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__

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  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. 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.