CDS 140a Winter 2013 Homework 5

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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_3^3-x_2^3\\ -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. 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
    <amsmath>k_1\|x\|^2\leq V(x)\leq k_2\|x\|^2,\quad \dot V(x)\leq -k_3\|x\|^2 </amsmath>

    for positive constants $k_1$, $k_2$ and $k_3$.

  4. 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.

  5. 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.

  6. Consider the following system
    <amsmath>\aligned

    \dot x&=x^2\\ \dot y&=-y

    \endaligned</amsmath>

    with stable manifold $W^s=\{(x,y):x=0\}$

    1. Show that the Taylor series approximation to the centre manifold gives $W^c=\{(x,y):y=0\}$.
    2. Show that the set $\{(x,y):y=h(x)\}$ with
      <amsmath>

      h(x)=\left\{\begin{array}{lr}ke^{1/x}&x<0\\0&x\geq 0\end{array}\right.

      </amsmath>

      describes a one-parameter set of invariant manifolds for any $k$. (Hint: what is $dy/dx$?)

    3. What are the dynamics describing the trajectories along any of these manifolds?
    4. (Optional): Plot the phase portrait for this system.