CDS 140a Winter 2013 Homework 5 - Revision history

Murray at 18:39, 19 February 2013

2013-02-19T18:39:55Z

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	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.		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>

			<!-- This problem was a bit tricky to grade. consider putting a different problem in next year? -->
	<li> Consider the following system		<li> Consider the following system
	<center><amsmath>\aligned		<center><amsmath>\aligned

2013-02-10T04:14:16Z

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	<li>		<li>
	An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\\|x(t)\\|\leq Me^{-\alpha t}\\|x(0)\\|$, $\forall\\|x(0)\\|\leq\epsilon,\,t\geq 0$.		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		Prove that an equilibrium point $x_0=0$ is exponentially stable if there is a function $V(x)$ satisfying
	<center><amsmath>k_1\\|x\\|^2\leq V(x)\leq k_2\\|x\\|^2,\quad DV(x)\leq -k_3\\|x\\|^2		<center><amsmath>k_1\\|x\\|^2\leq V(x)\leq k_2\\|x\\|^2,\quad \dot V(x)\leq -k_3\\|x\\|^2
	</amsmath></center>		</amsmath></center>
	for positive constants $k_1$, $k_2$ and $k_3$.		for positive constants $k_1$, $k_2$ and $k_3$.

2013-02-08T16:49:11Z

2013-02-05T20:16:36Z

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	~~<!--~~
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	~~</!--~~

	{{CDS homework		{{CDS homework
	\| instructor = R. Murray, D. MacMartin		\| instructor = R. Murray, D. MacMartin

2013-02-05T20:16:17Z

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			<!--
	{{warning\|This homework set is still being written. Do not start working on these problems until this banner is removed.}}		{{warning\|This homework set is still being written. Do not start working on these problems until this banner is removed.}}
			</!--

	{{CDS homework		{{CDS homework

2013-02-05T19:48:12Z

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 \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>
 </ol>

2013-02-04T23:58:24Z

2013-02-04T04:59:54Z

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	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?		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>
			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
			<center><amsmath>k_1\\|x\\|^2\leq V(x)\leq k_2\\|x\\|^2,\quad DV(x)\leq -k_3\\|x\\|^2
			</amsmath></center>
			for positive constants $k_1$, $k_2$ and $k_3$.
			</li>
	<li>'''Perko, Section 2.12, problem 2'''		<li>'''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		Use Theorem 1 [Centre Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system

2013-02-03T19:10:41Z

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

2013-02-03T04:43:24Z

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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).

			<ol>
			<li>'''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
			<center><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></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.
			</li>
			<li>'''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
			<center><amsmath>
			\aligned
			\dot{x}&=y\\
			\dot{y}&=-y+\alpha x^2+xy
			\endaligned
			</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.
			</li>
			</ol>