FoCM 2014 conference

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Workshop A1 - Computational Dynamics

December 11, 15:30 ~ 15:55 - Room B21

Coexistence of chaos and hyperchaos

Daniel Wilczak

Jagiellonian University, Poland   -   wilczak@ii.uj.edu.pl

Chaotic attractors can have a very nonuniform internal structure. Even on the plane the well known Newhouse phenomenon guarantees that small sinks of very high periods may be embedded in large chaotic zones. For higher dimensional systems one can expect coexistence of chaotic and hyperchaotic dynamics, i.e. topological horseshoes with more than one positive Lyapunov exponents.

Consider the classical 4D Rossler system

$$\begin{equation*} \dot x = -y-w,\quad \dot y = x+ay+z,\quad \dot z = dz+cw,\quad \dot w = xw+b \end{equation*}$$ with the parameter values $a=0.27857$, $b=3$, $c=-0.3$, $d=0.05$. Let $$\Pi=\{(x,0,z,w)\in\mathbb R^3, \dot y = x+z<0\}$$ be the Poincare section and let $P:\Pi\to\Pi$ be the associated Poincare map.

We prove that

1) there is an explicitly given trapping region $B\subset\Pi$ for $P$, i.e. $P(B)\subset B$,

2) the maximal invariant set $A = \mathrm{inv}(P,B)$ contains three invariant sets, say $H_1$, $H_2$, $H_3$, on which the dynamics is $\Sigma_2$ chaotic, i.e. it is semiconjugated to the Bernoulli shift on two symbols,

3) $H_1$ is a chaotic set with two positive Lyapunov exponents,

4) $H_2$ and $H_3$ are chaotic sets with one positive Lyapunov exponent,

5) there is a countable infinity of heteroclinic connections linking $H_1$ with $H_2$, $H_2$ with $H_3$ and $H_1$ with $H_3$,

6) there is countable infinity of periodic orbits and heteroclinic/homoclinic orbits inside each horseshoe.

Joint work with Roberto Barrio (Universidad de Zaragoza, Spain) and Sergio Serrano (Universidad de Zaragoza, Spain).

View abstract PDF

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