CDS 140a Winter 2014 Homework 5: Difference between revisions
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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). | ||
<!-- 2014 TA comments: As in last year, there were people who wanted to have their work checked with the center manifold calculations, the coefficients for their | |||
approximation, (problem 4), this year I happened to have my worked out solutions and a students solutions from last year to compare this student's calculations with, and the student's calculations compared with one of them, but basically I did tell the students their forms might look different depending on how they transformed the system, and how they simplified their algebra, etc. --> | |||
<ol> | <ol> | ||
<!-- 2014 TA comments: there were a few questions on how to actually show that the circles lie in planes. The students were told they could tell by the phase portrait, and they were also shown an algebraic way to show the circles lie in places. --> | |||
<li>'''Perko, Section 2.9, problem 3''' | <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 | 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 | ||
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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. | ||
</li> | </li> | ||
<!-- 2014 TA comments: there was some unclearness as to whether this Lyapunov function could show whether or not this system was asymptotically stable. For sure it could be shown to be stable. Picking a=... and b=... (or the opposite a=..., b=...), one could obtain Vdot = ... The problem was that if y=..., and x=..., Vdot=... still, even though the system was not necessarily at an equilibrium point. We did not have a good course solution for this problem from last year (we had half of an unlabeled solution that looking back does not correct). --> | |||
<li> | <li> | ||
Determine the stability of the system | Determine the stability of the system | ||
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<!-- Requires use of Lasalle; not covered in 2014 --> | <!-- Requires use of Lasalle; not covered in 2014 --> | ||
<!-- 2014 TA comments: Some of the students mentioned they couldn't use Gronwell's inequality because of a negative constant instead of positive. They were told they could do the problem without Gronwell's inequality, but if they could use Gronwell's inequality that would be good too. --> | |||
<li> | <li> | ||
Definition: An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\|x(t)\|\leq Me^{-\alpha t}\|x(0)\|$, $\forall\|x(0)\|\leq\epsilon,\,t\geq 0$. | Definition: An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\|x(t)\|\leq Me^{-\alpha t}\|x(0)\|$, $\forall\|x(0)\|\leq\epsilon,\,t\geq 0$. | ||
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</li> | </li> | ||
<!-- 2014 TA comments: some people tried to use the transform a1=..., and a2=..., instead of a1=.. and a2=.... The students were told them they are free to try the problem with that transform, but they were also told them that for sure they could come up with the answer we are looking for if they used the second transform cited here. It seemed people who used the first transform cited here became stuck at a certain point, I didn't know if it was because they didn't know how to do the problem, or if it was due to the form of the tranform they used (The first form cited here). I said go ahead and use any transform you would like, but try doing the problem and going through the process with the second transform cited here, and they were given more specific instructions on how to complete the problem using that transform (which they could then go and try to apply to the system tranformed with the first transform). In summary, it might save some students time if they were given the second transform to work with from the start, with me not knowing now if their other transforms are going to work to give the solutions that are expected. Some had questions about whether they could say that a stable manifold and unstable manifold didn't exist. Since there were not any nonzero eigenvalues for this system, they were told everything was on the center manifold, which didn't rule out their being stable and unstable parts of the center manifold, it was just that the usual theorems with the nonzero eigenvalues didn't apply here, so they needed to use a center manifold approximation, which in this case was to be a linear approximation. --> | |||
<li>'''Perko, Section 2.12, problem 2''' | <li>'''Perko, Section 2.12, problem 2''' | ||
Use Theorem 1 [Center Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system | Use Theorem 1 [Center Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system | ||
Latest revision as of 02:25, 25 January 2015
| R. Murray | Issued: 4 Feb 2014 (Tue) |
| ACM 101b/AM 125b/CDS 140a, Winter 2014 | Due: 12 Feb 2014 (Wed) @ noon |
__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_3^3-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.
-
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?
-
Definition: An equilibrium point is exponentially stable if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\|x(t)\|\leq Me^{-\alpha t}\|x(0)\|$, $\forall\|x(0)\|\leq\epsilon,\,t\geq 0$.
Let $\dot x = f(x)$ be a dynamical system with an equilibrium point at $x_e = 0$. Show that 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 $k_3$, then the equilibrium point at the origin is exponentially stable.
- Hint: you can use Gronwall's inequality from Section 2.3 of Perko.
- Perko, Section 2.12, problem 2
Use Theorem 1 [Center 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.
