ScommaMaruj

2022-07-04

Comparison function for the Rossler system
$\begin{array}{rl}& {\stackrel{˙}{x}}_{1}=-{x}_{2}-{x}_{3}\\ & {\stackrel{˙}{x}}_{2}={x}_{1}+\alpha {x}_{2}\\ & {\stackrel{˙}{x}}_{3}=\beta +{x}_{3}\left({x}_{1}-\gamma \right)\end{array}$
With $\alpha =\beta =0.1$ and $\gamma =14$, and ${x}_{3}\left(0\right)>0$, 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:
$\stackrel{˙}{W}\le 2\alpha W+2\beta$
This implies that the solutions of this system are well defined on the infinite time interval $\left[0,\mathrm{\infty }\right)$, and implies that solutions can not escape to infinity in finite time.

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pampatsha

Expert

One can easily confirm that ${x}_{3}>0\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}w\left(t\right)=W\left(x\left(t\right)\right)>0$ and thus
$\alpha w\left(t\right)+\beta \le \left(\alpha w\left(0\right)+\beta \right){e}^{2\alpha t}\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}w\left(t\right)\le \overline{w}\left(t\right)=w\left(0\right){e}^{2\alpha t}+\frac{\beta }{\alpha }\left({e}^{2\alpha t}-1\right).$
This means that over any finite time interval $\left[0,T\right]$ the solution is bounded by
$W\left(x\left(t\right)\right)\le \overline{w}\left(T\right).$
So if one considers the region $W\left(x\right)<2\overline{w}\left(T\right)$, it is bounded, and has $x\left(T\right)$ as inner point. Thus the ODE function has bound and a Lipschitz constant there, thus the IVP with the IC at $x\left(T\right)$ 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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