CDS 140a Winter 2014 Homework 4: Difference between revisions

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'''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
this homework set (including reading).
this homework set (including reading).
<!-- 2014 TA comments based on office hours:
* There were a ton of questions about problem 3 (manifold problem). A lot of people also were confused about solving for the unstable manifold by simply letting xdot = - f(x) on #1. There were surprisingly not that many questions about #5 (Hartman-Grobman successive approximations).
* Also, I told a couple of students that since their successive approximations converged exactly in problem 1, that they could simply state in problem 2 that their approximations to the stable and unstable manifold were just in fact exact.
* However, if you wanted them to do #2 by actually solving the system of differential equations analytically and then figuring out the stable and unstable manifolds by inspection, then maybe you could email the class and make that clear.
---
* The year I took the class we solved problem 2, (or 2.7.3), by solving the system of diff eqns, and then we looked at the limit at t goes to infinite, and then solved for the integration constants/initial conditions in terms each other, which then gave the stable/unstable manifolds (see the beginning of chapter 2.7 for an example in the book that does this).
* It seemed people weren't clear that the unstable manifold can be found by letting t go to -t and then solving for the stable manifold, or as Anandh said here, solving for -xdot = f(x), or xdot=-f(x) for an autonomous system.
* Most people didn't have any idea on how to solve for problem 3, or why finding the normal vector to the tangent of the stable manifold, or to the vector field, is important for this problem.  Perhaps for future years the hint for R^2, about the normal vector being orthogonal to the vector field could be explained as meaning that anytime a trajectory starts off on the stable manifold, it stays on the stable manifold since there is no verticle component to "lift" the trajectory off of the stable manifold.
-->


<ol>
<ol>
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<!-- 2014 TA comments: A lot of people also were confused about solving for the unstable manifold by simply letting xdot = - f(x) on #1.  -->
<li>'''Perko, Section 2.7, problem 2'''
<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
Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for
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</li>
</li>


<!-- TA comment: I told a couple of students that since their successive approximations converged exactly in problem 1, that they could simply state in problem 2 that their approximations to the stable and unstable manifold were just in fact exact.  if you wanted them to do #2 by actually solving the system of differential equations analytically and then figuring out the stable and unstable manifolds by inspection, then maybe you could email the class and make that clear. -->
<li>'''Perko, Section 2.7, problem 3'''
<li>'''Perko, Section 2.7, problem 3'''
Show that $S$ and $U$ for the previous problem are given by  
Show that $S$ and $U$ for the previous problem are given by  
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<!-- Lots of people didn't see the part about using this result to compute the stable manifold for the problem above -->
<!-- Lots of people didn't see the part about using this result to compute the stable manifold for the problem above -->
<!-- Most people didn't have any idea on how to solve for problem 3, or why finding the normal vector to the tangent of the stable manifold, or to the vector field, is important for this problem.  Perhaps for future years the hint for R^2, about the normal vector being orthogonal to the vector field could be explained as meaning that anytime a trajectory starts off on the stable manifold, it stays on the stable manifold since there is no verticle component to "lift" the trajectory off of the stable manifold. -->
<li> Consider a dynamical system with $x = (u, v) \in {\Bbb R}^n$.  For the case $n = 2$, prove that if
<li> Consider a dynamical system with $x = (u, v) \in {\Bbb R}^n$.  For the case $n = 2$, prove that if
<center><amsmath>
<center><amsmath>

Latest revision as of 00:15, 20 January 2015

R. Murray Issued: 28 Jan 2014 (Tue)
ACM 101b/AM 125b/CDS 140a, Winter 2014

(PDF)

Due: 4 Feb 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 2.7, problem 2 Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for
    <amsmath>

    \aligned \dot{x}_1&=-x_1,\\ \dot{x}_2&=x_2+x_1^2 \endaligned

    </amsmath>

    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 $S$.

  2. Perko, Section 2.7, problem 3 Show that $S$ and $U$ for the previous problem are given by
    <amsmath>

    S:\,x_2=-\frac{x_1^2}{3}

    </amsmath>
    <amsmath>

    U:\,x_1=0

    </amsmath>

    Sketch $S$, $U$, $E^s$ and $E^u$.

  3. Consider a dynamical system with $x = (u, v) \in {\Bbb R}^n$. For the case $n = 2$, prove that if
    <amsmath>

    \aligned \dot{u}&=f(u,v),\qquad u\in\mathbb{R}^k\\ \dot{v}&=g(u,v),\qquad v\in\mathbb{R}^{n-k} \endaligned

    </amsmath>

    then the manifold $S=\{(u,v)\in\mathbb R^k\times\mathbb R^{n-k}|v=h(u)\}$ is an invariant manifold of the system if

    <amsmath>

    g(u,h(u))=Dh(u)f(u,h(u))

    </amsmath>

    Use this result to compute the stable manifold for the system of problem 1 and 2 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 $h(x)$.

    • Note: The result holds for ${\Bbb R^n}$, but you only need to consider the case $n = 2$ and $k = 1$ (although you are free to prove the more general case if you prefer).
    • 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 $\mathbb R^2$.)
  4. Perko, Section 2.6, problem 3 Show that the continuous map $H:{\Bbb R}^3 \to {\Bbb R}^3$ defined by
    <amsmath>
    \begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix} = H(x) = \begin{bmatrix} x_1 \\ x_2 + x_1^2 \\ x_3 + x_1^2 / 3 \end{bmatrix}
    
    </amsmath>

    has a continuous inverse $H^{-1}:{\Bbb R}^3 \to {\Bbb R}^3$ and that the nonlinear system

    <amsmath>
     \frac{dx}{dt} = \begin{bmatrix} -x_1 \\ -x_2 + x_1^2 \\ x_3 + x_1^2 \end{bmatrix}
    
    </amsmath>

    is transformed into a linear system using this transformation, i.e., if $y = H(x)$, show that $\dot y = A x$ for an appropriate $A$. Use this transformation to compute and sketch the stable and unstable manifolds for the nonlinear system.

    • Note: since the nonlinear system can be transformed into a linear one, it follows that $S = H^{-1}(E^s)$ and $U = H^{-1}(E^u)$.
  5. Perko, Section 2.8, Problem 1 Solve the system
    <amsmath>

    \aligned \dot{y}_1&=-y_1\\ \dot{y}_2&=-y_2+z^2\\ \dot{z}&=z \endaligned

    </amsmath>

    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

    <amsmath>

    H (x) =\int_0^1 e^{-As} H_0 \bigl(\phi_s(x) \bigr) ds,

    </amsmath>

    where $A$ is the linearization of the nonlinear dynamics at the origin and $\phi_t(x)$ is the flow of the full system. Use the homemorphism $H$ to find the stable and unstable manifolds

    <amsmath>

    W^s(0)=H^{-1}(E^s)\quad {\mathrm{and}}\quad W^u(0)=H^{-1}(E^u)

    </amsmath>

    for this system.

    • Note: this problem involves some simple but somewhat tedious computations. If you know how to use Mathematica or a similar program, you may wish to carry out the computations for the successive approximations using that software. However, make sure to show the results at each step of the calculation.
    • Hint: You should find
    <amsmath>
     H(y_1,y_2,z) = \begin{bmatrix} y_1 \\ y_2-z^2/3 \\ z \end{bmatrix}, \qquad
     \aligned
       W^s(0) &=\{x\in{\mathbb{R}}|z=0\}, \\
       W^u(0) &=\{x\in{\mathbb{R}}|y_1=0,y_2=z^2/3\}.
     \endaligned
    
    </amsmath>

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.