The modified hyperchaotic Rössler system

Christophe LETELLIER

23/05/2009

The modified hyperchaotic Rössler system [1]

\left\{ \begin{array}{l} \dot{x}=-z-y \\[0.1cm] \dot{y}=x-0.75y+v \\[0.1cm] \dot{z}=b+xz \\[0.1cm] \dot{v}=x-0.8y-cz+1.05v \end{array} \right.

was proposed to ensure its
synchronization using a single variable. It corresponds to the original hyperchaotic
Rössler system [2] rewritten replacing (x,y,z,w) with (x,y,z,v=y+w).
This four dimensional system produces a hyperchaotic attractor (Fig. 1).

**Fig. 1 : Hyperchaotic attractor solution to the modified hyperchaotic Rössler system.**

Data set produced by the modified 4D Rössler system

This four dimensional system was numerically integrated to produce a data set corresponding to the attractor shown in Fig. 1. There are four columns associated with the time evolution of x, y, z, and v, respectively. Parameter values were b=3 and c=0.05. The sampling time was $\delta t=0.05$ s. Initial conditions were $$x_0=-10$$ ,
$y_0=-6$, $z_0=0$, and $v_0=10.1$.

The observability coefficients for this four dimensional system are
$\eta_x^3=0.88$,
$\eta_y^3 = 0.93$,
$\eta_z^3 = 0.84$,
$\eta_v^3 = 0.93$, that is,
the dynamical variables can be ranked as

$v = y \triangleright x \triangleright z$

according to the observability of the attractor they provide.

[1] A. Tamasevicius & A. Cenys,
Synchronizing hyperchaos with a single variable, Physical Review E, 55, 297, 1997.

[2] O. E. Rössler, An equation for hyperchaos,
Physics Letters A, 71, 155, 1979.

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Data set produced by the modified 4D (…)

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