Difference between revisions of "CDS 140a Winter 2013 Homework 5"

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</amsmath></center>
 
</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.
 
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>
 
<li>'''Perko, Section 2.12, problem 2'''
 
<li>'''Perko, Section 2.12, problem 2'''
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</amsmath></center>
 
</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.
 
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>
 
</ol>
 
</ol>

Revision as of 19:10, 3 February 2013

WARNING: This homework set is still being written. Do not start working on these problems until this banner is removed.


R. Murray, D. MacMartin Issued: 5 Feb 2013 (Tue)
ACM 101/AM 125b/CDS 140a, Winter 2013 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).

  1. 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
    <amsmath>

    \dot{x}=\begin{bmatrix}-x_2-x_1x_2^2+x_3^2-x_1^3\\ x_1+x_3^2-x_2^2\\ -x_1x_3-x_3x_1^2-x_2x_3^2-x_3^5\end{bmatrix}

    </amsmath>

    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.

  2. Determine the stability of the system
    <amsmath>

    \aligned \dot{x}&=-y-x^3\\ \dot{y}&=x^5 \endaligned

    </amsmath>

    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?

  3. 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
    <amsmath>

    \aligned \dot{x}&=y\\ \dot{y}&=-y+\alpha x^2+xy \endaligned

    </amsmath>

    for $\alpha\neq 0$ and for $\alpha=0$; i.e., follow the procedure in Example 1 after diagonalizing the system as in Example 3.

  4. Consider the following system in $\mathbb R^2$:
    <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>

    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.