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
Abstract
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)