# 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. 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 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 DV(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.