Difference between revisions of "CDS 140a Winter 2014 Homework 3"

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  | title = Problem Set #3
  | title = Problem Set #3
  | issued = 21 Jan 2014 (Tue)
  | issued = 21 Jan 2014 (Tue)
  | due = 29 Jan 2011 (Wed) @ noon <br>Turn in to box outside Steele House
  | due = 29 Jan 2014 (Wed) @ noon <br>Turn in to box outside Steele House
| pdf = cds140-wi14_hw3.pdf
}} __MATHJAX__
}} __MATHJAX__
{{warning|This homework set is still under preparation.  This banner will be removed when it is finalized}}


'''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
'''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
Line 28: Line 28:
</li>
</li>


<!-- Comments from 2014 TAs
Quite a people had questions on problem 2, and
the students were told that they could assume that continuity conditions
in both time and space. 
-->
<li> '''Perko, Section 2.2, problem 3''' <font color=blue>(if you have not had ACM 125a or CDS 201)</font>: Use the method of successive approximations to show that if $f(x,t)$ is continuous in $t$ in some interval containing $t = 0$ and continuously differentiable in $x$ for all $x$ in some open set $E \subset {\mathbb R}^n$ containing $x_0$, then there exists $a > 0$ such that the initial value problem
<li> '''Perko, Section 2.2, problem 3''' <font color=blue>(if you have not had ACM 125a or CDS 201)</font>: Use the method of successive approximations to show that if $f(x,t)$ is continuous in $t$ in some interval containing $t = 0$ and continuously differentiable in $x$ for all $x$ in some open set $E \subset {\mathbb R}^n$ containing $x_0$, then there exists $a > 0$ such that the initial value problem
<center><amsmath>
<center><amsmath>
Line 33: Line 38:
</amsmath></center>
</amsmath></center>
has a unique solution $x(t)$ on the interval $[-a, a]$.
has a unique solution $x(t)$ on the interval $[-a, a]$.
* Note: this problem is very similar to the case we worked out in class (and that you can find in Perko), but now $f(x,t)$ depends on the time $t$.  If you get stuck, there is a hint in Perko (not transcribed here).


<!-- Removted for 2014
<!-- Comments from 2014 TAs:
a lot of people asked about problem 3,
they didn't realize how qualitative of an answer they could give, they
were told they could draw what was going on at the point given, and/or
describe it in words.
-->
<li> '''Perko, Section 2.3, problem 1''':  
<li> '''Perko, Section 2.3, problem 1''':  
Use the fundamental theorem for linear systems in Chapter 1 of Perko to solve the initial value problem
Use the fundamental theorem for linear systems in Chapter 1 of Perko to solve the initial value problem
Line 48: Line 59:
   \dot \Phi = A \Phi, \qquad \Phi(0) = I.
   \dot \Phi = A \Phi, \qquad \Phi(0) = I.
</amsmath></center>
</amsmath></center>
* Note: this problem works through the more general result for nonlinear systems (Corallary on page 83) for the special case of a linear system.
</li>  
</li>  
<!-- 2014 TA comments:
* A lot of students also had questions about problem 4 I believe it was, kind of wondering if the solution was really as easy as it looked, and the students were told that yes, it was that easy.
* On problem 4, many students got the prinicipal axes wrong, and they were also frequently confused about the ... [rest of text missing]
-->
-->
<li> '''Perko, Section 2.5, problem 4''':  
<li> '''Perko, Section 2.5, problem 4''':  
Sketch the flow of the linear system  
Sketch the flow of the linear system  
Line 67: Line 83:
</li>
</li>


<!-- 2014 TA comments:
* Students didn't seem to understand what they were supposed to do.  They were showed an example of a 1-d system with a pitchfork bifurcation, and also a bifurcation plot of a 2-d system with a pitchfork bifurcation to help get intuition about what they were actually looking for, so they could apply the concepts to the 3-d system.  I didn't see that they had to classify the equilibrium points because that was in the Note at the end of the problem, and I didn't realize that Notes contained further instruction, so I told a few students they only needed to find mu where the single equilibrium split into several equilibriums.  So we decided to grade lightly for this problem.  The students were also showed how to do the stability analysis for the example of 1-d system (and qualitatively for the 2-d system with the phase portrait), in hopes it would give them more intuition about why this problem was important,  (not realizing that the Note at the end of the problem asked the students to actually classify the equilibrium points).
* On problem 6, students not performing stability analysis on the equilibrium points for mu > 1 as required.
-->
<li> '''Perko, Section 2.6, problem 2''':
Classify the equilibrium points of the Lorenz equation
<center><amsmath>
  \frac{d}{dt} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} =
  \begin{bmatrix} x_2 - x_1 \\ \mu x_1 - x_2 - x_1 x_3 \\ x_1 x_2 - x_3 \end{bmatrix}
</amsmath></center>
for $\mu > 0$.  At what value of the parameter $\mu$ do two new equilibrium points "bifurcate" from the equilibrium point at the origin?
* Note: the number and/or stability type of equilibrium points will change depending on the value of $\mu$.  Make sure to classify the equilibrium points for different ranges of $\mu$ and not just one value of $\mu$.  If you get stuck, there are some hints in problem 1(e) of Perko.
</li>
<!-- Dropped for 2014
<li> Choose ''one'' of the following systems and determine all of the equilibrium points for
<li> Choose ''one'' of the following systems and determine all of the equilibrium points for
the system, indicating whether each is a sync, source, or saddle. <br>
the system, indicating whether each is a sync, source, or saddle. <br>
Line 130: Line 162:
</li>
</li>
</ol>
</ol>
-->


<hr>
<hr>
Notes:  
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.
* 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.

Latest revision as of 23:40, 19 January 2015

R. Murray Issued: 21 Jan 2014 (Tue)
ACM 101b/AM 125b/CDS 140a, Winter 2014

(PDF)

Due: 29 Jan 2014 (Wed) @ noon
Turn in to box outside Steele House

__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).

  1. Perko, Section 2.2, problem 5 (only if you have had AM 125a or CDS 201): Let $V$ be a normed linear space. If $T:V \to V$ satisfies
    <amsmath>

    \|T(u) - T(v) \| \leq c\|u - v \|

    </amsmath>

    for all $u, v \in V$ with $0 < c < 1$ then $T$ is called a contraction mapping. It can be shown that contraction mappings give rise to unique solutions of the equation $T(u) = v$:

    Theorem (Contraction Mapping Principle) Let $V$ be a complete normed linear space and $T:V \to V$ a contraction mapping. Then there exists a unique $u \in V$ such that $T(u) = v$.

    Let $f \in C^1(E)$ and $x_0 \in E$. For $I = [-a, a]$ and $u \in C(I)$, let

    <amsmath>
     T(u)(t) = x_0 + \int_0^t f(u(s)) ds.
    
    </amsmath>

    Define a closed subset $V$ of $C(I)$ and apply the Contraction Mapping Principle to show that the integration equation (7) in Perko, Section 2.2 has a unique solution $u(t)$ for all $t \in [-a, a]$ provided the constant $a > 0$ is sufficiently small.

  2. Perko, Section 2.2, problem 3 (if you have not had ACM 125a or CDS 201): Use the method of successive approximations to show that if $f(x,t)$ is continuous in $t$ in some interval containing $t = 0$ and continuously differentiable in $x$ for all $x$ in some open set $E \subset {\mathbb R}^n$ containing $x_0$, then there exists $a > 0$ such that the initial value problem
    <amsmath>
     \dot x = f(x, t), \qquad x(0) = x_0
    
    </amsmath>

    has a unique solution $x(t)$ on the interval $[-a, a]$.

    • Note: this problem is very similar to the case we worked out in class (and that you can find in Perko), but now $f(x,t)$ depends on the time $t$. If you get stuck, there is a hint in Perko (not transcribed here).
  3. Perko, Section 2.3, problem 1: Use the fundamental theorem for linear systems in Chapter 1 of Perko to solve the initial value problem
    <amsmath>
     \dot x = A x, \qquad x(0) = y.
    
    </amsmath>

    Let $u(t, y)$ denote the solution and compute

    <amsmath>
     \Phi(t) = \frac{\partial u}{\partial y}(t, y).
    
    </amsmath>

    Show that $\Phi(t)$ is the fundamental matrix solution of

    <amsmath>
     \dot \Phi = A \Phi, \qquad \Phi(0) = I.
    
    </amsmath>
    • Note: this problem works through the more general result for nonlinear systems (Corallary on page 83) for the special case of a linear system.
  4. Perko, Section 2.5, problem 4: Sketch the flow of the linear system
    <amsmath>
     \dot x = A x \quad\text{with}\quad A = \begin{bmatrix} -1 & -3 \\ 0 & 2 \end{bmatrix}
    
    </amsmath>

    and describe $\phi_t(N_\epsilon(x_0))$ for $x_0 = (-3, 0)$, $\epsilon = 0.2$.

  5. Perko, Section 2.5, problem 5: Determine the flow $\phi_t:{\mathbb R}^2 \to {\mathbb R}^2$ for the nonlinear system
    <amsmath>
     \dot x = f(x) \quad\text{with}\quad f(x) = \begin{bmatrix} -x_1 \\ 2 x_2 + x_1^2 \end{bmatrix}
    
    </amsmath>

    and show that the set $S = \{x \in {\mathbb R}^2| x_2 = -x_1^2/4\}$ is invariant with respect to the flow $\phi_t$.

  6. Perko, Section 2.6, problem 2: Classify the equilibrium points of the Lorenz equation
    <amsmath>
     \frac{d}{dt} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = 
     \begin{bmatrix} x_2 - x_1 \\ \mu x_1 - x_2 - x_1 x_3 \\ x_1 x_2 - x_3 \end{bmatrix}
    
    </amsmath>

    for $\mu > 0$. At what value of the parameter $\mu$ do two new equilibrium points "bifurcate" from the equilibrium point at the origin?

    • Note: the number and/or stability type of equilibrium points will change depending on the value of $\mu$. Make sure to classify the equilibrium points for different ranges of $\mu$ and not just one value of $\mu$. If you get stuck, there are some hints in problem 1(e) of Perko.

  7. 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.