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we are studying the chaotic Rössler system: $$ \begin{align} & \dot{x}_1 = - x_2 - x_3\\ & \dot{x}_2 = x_1 +\alpha x_2\\ & \dot{x}_3 = \beta + x_3(x_1-\gamma) \end{align} $$ With $\alpha = \beta = 0.1$ and $\gamma = 14$, and $x_3(0)>0$, such that $x_3>0$ $\forall$ $t\geq0$.

we have found that for the comparison function $W=x_1^2+x_2^2+2x_3$ the following inequality holds: $$ \dot{W} \leq 2\alpha W + 2\beta $$

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

However, we are not able to derive this last conclusion, if anyone could point us in the right direction, it would greatly be appreciated. Thank you!

Bernard
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1 Answers1

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One can easily confirm that $x_3>0\implies w(t)=W(x(t))>0$ and thus $$ αw(t)+β\le (αw(0)+β)e^{2αt} \implies w(t)\le \bar w(t)=w(0)e^{2αt}+\frac{β}{α}(e^{2αt}-1). $$ This means that over any finite time interval $[0,T]$ the solution is bounded by $$W(x(t))\le \bar w(T).$$ So if one considers the region $W(x)<2\bar w(T)$, 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 $\bar w$.

Lutz Lehmann
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