Preservation of Synchronization Using a Tracy‐Singh Product in the Transformation on Their Linear Matrix (original) (raw)
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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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We examine synchronization of identical chaotic systems coupled in a drive/response manner. A rigorous criterion is presented which, if satisfied, guarantees that synchronization to the driving trajectory is linearly stable to perturbations. An easy to use approximate criterion for estimating linear stability is also presented. One major advantage of these criteria is that, for simple systems, many of the calculations needed to implement them can be performed analytically. Geometrical interpretations of the criterion are discussed, as well as how they may be used to investigate synchronization between mutual coupled systems and the stability of invariant manifolds within a dynamical system. Finally, the relationship between our criterion and results from control theory are discussed. Analytical and numerical results from tests of these criteria on four different dynamical systems are presented.
Recent Advances in Nonlinear Dynamics and Synchronization
Studies in Computational Intelligence, 2009
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
2016
Three important phenomena of chaos synchronization are considered in this paper, In detailed, complete synchronization, anti- synchronization and hybrid synchronization based on the nonlinear active control approach between two different (non-identical) 4D hyperchaotic systems, i. e. Modified Pan and Liu are study herein. The Modified hyperchaotic Pan system is taken as drive system and hyperchaotic Liu system as response. Stabilization of error dynamics for each phenomenon is realized by satisfying two analytical approaches; Lyapunov's second method and linear system theory. Controllers are designed by using the relevant variable of drive and response systems. Theoretical analysis and numerical simulations are shown to verify the results.
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