CDS 232, Winter 2027

Nonlinear Dynamics
Instructors Richard Murray (CDS/BE), murray@cds.caltech.edu Lectures: MW, 10-11:30 am, 213 Annenberg Office hours: Wed, 3-4 pm, Annenberg Lounge	Teaching Assistants TBD Office hours: TBD

This is the course homepage for CDS 232, Winter 2027.

Catalog Description

This course studies nonlinear dynamical systems beginning from first principles. Topics include: existence and uniqueness properties of solutions to nonlinear ODEs, stability of nonlinear systems from the perspective of Lyapunov, and behavior unique to nonlinear systems; for example: stability of periodic orbits, Poincare maps and stability/invariance of sets. The dynamics of robotic systems will be used as a motivating example.

Lecture Schedule

Date	Topic	Reading	Homework
Week 1 4 Jan 6 Jan	Introduction and solutions of nonlinear systems Linear vs. nonlinear dynamics, and phenomena unique to nonlinear systems Existence and uniqueness of solutions (Lipschitz continuity, Picard iteration) Discrete-time systems	AT Ch. 1, 2 Lecture materials: Lean proofs:	HW #1 Out: 4 Jan Due: 13 Jan, 10 am
Week 2 11 Jan 13 Jan	Continuity of solutions and linearization Gronwall and comparison lemmas Continuity of solutions in initial conditions and parameters Flows and orbital equivalence Linearization and the Hartman–Grobman theorem	AT Ch. 3, 4 Lecture materials: Lean proofs:	HW #2 Out: 11 Jan Due: 20 Jan, 10 am
Week 3 18 Jan 20 Jan	Structure near equilibria Stable, unstable and center subspaces Invariant manifolds and the stable manifold theorem	AT Ch. 5 Lecture materials: Lean proofs:	HW #3 Out: 18 Jan Due: 27 Jan, 10 am
Week 4 25 Jan 27 Jan*	Stability and Lyapunov's method Stability, asymptotic stability, instability, and regions of attraction Lyapunov functions and Lyapunov's theorem Global stability and radial unboundedness	AT Ch. 6, 7 Lecture materials: Lean proofs:	HW #4 Out: 25 Jan Due: 3 Feb, 10 am
Week 5 1 Feb 3 Feb	Exponential and converse Lyapunov theory Exponential stability and the Lyapunov equation for linear systems Stability of the linearization (indirect method) Time-varying systems and instability theorems Converse Lyapunov theorem	AT Ch. 8, 9 Lecture materials: Lean proofs:	HW #5 Out: 1 Feb Due: 10 Feb, 10 am
Week 6 8 Feb 10 Feb	Invariance and inputs LaSalle's invariance principle Robotic systems and their stabilization Input-to-state stability (ISS) and its Lyapunov characterizations	AT Ch. 10, 11 Lecture materials: Lean proofs:	HW #6 Out: 8 Feb Due: 17 Feb, 10 am
Week 7 15 Feb 17 Feb	Safety Penalty functions for safety Barrier functions and set invariance Relationship between barrier and Lyapunov functions	AT Ch. 12 Lecture materials: Lean proofs:	HW #7 Out: 15 Feb Due: 24 Feb, 10 am
Week 8 22 Feb 24 Feb*	Periodic orbits and Poincaré maps Periodic orbits and limit cycles Poincaré–Bendixson theorem and Bendixson's criterion Poincaré sections and maps Orbit stability as fixed-point stability of the Poincaré map	AT Ch. 13, 14 Lecture materials: Lean proofs:	HW #8 Out: 22 Feb Due: 3 Mar, 10 am
Week 9 1 Mar 3 Mar	Stability of periodic orbits and other attractors Time-varying linear systems and Floquet theory Linearization of the Poincaré map Lyapunov perspective on Poincaré maps Limit sets and more general attractors	AT Ch. 15 Lecture materials: Lean proofs:	HW #9 Out: 1 Mar Due: 10 Mar, 10 am
Week 10 8 Mar 10 Mar	Hybrid systems and review Hybrid systems, Zeno behavior and Zeno equilibria Lyapunov-like conditions for Zeno stability Final exam review (Wed)	AT Ch. 16 Lecture materials: Lean proofs:
Finals	None		Final exam

Grading

The final grade will be based on homework sets and a final exam:

• Homework (15%): Homework sets will be handed out weekly and due on Wednesdays by 10 am via Gradescope. Each student is allowed up to two extensions of no more than 2 days each over the course of the term. Homework turned in after Friday at 10 am or after the two extensions are exhausted will receive 50% credit.

• Final exam (85%): The final exam will be either an oral exam or a 1.5 hour, in-class, closed-book exam.

Collaboration Policy

Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor, but you cannot consult homework solutions from prior years and you must cite any use of material from outside references. All solutions that are handed in should be written up individually and should reflect your own understanding of the subject matter at the time of writing. Python (or MATLAB) scripts and plots are considered part of your writeup and should be done individually (you can share ideas, but not code).

AI tools may be used in the same manner as a fellow student in the class: you are allowed to consult online tools and use them to understand the topics, but all solutions should be guided by you.

No collaboration is allowed on the final exam.

Course Text and References

The primary course text is

• [NLD] Richard M. Murray, Feedback Systems: Notes on Nonlinear Dynamics, 2027. (Updated 10 Nov 2026)

This book is available via free download.

The following additional references, also available for free, may also be useful:

• [LST] Richard M. Murray, Feedback Systems: Notes on Linear Systems Theory, 2020. (Updated 30 Oct 2020)

In addition to the books above, the textbooks below may also be useful. They are available in the library (non-reserve), from other students, or you can order them online.

• [AT] Aaron D. Ames and Paulo Tabuada, Lectures on Nonlinear Dynamics and Control, 2024 (draft).