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Preserving synchronization under matrix product modifications

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Abstract

In this article we present a methodology under which stability and synchronization of a dynamical master/slave system configuration are preserved under modification through matrix multiplication. The objective is to show that under a defined multiplicative group, hyperbolic critical points are preserved along the stable and unstable manifolds. The properties of this multiplicative group were determined through the use of simultaneous Jordan decomposition. It is also shown that a consequence of this approach is the preservation of the signature of the Jacobian matrix associated with the dynamical system. To illustrate the results we present several examples of different modified systems.

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... These transformations may be interpreted as perturbations on the values of the physical parameters that describe the system. The problem of stability and synchronization preservation has been recently addressed for the case of hyperbolic, nonlinear systems with chaotic dynamics in [7] [8] [9]. However, the results reported in these articles deal with strictly linear modifications, i.e. ...
In this paper our aim is to show the viability of preserving the hyperbolicity of a master/salve pair of chaotic systems under different types of nonlinear modifications to its Jacobian matrix. Furthermore, we shall provide evidence to show that linear control methods used to achieve synchronization between master and slave systems are preserved under such transformations. We propose to modify both the coefficients of the Jacobian matrix’s associated characteristic polynomial through power evaluation as well as through matrix polynomial evaluation. To illustrate the results we present examples of several well known chaotic and hyperchaotic dynamical systems that have been modified using both methodologies.
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