notes on Lagrange points]
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw2-wi14.pdf HW 2] Due: 17 Apr (Thu)
|- valign=top
| 3
| 18 Apr 21 Apr RMM
| Advanced stability theory
* Non-autonomous (time-varying) systems
* Input-to-state stability, input-output stability, small gain theorem
|
* Khalil 4.5, 4.6-4.7 (skim), 5.1-5.4
| [[CDS 140b Spring 2014 Homework 3|HW 3]] 
Due 1 May (Thu)
|- valign=top
| 4
| 28 Apr '''(12-1 pm)''' 30 Apr RMM
| Stability of perturbed systems
* Vanishing perturbations
* Non-vanishing perturbations
* Slowiy varying systems
|
* Khalil, 9.1, 9.2, 9.6
| [[CDS 140b Spring 2014 Homework 4|HW 4]] 
Due 8 May (Thu)
|- valign=top
| 5
| 5 May 7 May RMM
| Perturbation theory
* Periodic perturbations
* Averaging
* Singular Perturbations
|
* Khalil, 10.3-10.4, 11.1-11.3, 11.5
| [[CDS 140b Spring 2014 Homework 5|HW 5]] 
Due 15 May (Thu)
|- valign=top
| 6
| 12 May 14 May RMM
| Interconnected systems
* Absolute stability
* Describing functions
|
* Khalil, 7.1-7.2
| [[CDS 140b Spring 2014 Homework 6|HW 6]] 
Due 22 May (Thu)
|- valign=top
| 7
| 19 May 21 May DGM
| Nonlinear control I
* Overview of techniques
* Controllability and Lie brackets
* Gain scheduling
* Feedback linearization
|
* Khalil 12.2,12.5 (gain scheduling), 13.1-13.3 (feedback linearization)
* Isidori (chapter 2) or Nijmeijer and van der Schaft, 3.1 (for more on controllability)
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/NonlinearControl1.pdf Lecture notes]
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw7-sp14.pdf HW 7] Due: 29 May (Thu)
|- valign=top
| 8
| 23 May* 30 May DGM
| Nonlinear control II
* Backstepping
* Sliding mode control
|
* Khalil 14.3, 14.1
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/Backstepping.pdf Lecture notes]
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw8-sp14.pdf HW 8] Due: 6 June (Thu)
|}

=== References: ===

'''Course Textbooks'''

* H. Khalil, Nonlinear Systems, Prentice Hall; 3rd edition, 2001. ISBN: 978-0130673893
* S. Strogatz, Nonlinear Dynamics And Chaos, Westview Press, 1994. ISBN: 978-0738204536
* F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer; 2ed Edition, 1996. ISBN: 978-3540609346

'''Additional Sources:'''
* L. Perko, Differential Equations and Dynamical Systems (3rd), Springer, 2001. ISBN: 978-0387951164
* S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer; 2nd edition, 2003. ISBN: 978-0387001777

=== Policies ===

'''Collaboration Policy'''

Homeworks are to be done and handed in individually. To improve the learning process, students are encouraged to discuss the problems with, provide guidance to and get help from other students, the TAs and instructors. However, to make sure each student understands the concepts, solutions must be written independently and should reﬂect your understanding of the subject matter at the time of writing. Copying solutions, using solutions from previous years, having someone else type or dictate any part of the solution manual or using publicly available solutions (from the Internet) are not allowed.

'''Grading Policy'''

The final grades will be evaluated based on homework assignments (75%) and final projects (25%).

== Projects ==

TBD

[[Category:Courses]]

CDS 140b, Spring 2014

2014-05-21T23:12:51Z

Macmardg:

{| width=100%
|-
| colspan=2 align=center |
Differential Equations and Dynamical Systems__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* Richard Murray (CDS/BE), murray@cds.caltech.edu
* Doug MacMartin (CDS), macmardg@cds.caltech.edu
* Lectures: MWF, 1-2 pm, 314 ANB
* Office hours: TBD
| width=50% |
'''Teaching Assistant'''
* Katie Broersma (CDS)
* Contact: cds140-tas@cds.caltech.edu
* Office hours: Wed, 3:30-4:30 pm, 101 STH
|}

=== Course Description ===
'''CDS 140b is a continuation of CDS 140a'''. A large part of the course will focus on tools from nonlinear dynamics, such as perturbation theory and averaging, advanced stability analysis, the existence of periodic orbits, bifurcation theory, chaos, etc. In addition, guest lecturers will give an introduction to current research topics in dynamical systems theory. There will be 5 homeworks throughout the term but no exams. Instead, the students are required to select a research topic and a journal paper related to CDS140b and present a brief review of the paper. The details of the projects will be discussed in the class.

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Topic'''
| '''Suggested Reading/Lecture Notes'''
| '''Homework'''
|- valign=top
| 0
| 31 Mar RMM
| Course overview
|
|
|- valign=top
| 1
| 2 Apr 4 Apr* DGM
| Lagrangian and Hamiltonian systems I
* Definition and properties
* Newtonian Systems
|
*Perko 2.14
* [http://www.cds.caltech.edu/~macmardg/courses/cds140a/MarsdenMechSystems.pdf Marsden, Mechanical systems]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_03_32.pdf Lecture notes]
| [[CDS 140b Spring 2014 Homework 1|HW 1]] Due: 10 Apr (Thu)
|- valign=top
| 2
| 9 Apr* 11 Apr DGM
| Lagrangian and Hamiltonian systems II
* Hamiltonian systems with 2 degrees of freedom
* Mechanical systems
|
* Perko 3.6
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_05_21.pdf Lecture notes]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/hetero_halos_4-12-00.pdf Notes on Lagrange points]
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw2-wi14.pdf HW 2] Due: 17 Apr (Thu)
|- valign=top
| 3
| 18 Apr 21 Apr RMM
| Advanced stability theory
* Non-autonomous (time-varying) systems
* Input-to-state stability, input-output stability, small gain theorem
|
* Khalil 4.5, 4.6-4.7 (skim), 5.1-5.4
| [[CDS 140b Spring 2014 Homework 3|HW 3]] 
Due 1 May (Thu)
|- valign=top
| 4
| 28 Apr '''(12-1 pm)''' 30 Apr RMM
| Stability of perturbed systems
* Vanishing perturbations
* Non-vanishing perturbations
* Slowiy varying systems
|
* Khalil, 9.1, 9.2, 9.6
| [[CDS 140b Spring 2014 Homework 4|HW 4]] 
Due 8 May (Thu)
|- valign=top
| 5
| 5 May 7 May RMM
| Perturbation theory
* Periodic perturbations
* Averaging
* Singular Perturbations
|
* Khalil, 10.3-10.4, 11.1-11.3, 11.5
| [[CDS 140b Spring 2014 Homework 5|HW 5]] 
Due 15 May (Thu)
|- valign=top
| 6
| 12 May 14 May RMM
| Interconnected systems
* Absolute stability
* Describing functions
|
* Khalil, 7.1-7.2
| [[CDS 140b Spring 2014 Homework 6|HW 6]] 
Due 22 May (Thu)
|- valign=top
| 7
| 19 May 21 May DGM
| Nonlinear control I
* Overview of techniques
* Controllability and Lie brackets
* Gain scheduling
* Feedback linearization
|
* Khalil 12.2,12.5 (gain scheduling), 13.1-13.3 (feedback linearization)
* Isidori (chapter 2) or Nijmeijer and van der Schaft, 3.1 (for more on controllability)
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/NonlinearControl1.pdf Lecture notes]
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw7-sp14.pdf HW 7] Due: 29 May (Thu)
|- valign=top
| 8
| 23 May* 30 May DGM
| Nonlinear control II
* Backstepping
* Sliding mode control
|
* Khalil 14.3, 14.1
| HW 8
|}

=== References: ===

'''Course Textbooks'''

* H. Khalil, Nonlinear Systems, Prentice Hall; 3rd edition, 2001. ISBN: 978-0130673893
* S. Strogatz, Nonlinear Dynamics And Chaos, Westview Press, 1994. ISBN: 978-0738204536
* F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer; 2ed Edition, 1996. ISBN: 978-3540609346

'''Additional Sources:'''
* L. Perko, Differential Equations and Dynamical Systems (3rd), Springer, 2001. ISBN: 978-0387951164
* S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer; 2nd edition, 2003. ISBN: 978-0387001777

=== Policies ===

'''Collaboration Policy'''

Homeworks are to be done and handed in individually. To improve the learning process, students are encouraged to discuss the problems with, provide guidance to and get help from other students, the TAs and instructors. However, to make sure each student understands the concepts, solutions must be written independently and should reﬂect your understanding of the subject matter at the time of writing. Copying solutions, using solutions from previous years, having someone else type or dictate any part of the solution manual or using publicly available solutions (from the Internet) are not allowed.

'''Grading Policy'''

The final grades will be evaluated based on homework assignments (75%) and final projects (25%).

== Projects ==

TBD

[[Category:Courses]]

CDS 140b, Spring 2014

2014-05-20T22:28:33Z

Macmardg:

{| width=100%
|-
| colspan=2 align=center |
Differential Equations and Dynamical Systems__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* Richard Murray (CDS/BE), murray@cds.caltech.edu
* Doug MacMartin (CDS), macmardg@cds.caltech.edu
* Lectures: MWF, 1-2 pm, 314 ANB
* Office hours: TBD
| width=50% |
'''Teaching Assistant'''
* Katie Broersma (CDS)
* Contact: cds140-tas@cds.caltech.edu
* Office hours: Wed, 3:30-4:30 pm, 101 STH
|}

=== Course Description ===
'''CDS 140b is a continuation of CDS 140a'''. A large part of the course will focus on tools from nonlinear dynamics, such as perturbation theory and averaging, advanced stability analysis, the existence of periodic orbits, bifurcation theory, chaos, etc. In addition, guest lecturers will give an introduction to current research topics in dynamical systems theory. There will be 5 homeworks throughout the term but no exams. Instead, the students are required to select a research topic and a journal paper related to CDS140b and present a brief review of the paper. The details of the projects will be discussed in the class.

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Topic'''
| '''Suggested Reading/Lecture Notes'''
| '''Homework'''
|- valign=top
| 0
| 31 Mar RMM
| Course overview
|
|
|- valign=top
| 1
| 2 Apr 4 Apr* DGM
| Lagrangian and Hamiltonian systems I
* Definition and properties
* Newtonian Systems
|
*Perko 2.14
* [http://www.cds.caltech.edu/~macmardg/courses/cds140a/MarsdenMechSystems.pdf Marsden, Mechanical systems]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_03_32.pdf Lecture notes]
| [[CDS 140b Spring 2014 Homework 1|HW 1]] Due: 10 Apr (Thu)
|- valign=top
| 2
| 9 Apr* 11 Apr DGM
| Lagrangian and Hamiltonian systems II
* Hamiltonian systems with 2 degrees of freedom
* Mechanical systems
|
* Perko 3.6
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_05_21.pdf Lecture notes]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/hetero_halos_4-12-00.pdf Notes on Lagrange points]
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw2-wi14.pdf HW 2] Due: 17 Apr (Thu)
|- valign=top
| 3
| 18 Apr 21 Apr RMM
| Advanced stability theory
* Non-autonomous (time-varying) systems
* Input-to-state stability, input-output stability, small gain theorem
|
* Khalil 4.5, 4.6-4.7 (skim), 5.1-5.4
| [[CDS 140b Spring 2014 Homework 3|HW 3]] 
Due 1 May (Thu)
|- valign=top
| 4
| 28 Apr '''(12-1 pm)''' 30 Apr RMM
| Stability of perturbed systems
* Vanishing perturbations
* Non-vanishing perturbations
* Slowiy varying systems
|
* Khalil, 9.1, 9.2, 9.6
| [[CDS 140b Spring 2014 Homework 4|HW 4]] 
Due 8 May (Thu)
|- valign=top
| 5
| 5 May 7 May RMM
| Perturbation theory
* Periodic perturbations
* Averaging
* Singular Perturbations
|
* Khalil, 10.3-10.4, 11.1-11.3, 11.5
| [[CDS 140b Spring 2014 Homework 5|HW 5]] 
Due 15 May (Thu)
|- valign=top
| 6
| 12 May 14 May RMM
| Interconnected systems
* Absolute stability
* Describing functions
|
* Khalil, 7.1-7.2
| [[CDS 140b Spring 2014 Homework 6|HW 6]] 
Due 22 May (Thu)
|- valign=top
| 7
| 19 May 21 May DGM
| Nonlinear control I
* Overview of techniques
* Controllability and Lie brackets
* Gain scheduling
* Feedback linearization
|
* Khalil 12.2,12.5 (gain scheduling), 13.1-13.3 (feedback linearization)
* Isidori (chapter 2) or Nijmeijer and van der Schaft, 3.1 (for more on controllability)
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw7-sp14.pdf HW 7] Due: 29 May (Thu)
|- valign=top
| 8
| 23 May* 30 May DGM
| Nonlinear control II
* Backstepping
* Sliding mode control
|
* Khalil 14.3, 14.1
| HW 8
|}

=== References: ===

'''Course Textbooks'''

* H. Khalil, Nonlinear Systems, Prentice Hall; 3rd edition, 2001. ISBN: 978-0130673893
* S. Strogatz, Nonlinear Dynamics And Chaos, Westview Press, 1994. ISBN: 978-0738204536
* F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer; 2ed Edition, 1996. ISBN: 978-3540609346

'''Additional Sources:'''
* L. Perko, Differential Equations and Dynamical Systems (3rd), Springer, 2001. ISBN: 978-0387951164
* S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer; 2nd edition, 2003. ISBN: 978-0387001777

=== Policies ===

'''Collaboration Policy'''

Homeworks are to be done and handed in individually. To improve the learning process, students are encouraged to discuss the problems with, provide guidance to and get help from other students, the TAs and instructors. However, to make sure each student understands the concepts, solutions must be written independently and should reﬂect your understanding of the subject matter at the time of writing. Copying solutions, using solutions from previous years, having someone else type or dictate any part of the solution manual or using publicly available solutions (from the Internet) are not allowed.

'''Grading Policy'''

The final grades will be evaluated based on homework assignments (75%) and final projects (25%).

== Projects ==

TBD

[[Category:Courses]]

CDS 140b, Spring 2014

2014-05-20T22:27:38Z

Macmardg:

{| width=100%
|-
| colspan=2 align=center |
Differential Equations and Dynamical Systems__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* Richard Murray (CDS/BE), murray@cds.caltech.edu
* Doug MacMartin (CDS), macmardg@cds.caltech.edu
* Lectures: MWF, 1-2 pm, 314 ANB
* Office hours: TBD
| width=50% |
'''Teaching Assistant'''
* Katie Broersma (CDS)
* Contact: cds140-tas@cds.caltech.edu
* Office hours: Wed, 3:30-4:30 pm, 101 STH
|}

=== Course Description ===
'''CDS 140b is a continuation of CDS 140a'''. A large part of the course will focus on tools from nonlinear dynamics, such as perturbation theory and averaging, advanced stability analysis, the existence of periodic orbits, bifurcation theory, chaos, etc. In addition, guest lecturers will give an introduction to current research topics in dynamical systems theory. There will be 5 homeworks throughout the term but no exams. Instead, the students are required to select a research topic and a journal paper related to CDS140b and present a brief review of the paper. The details of the projects will be discussed in the class.

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Topic'''
| '''Suggested Reading/Lecture Notes'''
| '''Homework'''
|- valign=top
| 0
| 31 Mar RMM
| Course overview
|
|
|- valign=top
| 1
| 2 Apr 4 Apr* DGM
| Lagrangian and Hamiltonian systems I
* Definition and properties
* Newtonian Systems
|
*Perko 2.14
* [http://www.cds.caltech.edu/~macmardg/courses/cds140a/MarsdenMechSystems.pdf Marsden, Mechanical systems]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_03_32.pdf Lecture notes]
| [[CDS 140b Spring 2014 Homework 1|HW 1]] Due: 10 Apr (Thu)
|- valign=top
| 2
| 9 Apr* 11 Apr DGM
| Lagrangian and Hamiltonian systems II
* Hamiltonian systems with 2 degrees of freedom
* Mechanical systems
|
* Perko 3.6
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_05_21.pdf Lecture notes]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/hetero_halos_4-12-00.pdf Notes on Lagrange points]
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw2-wi14.pdf HW 2] Due: 17 Apr (Thu)
|- valign=top
| 3
| 18 Apr 21 Apr RMM
| Advanced stability theory
* Non-autonomous (time-varying) systems
* Input-to-state stability, input-output stability, small gain theorem
|
* Khalil 4.5, 4.6-4.7 (skim), 5.1-5.4
| [[CDS 140b Spring 2014 Homework 3|HW 3]] 
Due 1 May (Thu)
|- valign=top
| 4
| 28 Apr '''(12-1 pm)''' 30 Apr RMM
| Stability of perturbed systems
* Vanishing perturbations
* Non-vanishing perturbations
* Slowiy varying systems
|
* Khalil, 9.1, 9.2, 9.6
| [[CDS 140b Spring 2014 Homework 4|HW 4]] 
Due 8 May (Thu)
|- valign=top
| 5
| 5 May 7 May RMM
| Perturbation theory
* Periodic perturbations
* Averaging
* Singular Perturbations
|
* Khalil, 10.3-10.4, 11.1-11.3, 11.5
| [[CDS 140b Spring 2014 Homework 5|HW 5]] 
Due 15 May (Thu)
|- valign=top
| 6
| 12 May 14 May RMM
| Interconnected systems
* Absolute stability
* Describing functions
|
* Khalil, 7.1-7.2
| [[CDS 140b Spring 2014 Homework 6|HW 6]] 
Due 22 May (Thu)
|- valign=top
| 7
| 19 May 21 May DGM
| Nonlinear control I
* Overview of techniques
* Controllability and Lie brackets
* Gain scheduling
* Feedback linearization
|
* Khalil 12.2,12.5 (gain scheduling), 13.1-13.3 (feedback linearization)
* Isidori (chapter 2) or Nijmeijer and van der Schaft, 3.1 (for more on controllability)
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw7-sp14.pdf HW 7]
|- valign=top
| 8
| 23 May* 30 May DGM
| Nonlinear control II
* Backstepping
* Sliding mode control
|
* Khalil 14.3, 14.1
| HW 8
|}

=== References: ===

'''Course Textbooks'''

* H. Khalil, Nonlinear Systems, Prentice Hall; 3rd edition, 2001. ISBN: 978-0130673893
* S. Strogatz, Nonlinear Dynamics And Chaos, Westview Press, 1994. ISBN: 978-0738204536
* F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer; 2ed Edition, 1996. ISBN: 978-3540609346

'''Additional Sources:'''
* L. Perko, Differential Equations and Dynamical Systems (3rd), Springer, 2001. ISBN: 978-0387951164
* S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer; 2nd edition, 2003. ISBN: 978-0387001777

=== Policies ===

'''Collaboration Policy'''

Homeworks are to be done and handed in individually. To improve the learning process, students are encouraged to discuss the problems with, provide guidance to and get help from other students, the TAs and instructors. However, to make sure each student understands the concepts, solutions must be written independently and should reﬂect your understanding of the subject matter at the time of writing. Copying solutions, using solutions from previous years, having someone else type or dictate any part of the solution manual or using publicly available solutions (from the Internet) are not allowed.

'''Grading Policy'''

The final grades will be evaluated based on homework assignments (75%) and final projects (25%).

== Projects ==

TBD

[[Category:Courses]]

CDS 140b, Spring 2014

2014-05-14T22:33:42Z

Macmardg:

{| width=100%
|-
| colspan=2 align=center |
Differential Equations and Dynamical Systems__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* Richard Murray (CDS/BE), murray@cds.caltech.edu
* Doug MacMartin (CDS), macmardg@cds.caltech.edu
* Lectures: MWF, 1-2 pm, 314 ANB
* Office hours: TBD
| width=50% |
'''Teaching Assistant'''
* Katie Broersma (CDS)
* Contact: cds140-tas@cds.caltech.edu
* Office hours: Wed, 3:30-4:30 pm, 101 STH
|}

=== Course Description ===
'''CDS 140b is a continuation of CDS 140a'''. A large part of the course will focus on tools from nonlinear dynamics, such as perturbation theory and averaging, advanced stability analysis, the existence of periodic orbits, bifurcation theory, chaos, etc. In addition, guest lecturers will give an introduction to current research topics in dynamical systems theory. There will be 5 homeworks throughout the term but no exams. Instead, the students are required to select a research topic and a journal paper related to CDS140b and present a brief review of the paper. The details of the projects will be discussed in the class.

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Topic'''
| '''Suggested Reading/Lecture Notes'''
| '''Homework'''
|- valign=top
| 0
| 31 Mar RMM
| Course overview
|
|
|- valign=top
| 1
| 2 Apr 4 Apr* DGM
| Lagrangian and Hamiltonian systems I
* Definition and properties
* Newtonian Systems
|
*Perko 2.14
* [http://www.cds.caltech.edu/~macmardg/courses/cds140a/MarsdenMechSystems.pdf Marsden, Mechanical systems]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_03_32.pdf Lecture notes]
| [[CDS 140b Spring 2014 Homework 1|HW 1]] Due: 10 Apr (Thu)
|- valign=top
| 2
| 9 Apr* 11 Apr DGM
| Lagrangian and Hamiltonian systems II
* Hamiltonian systems with 2 degrees of freedom
* Mechanical systems
|
* Perko 3.6
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_05_21.pdf Lecture notes]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/hetero_halos_4-12-00.pdf Notes on Lagrange points]
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw2-wi14.pdf HW 2] Due: 17 Apr (Thu)
|- valign=top
| 3
| 18 Apr 21 Apr RMM
| Advanced stability theory
* Non-autonomous (time-varying) systems
* Input-to-state stability, input-output stability, small gain theorem
|
* Khalil 4.5, 4.6-4.7 (skim), 5.1-5.4
| [[CDS 140b Spring 2014 Homework 3|HW 3]] 
Due 1 May (Thu)
|- valign=top
| 4
| 28 Apr '''(12-1 pm)''' 30 Apr RMM
| Stability of perturbed systems
* Vanishing perturbations
* Non-vanishing perturbations
* Slowiy varying systems
|
* Khalil, 9.1, 9.2, 9.6
| [[CDS 140b Spring 2014 Homework 4|HW 4]] 
Due 8 May (Thu)
|- valign=top
| 5
| 5 May 7 May RMM
| Perturbation theory
* Periodic perturbations
* Averaging
* Singular Perturbations
|
* Khalil, 10.3-10.4, 11.1-11.3, 11.5
| [[CDS 140b Spring 2014 Homework 5|HW 5]] 
Due 15 May (Thu)
|- valign=top
| 6
| 12 May 14 May RMM
| Interconnected systems
* Absolute stability
* Describing functions
|
* Khalil, 7.1-7.2
| [[CDS 140b Spring 2014 Homework 6|HW 6]] 
Due 22 May (Thu)
|- valign=top
| 7
| 19 May 21 May DGM
| Nonlinear control I
* Overview of techniques
* Controllability and Lie brackets
* Gain scheduling
* Feedback linearization
|
* Khalil 12.2,12.5 (gain scheduling), 13.1-13.3 (feedback linearization)
* Isidori (chapter 2) or Nijmeijer and van der Schaft, 3.1 (for more on controllability)
| HW 7
|- valign=top
| 8
| 23 May* 30 May DGM
| Nonlinear control II
* Backstepping
* Sliding mode control
|
* Khalil 14.3, 14.1
| HW 8
|}

=== References: ===

'''Course Textbooks'''

* H. Khalil, Nonlinear Systems, Prentice Hall; 3rd edition, 2001. ISBN: 978-0130673893
* S. Strogatz, Nonlinear Dynamics And Chaos, Westview Press, 1994. ISBN: 978-0738204536
* F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer; 2ed Edition, 1996. ISBN: 978-3540609346

'''Additional Sources:'''
* L. Perko, Differential Equations and Dynamical Systems (3rd), Springer, 2001. ISBN: 978-0387951164
* S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer; 2nd edition, 2003. ISBN: 978-0387001777

=== Policies ===

'''Collaboration Policy'''

Homeworks are to be done and handed in individually. To improve the learning process, students are encouraged to discuss the problems with, provide guidance to and get help from other students, the TAs and instructors. However, to make sure each student understands the concepts, solutions must be written independently and should reﬂect your understanding of the subject matter at the time of writing. Copying solutions, using solutions from previous years, having someone else type or dictate any part of the solution manual or using publicly available solutions (from the Internet) are not allowed.

'''Grading Policy'''

The final grades will be evaluated based on homework assignments (75%) and final projects (25%).

== Projects ==

TBD

[[Category:Courses]]

CDS 140b, Spring 2014

2014-05-14T22:33:09Z

Macmardg:

{| width=100%
|-
| colspan=2 align=center |
Differential Equations and Dynamical Systems__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* Richard Murray (CDS/BE), murray@cds.caltech.edu
* Doug MacMartin (CDS), macmardg@cds.caltech.edu
* Lectures: MWF, 1-2 pm, 314 ANB
* Office hours: TBD
| width=50% |
'''Teaching Assistant'''
* Katie Broersma (CDS)
* Contact: cds140-tas@cds.caltech.edu
* Office hours: Wed, 3:30-4:30 pm, 101 STH
|}

=== Course Description ===
'''CDS 140b is a continuation of CDS 140a'''. A large part of the course will focus on tools from nonlinear dynamics, such as perturbation theory and averaging, advanced stability analysis, the existence of periodic orbits, bifurcation theory, chaos, etc. In addition, guest lecturers will give an introduction to current research topics in dynamical systems theory. There will be 5 homeworks throughout the term but no exams. Instead, the students are required to select a research topic and a journal paper related to CDS140b and present a brief review of the paper. The details of the projects will be discussed in the class.

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Topic'''
| '''Suggested Reading/Lecture Notes'''
| '''Homework'''
|- valign=top
| 0
| 31 Mar RMM
| Course overview
|
|
|- valign=top
| 1
| 2 Apr 4 Apr* DGM
| Lagrangian and Hamiltonian systems I
* Definition and properties
* Newtonian Systems
|
*Perko 2.14
* [http://www.cds.caltech.edu/~macmardg/courses/cds140a/MarsdenMechSystems.pdf Marsden, Mechanical systems]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_03_32.pdf Lecture notes]
| [[CDS 140b Spring 2014 Homework 1|HW 1]] Due: 10 Apr (Thu)
|- valign=top
| 2
| 9 Apr* 11 Apr DGM
| Lagrangian and Hamiltonian systems II
* Hamiltonian systems with 2 degrees of freedom
* Mechanical systems
|
* Perko 3.6
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_05_21.pdf Lecture notes]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/hetero_halos_4-12-00.pdf Notes on Lagrange points]
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw2-wi14.pdf HW 2] Due: 17 Apr (Thu)
|- valign=top
| 3
| 18 Apr 21 Apr RMM
| Advanced stability theory
* Non-autonomous (time-varying) systems
* Input-to-state stability, input-output stability, small gain theorem
|
* Khalil 4.5, 4.6-4.7 (skim), 5.1-5.4
| [[CDS 140b Spring 2014 Homework 3|HW 3]] 
Due 1 May (Thu)
|- valign=top
| 4
| 28 Apr '''(12-1 pm)''' 30 Apr RMM
| Stability of perturbed systems
* Vanishing perturbations
* Non-vanishing perturbations
* Slowiy varying systems
|
* Khalil, 9.1, 9.2, 9.6
| [[CDS 140b Spring 2014 Homework 4|HW 4]] 
Due 8 May (Thu)
|- valign=top
| 5
| 5 May 7 May RMM
| Perturbation theory
* Periodic perturbations
* Averaging
* Singular Perturbations
|
* Khalil, 10.3-10.4, 11.1-11.3, 11.5
| [[CDS 140b Spring 2014 Homework 5|HW 5]] 
Due 15 May (Thu)
|- valign=top
| 6
| 12 May 14 May RMM
| Interconnected systems
* Absolute stability
* Describing functions
|
* Khalil, 7.1-7.2
| [[CDS 140b Spring 2014 Homework 6|HW 6]] 
Due 22 May (Thu)
|- valign=top
| 7
| 19 May 21 May* DGM
| Nonlinear control I
* Overview of techniques
* Controllability and Lie brackets
* Gain scheduling
* Feedback linearization
|
* Khalil 12.2,12.5 (gain scheduling), 13.1-13.3 (feedback linearization)
* Isidori (chapter 2) or Nijmeijer and van der Schaft, 3.1 (for more on controllability)
| HW 7
|- valign=top
| 8
| 23 May 30 May DGM
| Nonlinear control II
* Backstepping
* Sliding mode control
|
* Khalil 14.3, 14.1
| HW 8
|}

=== References: ===

'''Course Textbooks'''

* H. Khalil, Nonlinear Systems, Prentice Hall; 3rd edition, 2001. ISBN: 978-0130673893
* S. Strogatz, Nonlinear Dynamics And Chaos, Westview Press, 1994. ISBN: 978-0738204536
* F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer; 2ed Edition, 1996. ISBN: 978-3540609346

'''Additional Sources:'''
* L. Perko, Differential Equations and Dynamical Systems (3rd), Springer, 2001. ISBN: 978-0387951164
* S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer; 2nd edition, 2003. ISBN: 978-0387001777

=== Policies ===

'''Collaboration Policy'''

Homeworks are to be done and handed in individually. To improve the learning process, students are encouraged to discuss the problems with, provide guidance to and get help from other students, the TAs and instructors. However, to make sure each student understands the concepts, solutions must be written independently and should reﬂect your understanding of the subject matter at the time of writing. Copying solutions, using solutions from previous years, having someone else type or dictate any part of the solution manual or using publicly available solutions (from the Internet) are not allowed.

'''Grading Policy'''

The final grades will be evaluated based on homework assignments (75%) and final projects (25%).

== Projects ==

TBD

[[Category:Courses]]

CDS 140b, Spring 2014

2014-04-17T11:36:55Z

Macmardg:

{| width=100%
|-
| colspan=2 align=center |
Differential Equations and Dynamical Systems__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* Richard Murray (CDS/BE), murray@cds.caltech.edu
* Doug MacMartin (CDS), macmardg@cds.caltech.edu
* Lectures: MWF, 1-2 pm, 314 ANB
* Office hours: TBD
| width=50% |
'''Teaching Assistant'''
* Katie Broersma (CDS)
* Contact: cds140-tas@cds.caltech.edu
* Office hours: Wed, 3:30-4:30 pm, 101 STH
|}

=== Course Description ===
'''CDS 140b is a continuation of CDS 140a'''. A large part of the course will focus on tools from nonlinear dynamics, such as perturbation theory and averaging, advanced stability analysis, the existence of periodic orbits, bifurcation theory, chaos, etc. In addition, guest lecturers will give an introduction to current research topics in dynamical systems theory. There will be 5 homeworks throughout the term but no exams. Instead, the students are required to select a research topic and a journal paper related to CDS140b and present a brief review of the paper. The details of the projects will be discussed in the class.

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Topic'''
| '''Suggested Reading/Lecture Notes'''
| '''Homework'''
|- valign=top
| 0
| 31 Mar RMM
| Course overview
|
|
|- valign=top
| 1
| 2 Apr 4 Apr* DGM
| Lagrangian and Hamiltonian systems I
* Definition and properties
* Newtonian Systems
|
*Perko 2.14
* [http://www.cds.caltech.edu/~macmardg/courses/cds140a/MarsdenMechSystems.pdf Marsden, Mechanical systems]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_03_32.pdf Lecture notes]
| [[CDS 140b Spring 2014 Homework 1|HW 1]] Due: 10 Apr (Thu)
|- valign=top
| 2
| 9 Apr* 11 Apr DGM
| Lagrangian and Hamiltonian systems II
* Hamiltonian systems with 2 degrees of freedom
* Mechanical systems
|
* Perko 3.6
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/2014_04_04_14_05_21.pdf Lecture notes]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140b/hetero_halos_4-12-00.pdf Notes on Lagrange points]
| [http://www.cds.caltech.edu/~macmardg/courses/cds140b/cds140b-hw2-wi14.pdf HW 2] Due: 17 Apr (Thu)
|- valign=top
| 3
| 18 Apr 21 Apr RMM
| Advanced stability theory
* Time-varying systems
* I/O stability, passivity
|
* Khalil 4.5, 6.1-6.5
| HW 3
|- valign=top
| 4
| 28 Apr 30 Apr RMM
| Perturbation Theory
* Regular Perturbation
* Poincare-Lindstedt Method
* Periodically Perturbed Systems
|
* Khalil, 10.1-10.3
* Strogatz, 7.6
* Verhulst, 9.1, 10.1
| HW 4
|- valign=top
| 5
| 5 May 7 May RMM
| Averaging Method
* Periodic Case
* Periodic Solutions
* General Case
|
* Khalil, 10.4-10.6
* Strogatz, 7.6
* Verhulst, 11.1-11.3, 11.8
| HW 5
|- valign=top
| 6
| 12 May 14 May RMM
| Singular Perturbations
* Finite Interval
* Infinite Interval
* Stability Analysis
|
* Khalil, 11.1-11.3, 11.5
| HW 6
|- valign=top
| 7
| 21 May 23 May* DGM
| Nonlinear control I
* Overview of techniques
* Controllability and Lie brackets
* Gain scheduling
* Feedback linearization
|
* Khalil 12.2,12.5 (gain scheduling), 13.1-13.3 (feedback linearization)
* Isidori (chapter 2) or Nijmeijer and van der Schaft, 3.1 (for more on controllability)
| HW 7
|- valign=top
| 8
| 28 May 30 May DGM
| Nonlinear control II
* Backstepping
* Sliding mode control
|
* Khalil 14.3, 14.1
| HW 8

|- valign=top
| 9
| 2 Jun 3 Jun
| Project presentations
| 
| 
|}

=== References: ===

'''Course Textbooks'''

* H. Khalil, Nonlinear Systems, Prentice Hall; 3rd edition, 2001. ISBN: 978-0130673893
* S. Strogatz, Nonlinear Dynamics And Chaos, Westview Press, 1994. ISBN: 978-0738204536
* F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer; 2ed Edition, 1996. ISBN: 978-3540609346

'''Additional Sources:'''
* L. Perko, Differential Equations and Dynamical Systems (3rd), Springer, 2001. ISBN: 978-0387951164
* S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer; 2nd edition, 2003. ISBN: 978-0387001777

=== Policies ===

'''Collaboration Policy'''

Homeworks are to be done and handed in individually. To improve the learning process, students are encouraged to discuss the problems with, provide guidance to and get help from other students, the TAs and instructors. However, to make sure each student understands the concepts, solutions must be written independently and should reﬂect your understanding of the subject matter at the time of writing. Copying solutions, using solutions from previous years, having someone else type or dictate any part of the solution manual or using publicly available solutions (from the Internet) are not allowed.

'''Grading Policy'''

The final grades will be evaluated based on homework assignments (75%) and final projects (25%).

== Projects ==

TBD

[[Category:Courses]]

CDS 140b, Spring 2014

2014-04-10T15:00:47Z

Macmardg:

CDS 140b Spring 2014 Homework 2

2014-04-07T16:34:53Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = CDS 140b
| semester = Spring 2014
| title = Problem Set #2
| issued = 9 Feb 2014 (Wed)
| due = 16 Feb 2014 (Wed)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

{{warning|UNDER CONSTRUCTION, DO NOT START}}

<ol>
<li>
A planar pendulum (in the $x$-$z$ plane) of mass $m$ and length $\ell$ hangs from a support point that moves according to $x=a\cos (\omega t)$. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
</li>
<li>'''Perko, Section 3.3, problem 5.'''
Show that
<center><amsmath>\aligned
\dot x &=y+y(x^2+y^2)\\
\dot y &=x-x(x^2+y^2)
\endaligned</amsmath></center>
is a Hamiltonian system with $4H(x,y)=(x^2+y^2)^2-2(x^2-y^2)$. Show that $dH/dt=0$ along solution curves of this system and therefore that solution curves of this system are given by
<center><amsmath>
(x^2+y^2)^2-2(x^2-y^2)=C
</amsmath></center>
Show that the origin is a saddle for this system and that $(\pm 1,0)$ are centers for this system. (Note the symmetry with respect to the $x$-axis.) Sketch the two homoclinic orbits corresponding to $C=0$ and sketch the phase portrait for this system, noting the occurrence of a compound separatrix cycle.
</li>
<li>'''Perko, Section 3.6, problem 4.'''
</li>
<li>
Consider the stability of the Lagrange points (with some simplifying steps).
With the mass of the sun as $1-\mu$ and planet as $\mu$, then in the rotating coordinate system the Hamiltonian is given by
<center><amsmath>
H=\frac{(p_x+y)^2+(p_y-x)^2}{2}+\Omega(x,y)
</amsmath></center>
where the potential function
<center><amsmath>
\Omega(x,y)=\frac{x^2+y^2}{2}+\frac{1-\mu}{r_1}+\frac{\mu}{r_2}+\frac{\mu(1-\mu)}{2}
</amsmath></center>
with $r_1=\sqrt{(x+\mu)^2+y^2}$ and $r_2=\sqrt{(x-1+\mu)^2+y^2}$.
Show that the equilibrium points are given by the critical points of the (messy) function $\Omega$.
(This leads to 5 solutions, for L1 through L5, with L1, L2, and L3 collinear, i.e., y=0. If you are interested, the collinear solutions are not too difficult to solve for numerically for some $\mu$.)
To explore the linearized dynamics it is sufficient to retain only quadratic terms in $H$ (why?). For the collinear Lagrange points this leads to
<center><amsmath>
H=\frac{(p_x+y)^2+(p_y-x)^2}{2}-ax^2+by^2
</amsmath></center>
for $a>0$ and $b>0$. E.g., for the L1 point in the Earth-Jupiter system then a=9.892, b=3.446. Describe the linearized dynamics about this Lagrange point; are periodic orbits stable?
</li>
</ol>

CDS 140b Spring 2014 Homework 2

2014-04-07T16:33:51Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = CDS 140b
| semester = Spring 2014
| title = Problem Set #2
| issued = 9 Feb 2014 (Wed)
| due = 16 Feb 2014 (Wed)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

{{warning|UNDER CONSTRUCTION, DO NOT START}}

<ol>
<li>
A planar pendulum (in the $x$-$z$ plane) of mass $m$ and length $\ell$ hangs from a support point that moves according to $x=a\cos (\omega t)$. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
</li>
<li>'''Perko, Section 3.3, problem 5.'''
Show that
<center><amsmath>\aligned
\dot x &=y+y(x^2+y^2)\\
\dot y &=x-x(x^2+y^2)
\endaligned</amsmath></center>
is a Hamiltonian system with $4H(x,y)=(x^2+y^2)^2-2(x^2-y^2)$. Show that $dH/dt=0$ along solution curves of this system and therefore that solution curves of this system are given by
<center><amsmath>
(x^2+y^2)^2-2(x^2-y^2)=C
</amsmath></center>
Show that the origin is a saddle for this system and that $(\pm 1,0)$ are centers for this system. (Note the symmetry with respect to the $x$-axis.) Sketch the two homoclinic orbits corresponding to $C=0$ and sketch the phase portrait for this system, noting the occurrence of a compound separatrix cycle.
</li>
<li>'''Perko, Section 3.6, problem 4.'''
</li>
<li>
Consider the stability of the Lagrange points (with some simplifying steps).
With the mass of the sun as $1-\mu$ and planet as $\mu$, then in the rotating coordinate system the Hamiltonian is given by
<center><amsmath>
H=\frac{(p_x+y)^2+(p_y-x)^2}{2}+\Omega(x,y)
</amsmath></center>
where the potential function
<center><amsmath>
\Omega(x,y)=\frac{x^2+y^2}{2}+\frac{1-\mu}{r_1}+\frac{\mu}{r_2}+\frac{\mu(1-\mu){2}
</amsmath></center>
with $r_1=\sqrt{(x+\mu)^2+y^2}$ and $r_2=\sqrt{(x-1+\mu)^2+y^2}$.
Show that the equilibrium points are given by the critical points of the (messy) function $\Omega$.
(This leads to 5 solutions, for L1 through L5, with L1, L2, and L3 collinear, i.e., y=0. If you are interested, the collinear solutions are not too difficult to solve for numerically for some $\mu$.)
To explore the linearized dynamics it is sufficient to retain only quadratic terms in $H$ (why?). For the collinear Lagrange points this leads to
<center><amsmath>
H=\frac{(p_x+y)^2+(p_y-x)^2}{2}-ax^2+by^2
</amsmath></center>
for $a>0$ and $b>0$. E.g., for the L1 point in the Earth-Jupiter system then a=9.892, b=3.446. Describe the linearized dynamics about this Lagrange point; are periodic orbits stable?
</li>
</ol>

CDS 140b Spring 2014 Homework 2

2014-04-06T00:06:58Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = CDS 140b
| semester = Spring 2014
| title = Problem Set #2
| issued = 9 Feb 2014 (Wed)
| due = 16 Feb 2014 (Wed)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

{{warning|UNDER CONSTRUCTION, DO NOT START}}

<ol>
<li>
A planar pendulum (in the $x$-$z$ plane) of mass $m$ and length $\ell$ hangs from a support point that moves according to $x=a\cos (\omega t)$. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
</li>
<li>'''Perko, Section 3.3, problem 5.'''
Show that
<center><amsmath>\aligned
\dot x &=y+y(x^2+y^2)\\
\dot y &=x-x(x^2+y^2)
\endaligned</amsmath></center>
is a Hamiltonian system with $4H(x,y)=(x^2+y^2)^2-2(x^2-y^2)$. Show that $dH/dt=0$ along solution curves of this system and therefore that solution curves of this system are given by
<center><amsmath>
(x^2+y^2)^2-2(x^2-y^2)=C
</amsmath></center>
Show that the origin is a saddle for this system and that $(\pm 1,0)$ are centers for this system. (Note the symmetry with respect to the $x$-axis.) Sketch the two homoclinic orbits corresponding to $C=0$ and sketch the phase portrait for this system, noting the occurrence of a compound separatrix cycle.
</li>
<li>'''Perko, Section 3.6, problem 4.'''
</li>
<li>
Consider the stability of the Lagrange points (with some simplifying steps).
With the mass of the sun as $1-\mu$ and planet as $\mu$, then in the rotating coordinate system the Hamiltonian is given by
<center><amsmath>
H=\frac{(p_x+y)^2+(p_y-x)^2}{2}+\Omega(x,y)
</amsmath></center>
Show that the equilibrium points are given by the critical points of the (messy) function $\Omega$.
(This leads to 5 solutions, for L1 through L5, with L1, L2, and L3 collinear, i.e., y=0.)
To explore the linearized dynamics it is sufficient to retain only quadratic terms in $H$ (why?). For the collinear Lagrange points this leads to
<center><amsmath>
H=\frac{(p_x+y)^2+(p_y-x)^2}{2}-ax^2+by^2
</amsmath></center>
for $a>0$ and $b>0$. E.g., for the L1 point in the Earth-Jupiter system then a=9.892, b=3.446. Describe the linearized dynamics about this Lagrange point; are periodic orbits stable?
</li>
</ol>

CDS 140b, Spring 2014

2014-04-04T21:52:01Z

Macmardg:

CDS 140b, Spring 2014

2014-04-04T21:51:21Z

Macmardg:

CDS 140b Spring 2014 Homework 1

2014-04-02T03:29:59Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = CDS 140b
| semester = Spring 2014
| title = Problem Set #1
| issued = 2 Apr 2014 (Wed)
| due = 10 Apr 2014 (Thu)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

<ol>
<li>'''Perko, Section 2.14, problem 1'''
 
(a) Show that the system
<center><amsmath>\aligned
\dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\
\dot y&=a_{21}x-a_{11}y+Dx^2-2Axy+By^2
\endaligned</amsmath></center>
is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.
 
(b) Given $f\in C^2(E)$, where $E$ is an open, simply connected subset of $\mathbb R^2$, show that the system $\dot{x}=f(x)$ is a Hamiltonian system on $E$ iff $\nabla\cdot f(x)=0$ for all $x\in E$.
</li>
<li> '''Perko, Section 2.14, problem 4.'''
Given the function $U(x)$ in the text, sketch the phase portrait for the Newtonian system with Hamiltonian $H(x,y)=y^2/2+U(x)$
</li>
<li> '''Perko, Section 2.14, problem 5 a,b,c.'''
 
For each of the following Hamiltonian functions, sketch the phase portraits for the Hamiltonian system and the gradient system orthogonal to it. Draw both phase portraits on the same phase plane.
 
(a) $H(x,y)=x^2+2y^2$
 
(b) $H(x,y)=x^2-y^2$
 

</li>

<li>'''Perko, Section 2.14, problem 12.'''
Show that the flow defined by a Hamiltonian system with one degree of freedom is area preserving. '''Hint:''' Cf. Problem 6 in Section 2.3
</li>
<li> ''Preview for homework 2:''
 
A planar pendulum (in the $x$-$z$ plane) of mass $m$ and length $\ell$ hangs from a support point that moves according to $x=a\cos (\omega t)$. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
</li>
</ol>

CDS 140b Spring 2014 Homework 2

2014-03-29T22:42:54Z

Macmardg:

CDS 140b Spring 2014 Homework 1

2014-03-29T22:38:15Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = CDS 140b
| semester = Spring 2014
| title = Problem Set #1
| issued = 2 Apr 2014 (Wed)
| due = 9 Apr 2014 (Wed)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

<ol>
<li>'''Perko, Section 2.14, problem 1'''
 
(a) Show that the system
<center><amsmath>\aligned
\dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\
\dot y&=a_{21}x-a_{11}y+Dx^2-2Axy+By^2
\endaligned</amsmath></center>
is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.
 
(b) Given $f\in C^2(E)$, where $E$ is an open, simply connected subset of $\mathbb R^2$, show that the system $\dot{x}=f(x)$ is a Hamiltonian system on $E$ iff $\nabla\cdot f(x)=0$ for all $x\in E$.
</li>
<li> '''Perko, Section 2.14, problem 4.'''
Given the function $U(x)$ in the text, sketch the phase portrait for the Newtonian system with Hamiltonian $H(x,y)=y^2/2+U(x)$
</li>
<li> '''Perko, Section 2.14, problem 5 a,b,c.'''
 
For each of the following Hamiltonian functions, sketch the phase portraits for the Hamiltonian system and the gradient system orthogonal to it. Draw both phase portraits on the same phase plane.
 
(a) $H(x,y)=x^2+2y^2$
 
(b) $H(x,y)=x^2-y^2$
 
(c) $H(x,y)=y\sin{x}$
</li>
<li> '''Perko, Section 2.14, problem 7.'''
Show that if $x_0$ is a strict local minimum of $V(x)$ then the function $V(x)-V(x_0)$ is a strict Lyapunov function (i.e., $\dot{V}<0$ for $x\neq 0$) for the gradient system $\dot x=-\mathrm{grad}V(x)$.
<li>'''Perko, Section 2.14, problem 12.'''
Show that the flow defined by a Hamiltonian system with one degree of freedom is area preserving. '''Hint:''' Cf. Problem 6 in Section 2.3
</li>
<li> ''Preview for homework 2:''
 
A planar pendulum (in the $x$-$z$ plane) of mass $m$ and length $\ell$ hangs from a support point that moves according to $x=a\cos (\omega t)$. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
</li>
</ol>

CDS 140b Spring 2014 Homework 1

2014-03-29T22:37:08Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = CDS 140b
| semester = Spring 2014
| title = Problem Set #1
| issued = 2 Apr 2014 (Wed)
| due = 9 Apr 2014 (Wed)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

<ol>
<li>'''Perko, Section 2.14, problem 1'''
 
(a) Show that the system
<center><amsmath>\aligned
\dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\
\dot y&=a_{21}x-a_{11}y+Dx^2-2Axy+By^2
\endaligned</amsmath></center>
is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.
 
(b) Given $f\in C^2(E)$, where $E$ is an open, simply connected subset of $\mathbb R^2$, show that the system $\dot{x}=f(x)$ is a Hamiltonian system on $E$ iff $\nabla\cdot f(x)=0$ for all $x\in E$.
</li>
<li> '''Perko, Section 2.14, problem 4.'''
Given the function $U(x)$ in the text, sketch the phase portrait for the Newtonian system with Hamiltonian $H(x,y)=y^2/2+U(x)$
</li>
<li> '''Perko, Section 2.14, problem 5 a,b,c.'''
 
For each of the following Hamiltonian functions, sketch the phase portraits for the Hamiltonian system and the gradient system orthogonal to it. Draw both phase portraits on the same phase plane.
 
(a) $H(x,y)=x^2+2y^2$
 
(b) $H(x,y)=x^2-y^2$
 
(c) $H(x,y)=y\sin{x}$
</li>
<li> '''Perko, Section 2.14, problem 7.'''
Show that if $x_0$ is a strict local minimum of $V(x)$ then the function $V(x)-V(x_0)$ is a strict Lyapunov function (i.e., $\dot{V}<0$ for $x\neq 0$) for the gradient system $\dot x=-\mathrm{grad}V(x)$.
<li>'''Perko, Section 2.14, problem 12.'''
Show that the flow defined by a Hamiltonian system with one degree of freedom is area preserving. '''Hint:''' Cf. Problem 6 in Section 2.3
</li>
<li> A planar pendulum (in the $x$-$z$ plane) of mass $m$ and length $\ell$ hangs from a support point that moves according to $x=a\cos (\omega t)$. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
</li>
</ol>

CDS 140b Spring 2014 Homework 2

2014-03-25T22:27:18Z

Macmardg: Created page with " {{CDS homework | instructor = R. Murray, D. MacMartin | course = CDS 140b | semester = Spring 2014 | title = Problem Set #2 | issued = 9 Feb 2014 (Wed) | due = 16 Feb 2..."

CDS 140b, Spring 2014

2014-03-25T22:26:03Z

Macmardg:

CDS 140b Spring 2014 Homework 1

2014-03-25T22:25:11Z

Macmardg: Created page with " {{CDS homework | instructor = R. Murray, D. MacMartin | course = CDS 140b | semester = Spring 2014 | title = Problem Set #1 | issued = 2 Apr 2014 (Wed) | due = 9 Apr 20..."

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = CDS 140b
| semester = Spring 2014
| title = Problem Set #1
| issued = 2 Apr 2014 (Wed)
| due = 9 Apr 2014 (Wed)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

<ol>
<li>'''Perko, Section 2.14, problem 1'''
 
(a) Show that the system
<center><amsmath>\aligned
\dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\
\dot y&=a_{21}x-a_{11}y+Dx^2-2Axy+By^2
\endaligned</amsmath></center>
is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.
 
(b) Given $f\in C^2(E)$, where $E$ is an open, simply connected subset of $\mathbb R^2$, show that the system $\dot{x}=f(x)$ is a Hamiltonian system on $E$ iff $\nabla\cdot f(x)=0$ for all $x\in E$.
</li>
<li> '''Perko, Section 2.14, problem 7.'''
Show that if $x_0$ is a strict local minimum of $V(x)$ then the function $V(x)-V(x_0)$ is a strict Lyapunov function (i.e., $\dot{V}<0$ for $x\neq 0$) for the gradient system $\dot x=-\mathrm{grad}V(x)$.
<li>'''Perko, Section 2.14, problem 12.'''
Show that the flow defined by a Hamiltonian system with one degree of freedom is area preserving. '''Hint:''' Cf. Problem 6 in Section 2.3
</li>
<li> A planar pendulum (in the $x$-$z$ plane) of mass $m$ and length $\ell$ hangs from a support point that moves according to $x=a\cos (\omega t)$. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
</li>
</ol>

CDS 140b, Spring 2014

2014-03-25T22:22:54Z

Macmardg:

CDS 140b, Spring 2014

2014-03-24T20:41:43Z

Macmardg:

CDS 140b, Spring 2014

2014-03-18T22:29:05Z

Macmardg:

Dennis Bernstein, Jan 2014

2014-01-27T04:44:55Z

Macmardg:

Dennis Berstein and James Forbes from U. Michigan will be visiting Caltech on 29 January 2014. If you would like to meet with Dennis and Jim during their visit, please sign up below.

=== Schedule ===
* 12:30 - Lunch and meeting with Richard
* 2:00 - Seminar, 121 Annenberg
* 3:00 - CDS tea
* 3:30 - Doug MacMartin
* 4:00 - Open
* 4:30 - Matanya Horowitz
* 5:00 - Open
* 5:30 - Done for the day

=== Abstract ===
<center>
How Much Modeling Information Is Really Needed for Feedback Control?

Dennis Bernstein 
University of Michigan
</center>

Modeling for control is often expensive and time-consuming—not to mention futile, especially when a plant changes unpredictably. Our research is therefore aimed at the following fundamental question: What is the minimal modeling information (order, parameters, nonlinearities, noise spectra, etc.) that must be known—and how *well* must it be known—so that a controller can reliably meet performance specifications?
The approach we are developing is based on retrospective cost adaptive control (RCAC), which uses retrospective optimization for online learning. RCAC is easy to implement, and requires extremely limited modeling information. In this talk I will explain the rationale for RCAC, its applicability to various types of plants (stable/unstable, minimum-phase/NMP, SISO/MIMO, linear/nonlinear), the modeling information it can operate with and (especially) without, and the status of its theoretical foundation.
For flight control, we will apply RCAC to the extreme case of totally unknown control-surface faults, such as a stuck rudder or severe rate saturation. Additional examples are taken from missile control, noise and vibration control, and spacecraft attitude control with nonlinear actuation such as CMGs.

ACM 101/AM 125b/CDS 140a, Winter 2013

2013-02-21T20:28:52Z

Macmardg:

{| width=100%
|-
| colspan=2 align=center |
Differential Equations and Dynamical Systems__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* Richard Murray (CDS/BE), murray@cds.caltech.edu
* Doug MacMartin (CDS), macmardg@cds.caltech.edu
* Lectures: Tu/Th, 9-10:30, 105 ANB
* Office hours: Wed 2-3 pm (please e-mail to confirm)
| width=50% |
'''Teaching Assistants'''
* Katheryn Broersma (CDS), katheryn@caltech.edu
* Vanessa Jönsson (CDS), vjonsson@caltech.edu
* Office hours: Fridays, 11-12 in 314 ANB; Mondays, 12-1 in 243 ANB
* [https://piazza.com/caltech/winter2013/cds140 Piazza]
|}

=== Course Description ===
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.

===Announcements ===

* 16 Dec 2012: set up [https://piazza.com/caltech/winter2013/cds140 Piazza] page + added additional course info
* 23 Aug 2012: web page creation

=== Lecture Schedule ===

{| class="mw-collapsible " width=100% border=1 cellpadding=5
|-
| '''Date'''
| '''Topic'''
| '''Reading'''
| '''Homework'''
|- valign=top
|- valign=top
| 8 Jan 10 Jan RMM
| Linear Differential Equations I
* Course overview and administration
* Linear differential equations
* Matrix exponential, diagonalization
* Stable and unstable spaces
* Planar systems, behavior of solutions
|
Perko, 1.1-1.6 
Optional:
* J&S, Ch 1; Sec 2.1-2.5
* Ver, Sec 5.1-5.2; 6.1-6.2
| [[CDS 140a Winter 2013 Homework 1|HW 1]] Due: 15 Jan (Tue)
|- valign=top
| 15 Jan 17 Jan RMM
| Linear Differential Equations II
* S + N decomposition, Jordan form
* Stability theory
* Linear systems with inputs (nonhomogeneous systems)
* Boundary value problems (if time)
| Perko, 1.7-1.10 + notes
| [[CDS 140a Winter 2013 Homework 2|HW 2]] Due: 22 Jan (Tue)
|- valign=top
| 22 Jan 24 Jan RMM
| Nonlinear differential equations
* Existence and uniqueness
* Flow of a differential equation
* Linearization
| Perko, 2.1-2.6
| [[CDS 140a Winter 2013 Homework 3|HW 3]] Due: 29 Jan (Tue)
|- valign=top
| 29 Jan* 31 Jan DGM
| Behavior of differential equations
* Stable and unstable manifolds
* Stability of equilibrium points
| Perko, 2.7-2.10

| | [[CDS 140a Winter 2013 Homework 4|HW 4]] Due: 5 Feb (Tue)
|- valign=top
| 5 Feb* 7 Feb DGM
| Non-hyperbolic differential equations
* Lyapunov functions
* Center manifold theorem
| Perko, 2.11-2.13

| [[CDS 140a Winter 2013 Homework 5|HW 5]] Due: 12 Feb (Tue)
|- valign=top
| 12 Feb 14 Feb* DGM
| Hamiltonian systems
* Gradient and Hamiltonian systems
* Energy based stability methods
* Applications
| Perko 2.14 + notes
* [http://www.cds.caltech.edu/~macmardg/courses/cds140a/L6-Hamiltonian.pdf Scanned lecture notes]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140a/MarsdenMechSystems.pdf Marsden, Mechanical systems]

| [[CDS 140a Winter 2013 Homework 6|HW 6]] Due: 19 Feb (Tue)
|- valign=top
| 19 Feb 21 Feb* 26 Feb RMM
| Limit cycles
* Limit sets and attractors
* Periodic orbits and limit cycles
* Poincare' map
* Bendixson criterion for limit cycles in the plane
| Perko, 3.1-3.5, 3.9

| [[CDS 140a Winter 2013 Homework 7|HW 7]] Due: 5 Mar (Tue)
|- valign=top
|- valign=top
| 28 Feb 5 Mar 7 Mar* RMM
| Bifurcations
* Structural stability
* Bifurcation of equilibrium points
* Hopf bifurcation
| Perko 4.1-4.4 + notes

| [[CDS 140a Winter 2013 Homework 8|HW 8]] Due: 12 Mar (Tue)
|- valign=top
| 12 Mar 
| Course review
| 
| 
|}

=== Textbook ===

The primary text for the course (available via the online bookstore) is
{|
|- valign=top
| align=right |  [Perko] 
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.
|}

The following additional texts may be useful for some students (on reserve in SFL):
{|
|- valign=top
| align=right |  [J&S] 
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007.
|- valign=top
| align=right |  [Ver] 
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006.
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 8 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without ''prior'' permission from the instructor.''
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

== Old Announcements ==
[[Category:Courses]]

CDS 140a Winter 2013 Homework 6

2013-02-12T22:53:54Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = ACM 101/AM 125b/CDS 140a
| semester = Winter 2013
| title = Problem Set #6
| issued = 12 Feb 2013 (Tue)
| due = 19 Feb 2013 (Tue)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

<ol>
<li>'''Perko, Section 2.14, problem 1'''
 
(a) Show that the system
<center><amsmath>\aligned
\dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\
\dot y&=a_{21}x-a_{11}y+Dx^2-2Axy+By^2
\endaligned</amsmath></center>
is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.
 
(b) Given $f\in C^2(E)$, where $E$ is an open, simply connected subset of $\mathbb R^2$, show that the system $\dot{x}=f(x)$ is a Hamiltonian system on $E$ iff $\nabla\cdot f(x)=0$ for all $x\in E$.
</li>
<li> '''Perko, Section 2.14, problem 7.'''
Show that if $x_0$ is a strict local minimum of $V(x)$ then the function $V(x)-V(x_0)$ is a strict Lyapunov function (i.e., $\dot{V}<0$ for $x\neq 0$) for the gradient system $\dot x=-\mathrm{grad}V(x)$.
<li>'''Perko, Section 2.14, problem 12.'''
Show that the flow defined by a Hamiltonian system with one degree of freedom is area preserving. '''Hint:''' Cf. Problem 6 in Section 2.3
</li>
<li>'''Perko, Section 3.3, problem 5.'''
Show that
<center><amsmath>\aligned
\dot x &=y+y(x^2+y^2)\\
\dot y &=x-x(x^2+y^2)
\endaligned</amsmath></center>
is a Hamiltonian system with $4H(x,y)=(x^2+y^2)^2-2(x^2-y^2)$. Show that $dH/dt=0$ along solution curves of this system and therefore that solution curves of this system are given by
<center><amsmath>
(x^2+y^2)^2-2(x^2-y^2)=C
</amsmath></center>
Show that the origin is a saddle for this system and that $(\pm 1,0)$ are centers for this system. (Note the symmetry with respect to the $x$-axis.) Sketch the two homoclinic orbits corresponding to $C=0$ and sketch the phase portrait for this system. (You need not comment on the compound separatrix cycle.)
</li>
<li> A planar pendulum (in the $x$-$z$ plane) of mass $m$ and length $\ell$ hangs from a support point that moves according to $x=a\cos (\omega t)$. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
</li>
</ol>

CDS 140a Winter 2013 Homework 6

2013-02-12T22:51:41Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = ACM 101/AM 125b/CDS 140a
| semester = Winter 2013
| title = Problem Set #6
| issued = 12 Feb 2013 (Tue)
| due = 19 Feb 2013 (Tue)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

<ol>
<li>'''Perko, Section 2.14, problem 1'''
<ol type="a">
<li> Show that the system
<center><amsmath>\aligned
\dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\
\dot y&=a_{21}x-a_{11}y+Dx^2-2Axy+By^2
\endaligned</amsmath></center>
is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.
</li>
<li> Given $f\in C^2(E)$, where $E$ is an open, simply connected subset of $\mathbb R^2$, show that the system $\dot{x}=f(x)$ is a Hamiltonian system on $E$ iff $\nabla\cdot f(x)=0$ for all $x\in E$.
</li>
</ol>
</li>
<li> '''Perko, Section 2.14, problem 7.'''
Show that if $x_0$ is a strict local minimum of $V(x)$ then the function $V(x)-V(x_0)$ is a strict Lyapunov function (i.e., $\dot{V}<0$ for $x\neq 0$) for the gradient system $\dot x=-\mathrm{grad}V(x)$.
<li>'''Perko, Section 2.14, problem 12.'''
Show that the flow defined by a Hamiltonian system with one degree of freedom is area preserving. '''Hint:''' Cf. Problem 6 in Section 2.3
</li>
<li>'''Perko, Section 3.3, problem 5.'''
Show that
<center><amsmath>\aligned
\dot x &=y+y(x^2+y^2)\\
\dot y &=x-x(x^2+y^2)
\endaligned</amsmath></center>
is a Hamiltonian system with $4H(x,y)=(x^2+y^2)^2-2(x^2-y^2)$. Show that $dH/dt=0$ along solution curves of this system and therefore that solution curves of this system are given by
<center><amsmath>
(x^2+y^2)^2-2(x^2-y^2)=C
</amsmath></center>
Show that the origin is a saddle for this system and that $(\pm 1,0)$ are centers for this system. (Note the symmetry with respect to the $x$-axis.) Sketch the two homoclinic orbits corresponding to $C=0$ and sketch the phase portrait for this system. (You need not comment on the compound separatrix cycle.)
</li>
<li> A planar pendulum (in the $x$-$z$ plane) of mass $m$ and length $\ell$ hangs from a support point that moves according to $x=a\cos (\omega t)$. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
</li>
</ol>

CDS 140a Winter 2013 Homework 7

2013-02-12T22:44:24Z

Macmardg:

CDS 140a Winter 2013 Homework 7

2013-02-12T22:42:38Z

Macmardg: Created page with "{{warning|This homework set is still being written. Do not start working on these problems until this banner is removed.}}"

{{warning|This homework set is still being written. Do not start working on these problems until this banner is removed.}}

CDS 140a Winter 2013 Homework 6

2013-02-12T22:42:16Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = ACM 101/AM 125b/CDS 140a
| semester = Winter 2013
| title = Problem Set #6
| issued = 12 Feb 2013 (Tue)
| due = 19 Feb 2013 (Tue)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

<ol>
<li>'''Perko, Section 2.14, problem 1'''
<ol>
<li> Show that the system
<center><amsmath>\aligned
\dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\
\dot y&=a_{21}x-a_{11}y+Dx^2-2Axy+By^2
\endaligned</amsmath></center>
is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.
</li>
<li> Given $f\in C^2(E)$, where $E$ is an open, simply connected subset of $\mathbb R^2$, show that the system $\dot{x}=f(x)$ is a Hamiltonian system on $E$ iff $\nabla\cdot f(x)=0$ for all $x\in E$.
</li>
</ol>
</li>
<li> '''Perko, Section 2.14, problem 7.'''
Show that if $x_0$ is a strict local minimum of $V(x)$ then the function $V(x)-V(x_0)$ is a strict Lyapunov function (i.e., $\dot{V}<0$ for $x\neq 0$) for the gradient system $\dot x=-\mathrm{grad}V(x)$.
<li>'''Perko, Section 2.14, problem 12.'''
Show that the flow defined by a Hamiltonian system with one degree of freedom is area preserving. '''Hint:''' Cf. Problem 6 in Section 2.3
</li>
<li>'''Perko, Section 3.3, problem 5.'''
Show that
<center><amsmath>\aligned
\dot x &=y+y(x^2+y^2)\\
\dot y &=x-x(x^2+y^2)
\endaligned</amsmath></center>
is a Hamiltonian system with $4H(x,y)=(x^2+y^2)^2-2(x^2-y^2)$. Show that $dH/dt=0$ along solution curves of this system and therefore that solution curves of this system are given by
<center><amsmath>
(x^2+y^2)^2-2(x^2-y^2)=C
</amsmath></center>
Show that the origin is a saddle for this system and that $(\pm 1,0)$ are centers for this system. (Note the symmetry with respect to the $x$-axis.) Sketch the two homoclinic orbits corresponding to $C=0$ and sketch the phase portrait for this system. (You need not comment on the compound separatrix cycle.)
</li>
<li> A planar pendulum (in the $x$-$z$ plane) of mass $m$ and length $\ell$ hangs from a support point that moves according to $x=a\cos (\omega t)$. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
</li>
</ol>

CDS 140a Winter 2013 Homework 6

2013-02-11T04:45:15Z

Macmardg:

CDS 140a Winter 2013 Homework 6

2013-02-11T04:44:17Z

Macmardg:

CDS 140a Winter 2013 Homework 6

2013-02-11T04:39:40Z

Macmardg:

CDS 140a Winter 2013 Homework 5

2013-02-10T04:14:16Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = ACM 101/AM 125b/CDS 140a
| semester = Winter 2013
| title = Problem Set #5
| issued = 5 Feb 2013 (Tue)
| due = 12 Feb 2013 (Tue)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

<ol>
<li>'''Perko, Section 2.9, problem 3'''
Use the Lyapunov function $V(x)=x_1^2+x_2^2+x_3^2$ to show that the origin is an asymptotically stable equilibrium point of the system
<center><amsmath>
\dot{x}=\begin{bmatrix}-x_2-x_1x_2^2+x_3^2-x_1^3\\
x_1+x_3^3-x_2^3\\
-x_1x_3-x_3x_1^2-x_2x_3^2-x_3^5\end{bmatrix}
</amsmath></center>
Show that the trajectories of the linearized system $\dot{x}=Df(0)x$ for this problem lie on circles in planes parallel to the $x_1,\,x_2$ plane; hence, the origin is stable, but not asymptotically stable for the linearized system.
</li>
<li>
Determine the stability of the system
<center><amsmath>
\aligned
\dot{x}&=-y-x^3\\
\dot{y}&=x^5
\endaligned
</amsmath></center>
Motivated by the first equation, try a Lyapunov function of the form $V(x,y)=\alpha x^6+\beta y^2$. Is the origin asymptotically stable? Is the origin globally asymptotically stable?
</li>
<li>
An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\|x(t)\|\leq Me^{-\alpha t}\|x(0)\|$, $\forall\|x(0)\|\leq\epsilon,\,t\geq 0$.
Prove that an equilibrium point $x_0=0$ is exponentially stable if there is a function $V(x)$ satisfying
<center><amsmath>k_1\|x\|^2\leq V(x)\leq k_2\|x\|^2,\quad \dot V(x)\leq -k_3\|x\|^2
</amsmath></center>
for positive constants $k_1$, $k_2$ and $k_3$.
</li>
<li>'''Perko, Section 2.12, problem 2'''
Use Theorem 1 [Centre Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system
<center><amsmath>
\aligned
\dot{x}&=y\\
\dot{y}&=-y+\alpha x^2+xy
\endaligned
</amsmath></center>
for $\alpha\neq 0$ and for $\alpha=0$; i.e., follow the procedure in Example 1 after diagonalizing the system as in Example 3.
</li>
<li> Consider the following system in $\mathbb R^2$:
<center><amsmath>\aligned
\dot{x}&=-\frac{\alpha}{2}(x^2+y^2)+\alpha (x+y)-\alpha\\
\dot y&=-\alpha xy+\alpha (x+y)-\alpha
\endaligned</amsmath></center>
Determine the stable, unstable, and centre manifold of the equilibrium point at $(x,y)=(1,1)$, and determine the stability of this equilibrium point for $\alpha\neq 0$. For determining stability, note that near the equilibrium point there are two 1-dimensional invariant linear manifolds of the form $M=\{(a_1,a_2)\in\mathbb R^2\|a_2=ka_1\}$; determine the flow on these invariant manifolds.
</li>
<li> Consider the following system
<center><amsmath>\aligned
\dot x&=x^2\\
\dot y&=-y
\endaligned</amsmath></center>
with stable manifold $W^s=\{(x,y):x=0\}$
<ol>
<li> Show that the Taylor series approximation to the centre manifold gives $W^c=\{(x,y):y=0\}$.</li>
<li> Show that the set $\{(x,y):y=h(x)\}$ with
<center><amsmath>
h(x)=\left\{\begin{array}{lr}ke^{1/x}&x<0\\0&x\geq 0\end{array}\right.
</amsmath></center>
describes a one-parameter set of invariant manifolds for any $k$. (Hint: what is $dy/dx$?)
</li>
<li> What are the dynamics describing the trajectories along any of these manifolds?</li>
<li> (Optional): Plot the phase portrait for this system. </li>
</ol>
</li>
</ol>

CDS 140a Winter 2013 Homework 5

2013-02-08T16:49:11Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = ACM 101/AM 125b/CDS 140a
| semester = Winter 2013
| title = Problem Set #5
| issued = 5 Feb 2013 (Tue)
| due = 12 Feb 2013 (Tue)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

<ol>
<li>'''Perko, Section 2.9, problem 3'''
Use the Lyapunov function $V(x)=x_1^2+x_2^2+x_3^2$ to show that the origin is an asymptotically stable equilibrium point of the system
<center><amsmath>
\dot{x}=\begin{bmatrix}-x_2-x_1x_2^2+x_3^2-x_1^3\\
x_1+x_3^3-x_2^3\\
-x_1x_3-x_3x_1^2-x_2x_3^2-x_3^5\end{bmatrix}
</amsmath></center>
Show that the trajectories of the linearized system $\dot{x}=Df(0)x$ for this problem lie on circles in planes parallel to the $x_1,\,x_2$ plane; hence, the origin is stable, but not asymptotically stable for the linearized system.
</li>
<li>
Determine the stability of the system
<center><amsmath>
\aligned
\dot{x}&=-y-x^3\\
\dot{y}&=x^5
\endaligned
</amsmath></center>
Motivated by the first equation, try a Lyapunov function of the form $V(x,y)=\alpha x^6+\beta y^2$. Is the origin asymptotically stable? Is the origin globally asymptotically stable?
</li>
<li>
An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\|x(t)\|\leq Me^{-\alpha t}\|x(0)\|$, $\forall\|x(0)\|\leq\epsilon,\,t\geq 0$.
Prove that an equilibrium point is exponentially stable if there is a function $V(x)$ satisfying
<center><amsmath>k_1\|x\|^2\leq V(x)\leq k_2\|x\|^2,\quad DV(x)\leq -k_3\|x\|^2
</amsmath></center>
for positive constants $k_1$, $k_2$ and $k_3$.
</li>
<li>'''Perko, Section 2.12, problem 2'''
Use Theorem 1 [Centre Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system
<center><amsmath>
\aligned
\dot{x}&=y\\
\dot{y}&=-y+\alpha x^2+xy
\endaligned
</amsmath></center>
for $\alpha\neq 0$ and for $\alpha=0$; i.e., follow the procedure in Example 1 after diagonalizing the system as in Example 3.
</li>
<li> Consider the following system in $\mathbb R^2$:
<center><amsmath>\aligned
\dot{x}&=-\frac{\alpha}{2}(x^2+y^2)+\alpha (x+y)-\alpha\\
\dot y&=-\alpha xy+\alpha (x+y)-\alpha
\endaligned</amsmath></center>
Determine the stable, unstable, and centre manifold of the equilibrium point at $(x,y)=(1,1)$, and determine the stability of this equilibrium point for $\alpha\neq 0$. For determining stability, note that near the equilibrium point there are two 1-dimensional invariant linear manifolds of the form $M=\{(a_1,a_2)\in\mathbb R^2\|a_2=ka_1\}$; determine the flow on these invariant manifolds.
</li>
<li> Consider the following system
<center><amsmath>\aligned
\dot x&=x^2\\
\dot y&=-y
\endaligned</amsmath></center>
with stable manifold $W^s=\{(x,y):x=0\}$
<ol>
<li> Show that the Taylor series approximation to the centre manifold gives $W^c=\{(x,y):y=0\}$.</li>
<li> Show that the set $\{(x,y):y=h(x)\}$ with
<center><amsmath>
h(x)=\left\{\begin{array}{lr}ke^{1/x}&x<0\\0&x\geq 0\end{array}\right.
</amsmath></center>
describes a one-parameter set of invariant manifolds for any $k$. (Hint: what is $dy/dx$?)
</li>
<li> What are the dynamics describing the trajectories along any of these manifolds?</li>
<li> (Optional): Plot the phase portrait for this system. </li>
</ol>
</li>
</ol>

CDS 140a Winter 2013 Homework 6

2013-02-05T20:17:07Z

Macmardg: Created page with "{{warning|This homework set is still being written. Do not start working on these problems until this banner is removed.}}"

{{warning|This homework set is still being written. Do not start working on these problems until this banner is removed.}}

CDS 140a Winter 2013 Homework 5

2013-02-05T20:16:36Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = ACM 101/AM 125b/CDS 140a
| semester = Winter 2013
| title = Problem Set #5
| issued = 5 Feb 2013 (Tue)
| due = 12 Feb 2013 (Tue)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

<ol>
<li>'''Perko, Section 2.9, problem 3'''
Use the Lyapunov function $V(x)=x_1^2+x_2^2+x_3^2$ to show that the origin is an asymptotically stable equilibrium point of the system
<center><amsmath>
\dot{x}=\begin{bmatrix}-x_2-x_1x_2^2+x_3^2-x_1^3\\
x_1+x_3^3-x_2^2\\
-x_1x_3-x_3x_1^2-x_2x_3^2-x_3^5\end{bmatrix}
</amsmath></center>
Show that the trajectories of the linearized system $\dot{x}=Df(0)x$ for this problem lie on circles in planes parallel to the $x_1,\,x_2$ plane; hence, the origin is stable, but not asymptotically stable for the linearized system.
</li>
<li>
Determine the stability of the system
<center><amsmath>
\aligned
\dot{x}&=-y-x^3\\
\dot{y}&=x^5
\endaligned
</amsmath></center>
Motivated by the first equation, try a Lyapunov function of the form $V(x,y)=\alpha x^6+\beta y^2$. Is the origin asymptotically stable? Is the origin globally asymptotically stable?
</li>
<li>
An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\|x(t)\|\leq Me^{-\alpha t}\|x(0)\|$, $\forall\|x(0)\|\leq\epsilon,\,t\geq 0$.
Prove that an equilibrium point is exponentially stable if there is a function $V(x)$ satisfying
<center><amsmath>k_1\|x\|^2\leq V(x)\leq k_2\|x\|^2,\quad DV(x)\leq -k_3\|x\|^2
</amsmath></center>
for positive constants $k_1$, $k_2$ and $k_3$.
</li>
<li>'''Perko, Section 2.12, problem 2'''
Use Theorem 1 [Centre Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system
<center><amsmath>
\aligned
\dot{x}&=y\\
\dot{y}&=-y+\alpha x^2+xy
\endaligned
</amsmath></center>
for $\alpha\neq 0$ and for $\alpha=0$; i.e., follow the procedure in Example 1 after diagonalizing the system as in Example 3.
</li>
<li> Consider the following system in $\mathbb R^2$:
<center><amsmath>\aligned
\dot{x}&=-\frac{\alpha}{2}(x^2+y^2)+\alpha (x+y)-\alpha\\
\dot y&=-\alpha xy+\alpha (x+y)-\alpha
\endaligned</amsmath></center>
Determine the stable, unstable, and centre manifold of the equilibrium point at $(x,y)=(1,1)$, and determine the stability of this equilibrium point for $\alpha\neq 0$. For determining stability, note that near the equilibrium point there are two 1-dimensional invariant linear manifolds of the form $M=\{(a_1,a_2)\in\mathbb R^2\|a_2=ka_1\}$; determine the flow on these invariant manifolds.
</li>
<li> Consider the following system
<center><amsmath>\aligned
\dot x&=x^2\\
\dot y&=-y
\endaligned</amsmath></center>
with stable manifold $W^s=\{(x,y):x=0\}$
<ol>
<li> Show that the Taylor series approximation to the centre manifold gives $W^c=\{(x,y):y=0\}$.</li>
<li> Show that the set $\{(x,y):y=h(x)\}$ with
<center><amsmath>
h(x)=\left\{\begin{array}{lr}ke^{1/x}&x<0\\0&x\geq 0\end{array}\right.
</amsmath></center>
describes a one-parameter set of invariant manifolds for any $k$. (Hint: what is $dy/dx$?)
</li>
<li> What are the dynamics describing the trajectories along any of these manifolds?</li>
<li> (Optional): Plot the phase portrait for this system. </li>
</ol>
</li>
</ol>

CDS 140a Winter 2013 Homework 5

2013-02-05T20:16:17Z

Macmardg:

CDS 140a Winter 2013 Homework 5

2013-02-05T19:48:12Z

Macmardg:

CDS 140a Winter 2013 Homework 5

2013-02-04T23:58:24Z

Macmardg:

CDS 140a Winter 2013 Homework 5

2013-02-04T04:59:54Z

Macmardg:

CDS 140a Winter 2013 Homework 4

2013-02-04T03:09:43Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = ACM 101/AM 125b/CDS 140a
| semester = Winter 2013
| title = Problem Set #3
| issued = 29 Jan 2013 (Tue)
| due = 5 Feb 2013 (Tue)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

<ol>
<li>'''Perko, Section 2.7, problem 1'''
Write the system
<center><amsmath>
\aligned
\dot{x}_1&=x_1+6x_2+x_1x_2,\\
\dot{x}_2&=4x_1+3x_2-x_1^2
\endaligned
</amsmath></center>
in the form
<center><amsmath>
\dot{y}=By+G(y)
</amsmath></center>
where
<center><amsmath>
B=\begin{bmatrix}\lambda_1&0\\0&\lambda_2\end{bmatrix}
</amsmath></center>
with $\lambda_1<0$, $\lambda_2>0$ and $G(y)$ is quadratic in $y_1$ and $y_2$.
</li>

<li>'''Perko, Section 2.7, problem 2'''
Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for
<center><amsmath>
\aligned
\dot{x}_1&=-x_1,\\
\dot{x}_2&=x_2+x_1^2
\endaligned
</amsmath></center>
and use $u^{(3)}(t,a)$ to approximate $S$ near the origin. Also approximate the unstable manifold $U$ near the origin for this system. Note that $u^{(2)}(t,a)=u^{(3)}(t,a)$ and therefore $u^{(j+1)}(t,a)=u^{(j)}(t,a)$ for $j\geq 2$. Thus $u(t,a)=u^{(2)}(t,a)$ which gives the exact function defining $S$.
</li>

<li>'''Perko, Section 2.7, problem 3'''
Solve the system in Problem 2 and show that $S$ and $U$ are given by
<center><amsmath>
S:\,x_2=-\frac{x_1^2}{3}
</amsmath></center>
<center><amsmath>
U:\,x_1=0
</amsmath></center>
Sketch $S$, $U$, $E^s$ and $E^u$.
</li>

<li> Prove that if
<center><amsmath>
\aligned
\dot{x}&=f(x,y),\qquad x\in\mathbb{R}^k\\
\dot{y}&=g(x,y),\qquad g\in\mathbb{R}^m
\endaligned
</amsmath></center>
then the manifold $S=\{(x,y)\in\mathbb R^k\times\mathbb R^m|y=h(x)\}$ is an invariant manifold of the system if
<center><amsmath>
g(x,h(x))=Dh(x)f(x,h(x))
</amsmath></center>
Use this result to compute the stable manifold for the system of problem 2 and 3 above using the Taylor series for $h(x)$ to define $S=\{(x_1,x_2)|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$.

'''Hint:''' One way to show $S$ is an invariant manifold in $\mathbb R^2$ is to show that the normal vector (orthogonal to the tangent to $S$ at $(x,h(x))$) is orthogonal to the vector field $(f,g)$ at that point. (It is sufficient to prove the result for $\mathbb R^2$.)
</li>
<li> '''Perko, Section 2.7, Problem 6'''
Let $E$ be an open subset of $\mathbb{R}^n$ containing the origin. Use the fact that if $F\in C^1(E)$ then for all $x, y\in N_\delta(0)\subset E$ there exists a $\xi\in N_\delta(0)$ such that
<center><amsmath>
|F(x)-F(y)|\leq\|DF(\xi)\|\,|x-y|
</amsmath></center>
(cf. Theorem 4.19 and the proof of Theorem 9.19 in [R]) to prove that if $F\in C^1(E)$ and $F(0)=DF(0)=0$ then given any $\epsilon>0$ there exists a $\delta>0$ such that for all $x, y\in N_\delta(0)$ we have
<center><amsmath>
|F(x)-F(y)|<\epsilon |x-y|
</amsmath></center>
</li>
<li>'''Perko, Section 2.9, problem 2(a)(b)'''
Determine the stability of the equilibrium points of the system (1) with $f(x)$ given by
<center><amsmath>
(a)\quad\begin{bmatrix}x_1^2-x_2^2-1\\2x_2\end{bmatrix}
</amsmath></center>
<center><amsmath>
(b)\quad\begin{bmatrix}x_2-x_1^2+2\\2x_2^2-2x_1x_2\end{bmatrix}
</amsmath></center>
</li>

<!--
And any two of the following:
<li>
'''Perko, Section 2.7, problem 4'''
Find the first four successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, $u^{(3)}(t,a)$, and $u^{(4)}(t,a)$ for the system
<center><amsmath>
\aligned
\dot{x}_1&=-x_1\\
\dot{x}_2&=-x_2+x_1^2\\
\dot{x}_3&=x_3+x_2^2
\endaligned
</amsmath></center>
Show that $u^{(3)}(t,a)=u^{(4)}(t,a)=\cdots$ and hence $u(t,a)=u^{(3)}(t,a)$. Find $S$ and $U$ for this problem.
</li>
<li> '''Perko, Section 2.7, problem 5'''
Solve the system and show that
<center><amsmath>
S:\,\,x_3=-\frac{1}{3}x_2^2-\frac{1}{6}x_1^2x_2-\frac{1}{30}x_1^4
</amsmath></center>
and
<center><amsmath>
U:\,\,x_1=x_2=0
</amsmath></center>
Note that these formulae actually determine the global stable and unstable manifolds $W^s(0)$ and $W^u(0)$ respectively.
</li>

<li>
'''Perko, Section 2.8, Problem 1'''
Solve the system
<center><amsmath>
\aligned
\dot{y}_1&=-y_1\\
\dot{y}_2&=-y_2+z^2\\
\dot{z}&=z
\endaligned
</amsmath></center>
and show that the successive approximations $\Phi_k\rightarrow\Phi$ and $\Psi_k\rightarrow\Psi$ as $k\rightarrow\infty$ for all $x=(y_1,y_2,z)\in{\mathbb R}^3$. Define $H_0=(\Phi,\Psi)^T$ and use this homeomorphism to find
<center><amsmath>
H=\int_0^1L^{-s}H_0T^sds.
</amsmath></center>
Use the homemorphism $H$ to find the stable and unstable manifolds
<center><amsmath>
W^s(0)=H^{-1}(E^s)\quad {\mathrm{and}}\quad W^u(0)=H^{-1}(E^u)
</amsmath></center>
for this system.

HINT: You should find
<center><amsmath>\aligned
H(y_1,y_2,z)&=(y_1,y_2-z^2/3,z)^T\\
W^s(0)&=\{x\in{\mathbb{R}}|z=0\}
\endaligned
</amsmath></center>
and
<center><amsmath>
W^u(0)=\{x\in{\mathbb{R}}|y_1=0,y_2=z^2/3\}.
</amsmath></center>
</li>
</!--
</ol>

<hr>
Notes:
* The problems are transcribed above in case you don't have access to Perko. However, in the case of discrepancy, you should use Perko as the definitive source of the problem statement.

CDS 140a Winter 2013 Homework 5

2013-02-03T19:10:41Z

Macmardg:

CDS 140a Winter 2013 Homework 5

2013-02-03T04:43:24Z

Macmardg:

CDS 140a Winter 2013 Homework 5

2013-01-30T16:00:53Z

Macmardg: Created page with "{{warning|This homework set is still being written. Do not start working on these problems until this banner is removed.}} {{CDS homework | instructor = R. Murray, D. MacM..."

CDS 140a Winter 2013 Homework 4

2013-01-30T16:00:09Z

Macmardg:

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = ACM 101/AM 125b/CDS 140a
| semester = Winter 2013
| title = Problem Set #3
| issued = 29 Jan 2013 (Tue)
| due = 5 Feb 2013 (Tue)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

<ol>
<li>'''Perko, Section 2.7, problem 1'''
Write the system
<center><amsmath>
\aligned
\dot{x}_1&=x_1+6x_2+x_1x_2,\\
\dot{x}_2&=4x_1+3x_2-x_1^2
\endaligned
</amsmath></center>
in the form
<center><amsmath>
\dot{y}=By+G(y)
</amsmath></center>
where
<center><amsmath>
B=\begin{bmatrix}\lambda_1&0\\0&\lambda_2\end{bmatrix}
</amsmath></center>
with $\lambda_1<0$, $\lambda_2>0$ and $G(y)$ is quadratic in $y_1$ and $y_2$.
</li>

<li>'''Perko, Section 2.7, problem 2'''
Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for
<center><amsmath>
\aligned
\dot{x}_1&=-x_1,\\
\dot{x}_2&=x_2+x_1^2
\endaligned
</amsmath></center>
and use $u^{(3)}(t,a)$ to approximate $S$ near the origin. Also approximate the unstable manifold $U$ near the origin for this system. Note that $u^{(2)}(t,a)=u^{(3)}(t,a)$ and therefore $u^{(j+1)}(t,a)=u^{(j)}(t,a)$ for $j\geq 2$. Thus $u(t,a)=u^{(2)}(t,a)$ which gives the exact function defining $S$.
</li>

<li>'''Perko, Section 2.7, problem 3'''
Solve the system in Problem 2 and show that $S$ and $U$ are given by
<center><amsmath>
S:\,x_2=-\frac{x_1^2}{3}
</amsmath></center>
<center><amsmath>
U:\,x_1=0
</amsmath></center>
Sketch $S$, $U$, $E^s$ and $E^u$.
</li>

<li> Prove that if
<center><amsmath>
\aligned
\dot{x}&=f(x,y),\qquad x\in\mathbb{R}^k\\
\dot{y}&=g(x,y),\qquad g\in\mathbb{R}^m
\endaligned
</amsmath></center>
then the manifold $S=\{(x,y)\in\mathbb R^k\times\mathbb R^m|y=h(x)\}$ is an invariant manifold of the system if
<center><amsmath>
g(x,h(x))=Dh(x)f(x,h(x))
</amsmath></center>
Use this result to compute the stable manifold for the system of problem 2 and 3 above using the Taylor series for $h(x)$ to define $S=\{(x_1,x_2)|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$.

'''Hint:''' to show $S$ is an invariant manifold, show that the normal vector (orthogonal to the tangent to $S$ at $(x,h(x))$) is orthogonal to the vector field $(f,g)$ at that point.
</li>
<li> '''Perko, Section 2.7, Problem 6'''
Let $E$ be an open subset of $\mathbb{R}^n$ containing the origin. Use the fact that if $F\in C^1(E)$ then for all $x, y\in N_\delta(0)\subset E$ there exists a $\xi\in N_\delta(0)$ such that
<center><amsmath>
|F(x)-F(y)|\leq\|DF(\xi)\|\,|x-y|
</amsmath></center>
(cf. Theorem 4.19 and the proof of Theorem 9.19 in [R]) to prove that if $F\in C^1(E)$ and $F(0)=DF(0)=0$ then given any $\epsilon>0$ there exists a $\delta>0$ such that for all $x, y\in N_\delta(0)$ we have
<center><amsmath>
|F(x)-F(y)|<\epsilon |x-y|
</amsmath></center>
</li>
<li>'''Perko, Section 2.9, problem 2(a)(b)'''
Determine the stability of the equilibrium points of the system (1) with $f(x)$ given by
<center><amsmath>
(a)\quad\begin{bmatrix}x_1^2-x_2^2-1\\2x_2\end{bmatrix}
</amsmath></center>
<center><amsmath>
(b)\quad\begin{bmatrix}x_2-x_1^2+2\\2x_2^2-2x_1x_2\end{bmatrix}
</amsmath></center>
</li>

<!--
And any two of the following:
<li>
'''Perko, Section 2.7, problem 4'''
Find the first four successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, $u^{(3)}(t,a)$, and $u^{(4)}(t,a)$ for the system
<center><amsmath>
\aligned
\dot{x}_1&=-x_1\\
\dot{x}_2&=-x_2+x_1^2\\
\dot{x}_3&=x_3+x_2^2
\endaligned
</amsmath></center>
Show that $u^{(3)}(t,a)=u^{(4)}(t,a)=\cdots$ and hence $u(t,a)=u^{(3)}(t,a)$. Find $S$ and $U$ for this problem.
</li>
<li> '''Perko, Section 2.7, problem 5'''
Solve the system and show that
<center><amsmath>
S:\,\,x_3=-\frac{1}{3}x_2^2-\frac{1}{6}x_1^2x_2-\frac{1}{30}x_1^4
</amsmath></center>
and
<center><amsmath>
U:\,\,x_1=x_2=0
</amsmath></center>
Note that these formulae actually determine the global stable and unstable manifolds $W^s(0)$ and $W^u(0)$ respectively.
</li>

<li>
'''Perko, Section 2.8, Problem 1'''
Solve the system
<center><amsmath>
\aligned
\dot{y}_1&=-y_1\\
\dot{y}_2&=-y_2+z^2\\
\dot{z}&=z
\endaligned
</amsmath></center>
and show that the successive approximations $\Phi_k\rightarrow\Phi$ and $\Psi_k\rightarrow\Psi$ as $k\rightarrow\infty$ for all $x=(y_1,y_2,z)\in{\mathbb R}^3$. Define $H_0=(\Phi,\Psi)^T$ and use this homeomorphism to find
<center><amsmath>
H=\int_0^1L^{-s}H_0T^sds.
</amsmath></center>
Use the homemorphism $H$ to find the stable and unstable manifolds
<center><amsmath>
W^s(0)=H^{-1}(E^s)\quad {\mathrm{and}}\quad W^u(0)=H^{-1}(E^u)
</amsmath></center>
for this system.

HINT: You should find
<center><amsmath>\aligned
H(y_1,y_2,z)&=(y_1,y_2-z^2/3,z)^T\\
W^s(0)&=\{x\in{\mathbb{R}}|z=0\}
\endaligned
</amsmath></center>
and
<center><amsmath>
W^u(0)=\{x\in{\mathbb{R}}|y_1=0,y_2=z^2/3\}.
</amsmath></center>
</li>
</!--
</ol>

<hr>
Notes:
* The problems are transcribed above in case you don't have access to Perko. However, in the case of discrepancy, you should use Perko as the definitive source of the problem statement.

CDS 140a Winter 2013 Homework 4

2013-01-30T15:58:52Z

Macmardg:

<!--
{{warning|This homework set is still being written. Do not start working on these problems until this banner is removed.}}
</!--

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = ACM 101/AM 125b/CDS 140a
| semester = Winter 2013
| title = Problem Set #3
| issued = 29 Jan 2013 (Tue)
| due = 5 Feb 2013 (Tue)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

(Not yet edited from 2011)

<ol>
<li>'''Perko, Section 2.7, problem 1'''
Write the system
<center><amsmath>
\aligned
\dot{x}_1&=x_1+6x_2+x_1x_2,\\
\dot{x}_2&=4x_1+3x_2-x_1^2
\endaligned
</amsmath></center>
in the form
<center><amsmath>
\dot{y}=By+G(y)
</amsmath></center>
where
<center><amsmath>
B=\begin{bmatrix}\lambda_1&0\\0&\lambda_2\end{bmatrix}
</amsmath></center>
with $\lambda_1<0$, $\lambda_2>0$ and $G(y)$ is quadratic in $y_1$ and $y_2$.
</li>

<li>'''Perko, Section 2.7, problem 2'''
Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for
<center><amsmath>
\aligned
\dot{x}_1&=-x_1,\\
\dot{x}_2&=x_2+x_1^2
\endaligned
</amsmath></center>
and use $u^{(3)}(t,a)$ to approximate $S$ near the origin. Also approximate the unstable manifold $U$ near the origin for this system. Note that $u^{(2)}(t,a)=u^{(3)}(t,a)$ and therefore $u^{(j+1)}(t,a)=u^{(j)}(t,a)$ for $j\geq 2$. Thus $u(t,a)=u^{(2)}(t,a)$ which gives the exact function defining $S$.
</li>

<li>'''Perko, Section 2.7, problem 3'''
Solve the system in Problem 2 and show that $S$ and $U$ are given by
<center><amsmath>
S:\,x_2=-\frac{x_1^2}{3}
</amsmath></center>
<center><amsmath>
U:\,x_1=0
</amsmath></center>
Sketch $S$, $U$, $E^s$ and $E^u$.
</li>

<li> Prove that if
<center><amsmath>
\aligned
\dot{x}&=f(x,y),\qquad x\in\mathbb{R}^k\\
\dot{y}&=g(x,y),\qquad g\in\mathbb{R}^m
\endaligned
</amsmath></center>
then the manifold $S=\{(x,y)\in\mathbb R^k\times\mathbb R^m|y=h(x)\}$ is an invariant manifold of the system if
<center><amsmath>
g(x,h(x))=Dh(x)f(x,h(x))
</amsmath></center>
Use this result to compute the stable manifold for the system of problem 2 and 3 above using the Taylor series for $h(x)$ to define $S=\{(x_1,x_2)|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$.

'''Hint:''' to show $S$ is an invariant manifold, show that the normal vector (orthogonal to the tangent to $S$ at $(x,h(x))$) is orthogonal to the vector field $(f,g)$ at that point.
</li>
<li> '''Perko, Section 2.7, Problem 6'''
Let $E$ be an open subset of $\mathbb{R}^n$ containing the origin. Use the fact that if $F\in C^1(E)$ then for all $x, y\in N_\delta(0)\subset E$ there exists a $\xi\in N_\delta(0)$ such that
<center><amsmath>
|F(x)-F(y)|\leq\|DF(\xi)\|\,|x-y|
</amsmath></center>
(cf. Theorem 4.19 and the proof of Theorem 9.19 in [R]) to prove that if $F\in C^1(E)$ and $F(0)=DF(0)=0$ then given any $\epsilon>0$ there exists a $\delta>0$ such that for all $x, y\in N_\delta(0)$ we have
<center><amsmath>
|F(x)-F(y)|<\epsilon |x-y|
</amsmath></center>
</li>
<li>'''Perko, Section 2.9, problem 2(a)(b)'''
Determine the stability of the equilibrium points of the system (1) with $f(x)$ given by
<center><amsmath>
(a)\quad\begin{bmatrix}x_1^2-x_2^2-1\\2x_2\end{bmatrix}
</amsmath></center>
<center><amsmath>
(b)\quad\begin{bmatrix}x_2-x_1^2+2\\2x_2^2-2x_1x_2\end{bmatrix}
</amsmath></center>
</li>

<!--
And any two of the following:
<li>
'''Perko, Section 2.7, problem 4'''
Find the first four successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, $u^{(3)}(t,a)$, and $u^{(4)}(t,a)$ for the system
<center><amsmath>
\aligned
\dot{x}_1&=-x_1\\
\dot{x}_2&=-x_2+x_1^2\\
\dot{x}_3&=x_3+x_2^2
\endaligned
</amsmath></center>
Show that $u^{(3)}(t,a)=u^{(4)}(t,a)=\cdots$ and hence $u(t,a)=u^{(3)}(t,a)$. Find $S$ and $U$ for this problem.
</li>
<li> '''Perko, Section 2.7, problem 5'''
Solve the system and show that
<center><amsmath>
S:\,\,x_3=-\frac{1}{3}x_2^2-\frac{1}{6}x_1^2x_2-\frac{1}{30}x_1^4
</amsmath></center>
and
<center><amsmath>
U:\,\,x_1=x_2=0
</amsmath></center>
Note that these formulae actually determine the global stable and unstable manifolds $W^s(0)$ and $W^u(0)$ respectively.
</li>

<li>
'''Perko, Section 2.8, Problem 1'''
Solve the system
<center><amsmath>
\aligned
\dot{y}_1&=-y_1\\
\dot{y}_2&=-y_2+z^2\\
\dot{z}&=z
\endaligned
</amsmath></center>
and show that the successive approximations $\Phi_k\rightarrow\Phi$ and $\Psi_k\rightarrow\Psi$ as $k\rightarrow\infty$ for all $x=(y_1,y_2,z)\in{\mathbb R}^3$. Define $H_0=(\Phi,\Psi)^T$ and use this homeomorphism to find
<center><amsmath>
H=\int_0^1L^{-s}H_0T^sds.
</amsmath></center>
Use the homemorphism $H$ to find the stable and unstable manifolds
<center><amsmath>
W^s(0)=H^{-1}(E^s)\quad {\mathrm{and}}\quad W^u(0)=H^{-1}(E^u)
</amsmath></center>
for this system.

HINT: You should find
<center><amsmath>\aligned
H(y_1,y_2,z)&=(y_1,y_2-z^2/3,z)^T\\
W^s(0)&=\{x\in{\mathbb{R}}|z=0\}
\endaligned
</amsmath></center>
and
<center><amsmath>
W^u(0)=\{x\in{\mathbb{R}}|y_1=0,y_2=z^2/3\}.
</amsmath></center>
</li>
</!--
</ol>

<hr>
Notes:
* The problems are transcribed above in case you don't have access to Perko. However, in the case of discrepancy, you should use Perko as the definitive source of the problem statement.

CDS 140a Winter 2013 Homework 4

2013-01-30T03:19:52Z

Macmardg:

{{warning|This homework set is still being written. Do not start working on these problems until this banner is removed.}}

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = ACM 101/AM 125b/CDS 140a
| semester = Winter 2013
| title = Problem Set #3
| issued = 29 Jan 2013 (Tue)
| due = 5 Feb 2013 (Tue)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

(Not yet edited from 2011)

<ol>
<li>'''Perko, Section 2.7, problem 1'''
Write the system
<center><amsmath>
\aligned
\dot{x}_1&=x_1+6x_2+x_1x_2,\\
\dot{x}_2&=4x_1+3x_2-x_1^2
\endaligned
</amsmath></center>
in the form
<center><amsmath>
\dot{y}=By+G(y)
</amsmath></center>
where
<center><amsmath>
B=\begin{bmatrix}\lambda_1&0\\0&\lambda_2\end{bmatrix}
</amsmath></center>
with $\lambda_1<0$, $\lambda_2>0$ and $G(y)$ is quadratic in $y_1$ and $y_2$.
</li>

<li>'''Perko, Section 2.7, problem 2'''
Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for
<center><amsmath>
\aligned
\dot{x}_1&=-x_1,\\
\dot{x}_2&=x_2+x_1^2
\endaligned
</amsmath></center>
and use $u^{(3)}(t,a)$ to approximate $S$ near the origin. Also approximate the unstable manifold $U$ near the origin for this system. Note that $u^{(2)}(t,a)=u^{(3)}(t,a)$ and therefore $u^{(j+1)}(t,a)=u^{(j)}(t,a)$ for $j\geq 2$. Thus $u(t,a)=u^{(2)}(t,a)$ which gives the exact function defining $S$.
</li>

<li>'''Perko, Section 2.7, problem 3'''
Solve the system in Problem 2 and show that $S$ and $U$ are given by
<center><amsmath>
S:\,x_2=-\frac{x_1^2}{3}
</amsmath></center>
<center><amsmath>
U:\,x_1=0
</amsmath></center>
Sketch $S$, $U$, $E^s$ and $E^u$.
</li>

<li> Prove that if
<center><amsmath>
\aligned
\dot{x}&=f(x,y),\qquad x\in\mathbb{R}^k\\
\dot{y}&=g(x,y),\qquad g\in\mathbb{R}^m
\endaligned
</amsmath></center>
then the manifold $S=\{(x,y)\in\mathbb R^k\times\mathbb R^m|y=h(x)\}$ is an invariant manifold of the system if
<center><amsmath>
g(x,h(x))=Dh(x)f(x,h(x))
</amsmath></center>
Use this result to compute the stable manifold for the system of problem 2 and 3 above using the Taylor series for $h(x)$ to define $S=\{(x_1,x_2)|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$.

'''Hint:''' to show $S$ is an invariant manifold, show that the normal vector (orthogonal to the tangent to $S$ at $(x,h(x))$) is orthogonal to the vector field $(f,g)$ at that point.
</li>
<li> '''Perko, Section 2.7, Problem 6'''
Let $E$ be an open subset of $\mathbb{R}^n$ containing the origin. Use the fact that if $F\in C^1(E)$ then for all $x, y\in N_\delta(0)\subset E$ there exists a $\xi\in N_\delta(0)$ such that
<center><amsmath>
|F(x)-F(y)|\leq\|DF(\xi)\|\,|x-y|
</amsmath></center>
(cf. Theorem 4.19 and the proof of Theorem 9.19 in [R]) to prove that if $F\in C^1(E)$ and $F(0)=DF(0)=0$ then given any $\epsilon>0$ there exists a $\delta>0$ such that for all $x, y\in N_\delta(0)$ we have
<center><amsmath>
|F(x)-F(y)|<\epsilon |x-y|
</amsmath></center>
</li>
<li>'''Perko, Section 2.9, problem 2(a)(b)'''
Determine the stability of the equilibrium points of the system (1) with $f(x)$ given by
<center><amsmath>
(a)\quad\begin{bmatrix}x_1^2-x_2^2-1\\2x_2\end{bmatrix}
</amsmath></center>
<center><amsmath>
(b)\quad\begin{bmatrix}x_2-x_1^2+2\\2x_2^2-2x_1x_2\end{bmatrix}
</amsmath></center>
</li>

<!--
And any two of the following:
<li>
'''Perko, Section 2.7, problem 4'''
Find the first four successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, $u^{(3)}(t,a)$, and $u^{(4)}(t,a)$ for the system
<center><amsmath>
\aligned
\dot{x}_1&=-x_1\\
\dot{x}_2&=-x_2+x_1^2\\
\dot{x}_3&=x_3+x_2^2
\endaligned
</amsmath></center>
Show that $u^{(3)}(t,a)=u^{(4)}(t,a)=\cdots$ and hence $u(t,a)=u^{(3)}(t,a)$. Find $S$ and $U$ for this problem.
</li>
<li> '''Perko, Section 2.7, problem 5'''
Solve the system and show that
<center><amsmath>
S:\,\,x_3=-\frac{1}{3}x_2^2-\frac{1}{6}x_1^2x_2-\frac{1}{30}x_1^4
</amsmath></center>
and
<center><amsmath>
U:\,\,x_1=x_2=0
</amsmath></center>
Note that these formulae actually determine the global stable and unstable manifolds $W^s(0)$ and $W^u(0)$ respectively.
</li>

<li>
'''Perko, Section 2.8, Problem 1'''
Solve the system
<center><amsmath>
\aligned
\dot{y}_1&=-y_1\\
\dot{y}_2&=-y_2+z^2\\
\dot{z}&=z
\endaligned
</amsmath></center>
and show that the successive approximations $\Phi_k\rightarrow\Phi$ and $\Psi_k\rightarrow\Psi$ as $k\rightarrow\infty$ for all $x=(y_1,y_2,z)\in{\mathbb R}^3$. Define $H_0=(\Phi,\Psi)^T$ and use this homeomorphism to find
<center><amsmath>
H=\int_0^1L^{-s}H_0T^sds.
</amsmath></center>
Use the homemorphism $H$ to find the stable and unstable manifolds
<center><amsmath>
W^s(0)=H^{-1}(E^s)\quad {\mathrm{and}}\quad W^u(0)=H^{-1}(E^u)
</amsmath></center>
for this system.

HINT: You should find
<center><amsmath>\aligned
H(y_1,y_2,z)&=(y_1,y_2-z^2/3,z)^T\\
W^s(0)&=\{x\in{\mathbb{R}}|z=0\}
\endaligned
</amsmath></center>
and
<center><amsmath>
W^u(0)=\{x\in{\mathbb{R}}|y_1=0,y_2=z^2/3\}.
</amsmath></center>
</li>
</!--
</ol>

<hr>
Notes:
* The problems are transcribed above in case you don't have access to Perko. However, in the case of discrepancy, you should use Perko as the definitive source of the problem statement.

CDS 140a Winter 2013 Homework 4

2013-01-27T20:53:47Z

Macmardg:

{{warning|This homework set is still being written. Do not start working on these problems until this banner is removed.}}

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = ACM 101/AM 125b/CDS 140a
| semester = Winter 2013
| title = Problem Set #3
| issued = 29 Jan 2013 (Tue)
| due = 5 Feb 2013 (Tue)
}} __MATHJAX__

'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

(Not yet edited from 2011)

<ol>
<li>'''Perko, Section 2.7, problem 1'''
Write the system
<center><amsmath>
\aligned
\dot{x}_1&=x_1+6x_2+x_1x_2,\\
\dot{x}_2&=4x_1+3x_2-x_1^2
\endaligned
</amsmath></center>
in the form
<center><amsmath>
\dot{y}=By+G(y)
</amsmath></center>
where
<center><amsmath>
B=\begin{bmatrix}\lambda_1&0\\0&\lambda_2\end{bmatrix}
</amsmath></center>
with $\lambda_1<0$, $\lambda_2>0$ and $G(y)$ is quadratic in $y_1$ and $y_2$.
</li>

<li>'''Perko, Section 2.7, problem 2'''
Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for
<center><amsmath>
\aligned
\dot{x}_1&=-x_1,\\
\dot{x}_2&=x_2+x_1^2
\endaligned
</amsmath></center>
and use $u^{(3)}(t,a)$ to approximate $S$ near the origin. Also approximate the unstable manifold $U$ near the origin for this system. Note that $u^{(2)}(t,a)=u^{(3)}(t,a)$ and therefore $u^{(j+1)}(t,a)=u^{(j)}(t,a)$ for $j\geq 2$. Thus $u(t,a)=u^{(2)}(t,a)$ which gives the exact function defining $S$.
</li>

<li>'''Perko, Section 2.7, problem 3'''
Solve the system in Problem 2 and show that $S$ and $U$ are given by
<center><amsmath>
S:\,x_2=-\frac{x_1^2}{3}
</amsmath></center>
<center><amsmath>
U:\,x_1=0
</amsmath></center>
Sketch $S$, $U$, $E^s$ and $E^u$.
</li>

<li> Compute the stable manifold for the system of problem 2 and 3 above using the Taylor series for $h(x)$ to define $S=\{(x_1,x_2)|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$.
</li>

<li>'''Perko, Section 2.9, problem 2(a)(b)'''
Determine the stability of the equilibrium points of the system (1) with $f(x)$ given by
<center><amsmath>
(a)\quad\begin{bmatrix}x_1^2-x_2^2-1\\2x_2\end{bmatrix}
</amsmath></center>
<center><amsmath>
(b)\quad\begin{bmatrix}x_2-x_1^2+2\\2x_2^2-2x_1x_2\end{bmatrix}
</amsmath></center>
</li>

And any two of the following:
<li>
'''Perko, Section 2.7, problem 4'''
Find the first four successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, $u^{(3)}(t,a)$, and $u^{(4)}(t,a)$ for the system
<center><amsmath>
\aligned
\dot{x}_1&=-x_1\\
\dot{x}_2&=-x_2+x_1^2\\
\dot{x}_3&=x_3+x_2^2
\endaligned
</amsmath></center>
Show that $u^{(3)}(t,a)=u^{(4)}(t,a)=\cdots$ and hence $u(t,a)=u^{(3)}(t,a)$. Find $S$ and $U$ for this problem.
</li>
<li> '''Perko, Section 2.7, problem 5'''
Solve the system and show that
<center><amsmath>
S:\,\,x_3=-\frac{1}{3}x_2^2-\frac{1}{6}x_1^2x_2-\frac{1}{30}x_1^4
</amsmath></center>
and
<center><amsmath>
U:\,\,x_1=x_2=0
</amsmath></center>
Note that these formulae actually determine the global stable and unstable manifolds $W^s(0)$ and $W^u(0)$ respectively.
</li>
<li> '''Perko, Section 2.7, Problem 6'''
Let $E$ be an open subset of $\mathbb{R}^n$ containing the origin. Use the fact that if $F\in C^1(E)$ then for all $x, y\in N_\delta(0)\subset E$ there exists a $\xi\in N_\delta(0)$ such that
<center><amsmath>
|F(x)-F(y)|\leq\|DF(\xi)\|\,|x-y|
</amsmath></center>
(cf. Theorem 4.19 and the proof of Theorem 9.19 in [R]) to prove that if $F\in C^1(E)$ and $F(0)=DF(0)=0$ then given any $\epsilon>0$ there exists a $\delta>0$ such that for all $x, y\in N_\delta(0)$ we have
<center><amsmath>
|F(x)-F(y)|<\epsilon |x-y|
</amsmath></center>
</li>
<li>
'''Perko, Section 2.8, Problem 1'''
Solve the system
<center><amsmath>
\aligned
\dot{y}_1&=-y_1\\
\dot{y}_2&=-y_2+z^2\\
\dot{z}&=z
\endaligned
</amsmath></center>
and show that the successive approximations $\Phi_k\rightarrow\Phi$ and $\Psi_k\rightarrow\Psi$ as $k\rightarrow\infty$ for all $x=(y_1,y_2,z)\in{\mathbb R}^3$. Define $H_0=(\Phi,\Psi)^T$ and use this homeomorphism to find
<center><amsmath>
H=\int_0^1L^{-s}H_0T^sds.
</amsmath></center>
Use the homemorphism $H$ to find the stable and unstable manifolds
<center><amsmath>
W^s(0)=H^{-1}(E^s)\quad {\mathrm{and}}\quad W^u(0)=H^{-1}(E^u)
</amsmath></center>
for this system.

HINT: You should find
<center><amsmath>\aligned
H(y_1,y_2,z)&=(y_1,y_2-z^2/3,z)^T\\
W^s(0)&=\{x\in{\mathbb{R}}|z=0\}
\endaligned
</amsmath></center>
and
<center><amsmath>
W^u(0)=\{x\in{\mathbb{R}}|y_1=0,y_2=z^2/3\}.
</amsmath></center>

</li>
</ol>

<hr>
Notes:
* The problems are transcribed above in case you don't have access to Perko. However, in the case of discrepancy, you should use Perko as the definitive source of the problem statement.