CDS 140a Winter 2013 Homework 5: Difference between revisions
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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. | ||
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<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? | |||
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<li>'''Perko, Section 2.12, problem 2''' | <li>'''Perko, Section 2.12, problem 2''' | ||
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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. | ||
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<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. | |||
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Revision as of 19:10, 3 February 2013
| 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).
- 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.
-
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?
- 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.
- 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.
