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Consider the following system of differential equations: $$\dot {x}=y-x+x^3,\qquad \dot{y}=-x.$$ By linearization, it's easy to see that $(0,0)$ is a (nonlinear) sink.

Show that there exists an open connected set $D$ such that if $\phi^t:\mathbb{R}^2\rightarrow \mathbb{R}^2$ is the solution flux, then $\lim\limits_{t\rightarrow +\infty} \phi^t(x_0,y_0) = (0,0)$ iff $(x_0,y_0) \in D$ and such that $\partial D$ is compact and invariant, that is, such that $\phi^t(x_0,y_0) \in \partial D$ for all $t \in \mathbb{R}$ and $(x_0,y_0) \in \partial D$.

Maybe this can be done with a substitution (to show that this system behaves like for instance $\dot {r}=r(r-1)$, $\dot{\theta}=1$) or by finding a function $f \in C^1(\mathbb{R}^2,\mathbb{R})$ such that $(0,0)$ is a minimum for $f$, $\nabla f(z) \cdot \dot{z}<0$ for every $z=(x,y)$ in $D$, and $\nabla f(z) \cdot \dot{z}=0$ for every $z=(x,y)$ in $\partial D$.

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  • $\begingroup$ Can you find a Lyapunov function for the linearized system? Can you prove by hands that the same function works for some neighbourhood of origin in nonlinear system? $\endgroup$
    – Evgeny
    Jan 30 '15 at 6:59
  • $\begingroup$ $x^2+y^2$ is a Lyapunov function for $(0,0)$ in a neighbourhood $D'$ of the origin, but here we need a special Lyapunov function (such that $\nabla f(x) \cdot \dot{x}=0$ in $\partial D'$). $\endgroup$
    – d. zeffiro
    Jan 30 '15 at 11:42
  • $\begingroup$ Sorry then, I've paid less attention than I should. Roughly speaking, we must say about existence of limit cycle. Interesting. $\endgroup$
    – Evgeny
    Jan 30 '15 at 21:57
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The boundary region $\partial D$ is quite visible, and yet it seems that finding a mathematical proof is not so direct...

enter image description here

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  • $\begingroup$ What software did you use to get the phase portrait? $\endgroup$
    – d. zeffiro
    Feb 5 '15 at 22:58
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    $\begingroup$ wolframalpha.com $\endgroup$
    – Did
    Feb 5 '15 at 23:21
  • $\begingroup$ Thanks! I also tried it, but for some reason it ran out of time $\endgroup$
    – d. zeffiro
    Feb 6 '15 at 10:29
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    $\begingroup$ The line streamplot y−x+x^3,-x produces an answer in a few seconds. $\endgroup$
    – Did
    Feb 6 '15 at 11:28
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If you make the change of variables $\tau=-t, X(\tau)=x(t), Y(\tau)=-y(t)$, your system becomes $$ \dot X=Y+X-X^3,\\ \dot Y=-X, $$ where the dot now denotes the derivative with respect to $\tau$. Now this is a classical examples of the so-called Lienard's system, for which it can be readily proved (see, e.g., the textbook by Perko on dynamical systems) that there exists a unique asymptotically stable limit cycle. Reversing the time back you get that in your original system there is a unique unstable limit cycle, and hence there exists $D$ with the required properties.

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