telech

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).

JPG - 22.4 ko
Fig. 1 : Hyperchaotic attractor solution to the modified hyperchaotic Rössler system.
Zip - 418.8 ko
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.

Documents

Data set produced by the modified 4D Rössler system
Zip · 418.8 ko
2097 - 20/07/18

ATOMOSYD © 2007-2018 |  Suivre la vie du site  |  SPIP  |  scoty  |  MàJ . 25/05/2018