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1992 The Genesio-Tesi system

Christophe LETELLIER
07/03/2015

While investigating the algebraic conditions for getting chaos in dynamical systems, Roberto Genesio and Alberto Tesi proposed a new three-dimensional system producing chaotic solutions [1]. The system results from the jerk equation

\stackrel{...}{x} + a \ddot{x} + b \dot{x} + x (1+x) = 0

which can be rewritten as the three-dimensional system

  \left\{
    \begin{array}{l}
      \dot{x} = y \\
      \dot{y} = z \\
      \dot{z} = -az -by -x(1+x) 
    \end{array}
  \right.

where a and b are the bifurcation parameters. This system has two singular points : point F0 is located at the origin of the state space and point F1 is located at (-1,0,0). Singular point F0 is saddle-focus with two complex conjugated eigenvalues with a positive real parts and point F1 is also saddle-focus but with complex eigenvalues with negative real parts. Such a configuration with two fixed points is very similar to the configuration of the Rössler system [2]. A chaotic attractor is obtained, for instance, when a=0.446 and b=1.1 as shown in Fig. 1.

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Fig. 1. Chaotic attractor solution to the Genesio-Tesi system and a first-return map to a Poincaré section.

Forgetting the layered structure of its map, the chaotic attractor is structured by a smooth unimodal map as always observed after a period-doubling cascade. Such a cascade is evidenced by the bifurcation diagram shown in Fig. 2 (for b=1.1). The chaotic attractor is topologically equivalent to the spiral Rössler attractor as shown in [3].

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Fig. 2. Bifurcation diagram of the Genesio-Tesi system.

[1] R. Genesio & A. Tesi, Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems, Automatica, 28 (3), 531-548, 1992.

[2] O. E. Rössler, An equation for continuous chaos, Physics Letters A, 57 (5), 397-398, 1976.

[3] E. Mendes & C. Letellier, Displacement in the parameter space versus spurious solution of discretization with large time step, Journal of Physics A, 37, 1203-1218, 2004.

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