ScommaMaruj

Answered

2022-07-04

Comparison function for the Rossler system

$\begin{array}{rl}& {\dot{x}}_{1}=-{x}_{2}-{x}_{3}\\ & {\dot{x}}_{2}={x}_{1}+\alpha {x}_{2}\\ & {\dot{x}}_{3}=\beta +{x}_{3}({x}_{1}-\gamma )\end{array}$

With $\alpha =\beta =0.1$ and $\gamma =14$, and ${x}_{3}(0)>0$, such that ${x}_{3}>0$ $\mathrm{\forall}$ $t\ge 0$.

we have found that for the comparison function $W={x}_{1}^{2}+{x}_{2}^{2}+2{x}_{3}$ the following inequality holds:

$\dot{W}\le 2\alpha W+2\beta $

This implies that the solutions of this system are well defined on the infinite time interval $[0,\mathrm{\infty})$, and implies that solutions can not escape to infinity in finite time.

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Answer & Explanation

pampatsha

Expert

2022-07-05Added 15 answers

One can easily confirm that ${x}_{3}>0\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}w(t)=W(x(t))>0$ and thus

$\alpha w(t)+\beta \le (\alpha w(0)+\beta ){e}^{2\alpha t}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}w(t)\le \overline{w}(t)=w(0){e}^{2\alpha t}+\frac{\beta}{\alpha}({e}^{2\alpha t}-1).$

This means that over any finite time interval $[0,T]$ the solution is bounded by

$W(x(t))\le \overline{w}(T).$

So if one considers the region $W(x)<2\overline{w}(T)$, it is bounded, and has $x(T)$ as inner point. Thus the ODE function has bound and a Lipschitz constant there, thus the IVP with the IC at $x(T)$ can be solved locally in forwar direction and the solution $x$ thus continued.

Note that the movement forward in time is essential for the claim that ${x}_{3}$ stays positive, backwards in time ${x}_{3}$ can become negative and thus unbounded by $\overline{w}$.

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