Hyperbolification of Dynamical Systems:
The Case of Continuous-Time Systems

Z. Elhadja and J. C.Sprottb
aDepartment of Mathematics, University of Tébessa 12002, Algeria
bDepartment of Physics, University of Wisconsin, Madison WI 53706, USA
e-mail: zeraoulia@mail.univ-tebessa.dz, zelhadj12@yahoo.fr, sprott@physics.wisc.edu

Received December 9, 2011


We present a new method to generate chaotic hyperbolic systems. The method is based on the knowledge of a chaotic hyperbolic system and the use of a synchronization technique. This procedure is called hyperbolification of dynamical systems. The aim of this process is to create or enhance the hyperbolicity of a dynamical system. In other words, hyperbolification of dynamical systems produces chaotic hyperbolic (structurally stable) behaviors in a system that would not otherwise be hyperbolic. The method of hyperbolification can be outlined as follows. We consider a known ndimensional hyperbolic chaotic system as a drive system and another ndimensional system as the response system plus a feedback control function to be determined in accordance with a specific synchronization criterion. We then consider the error system and
apply a synchronization method, and find sufficient conditions for the errors to converge to zero and hence the synchronization between the two systems to be established. This means that we construct a 2n-dimensional continuoustime system that displays a robust hyperbolic chaotic attractor. An illustrative example is given to show the effectiveness of the proposed hyperbolification method.

Ref: E. Zeraoulia and  J. C. Sprott, Journal of Experimental and Theoretical Physics 115, 356-360 (2012)

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