In this paper, we present a novel explicit analytical solution for the normalized state equations of mutually-coupled simple chaotic systems. A generalized analytical solution is obtained for a class of simple nonlinear electronic circuits with two different nonlinear elements. The synchronization dynamics of the circuit systems were studied using the analytical solutions. the analytical results thus obtained have been validated through numerical simulation results. Further, we provide a sufficient condition for synchronization in mutually-coupled, second-order simple chaotic systems through an analysis on the eigenvalues of the difference system. The bifurcation of the eigenvalues of the difference system as functions of the coupling parameter in each of the piecewise-linear regions, revealing the existence of stable synchronized states is presented. The stability of synchronized states are studied using the Master Stability Function. Finally, the electronic circuit experimental results confirming the phenomenon of complete synchronization in each of the circuit system is presented.
We report in this paper a complete analytical study on the bifurcations and chaotic phenomena observed in certain second-order, non-autonomous, dissipative chaotic systems. One-parameter bifurcation diagrams obtained from the analytical solutions proving the numerically observed chaotic phenomena such as antimonotonicity, period-doubling sequences, Feignbaum remerging have been presented. Further, the analytical solutions are used to obtain the basins of attraction, phaseportraits and Poincare maps for different chaotic systems. Experimentally observed chaotic attractors in some of the systems are presented to confirm the analytical results. The bifurcations and chaotic phenomena studied through explicit analytical solutions is reported in the literature for the first time.
In this paper, we report the enhanced stability of induced synchronization by transient uncoupling observed in certain unidirectionally coupled second-order chaotic systems. The stability of synchronization observed in the coupled systems subjected to transient uncoupling is analyzed using the Master Stability Function. The existence of the coupled systems in stable synchronized states over a certain range of the clipping fraction of the driven system is identified. The enhanced stable synchronized states are obtained for fixed values of clipping fraction in certain second-order chaotic systems. The two-parameter bifurcation diagram indicating the parameter regions over which stable synchronization occurs is presented. The negative eigenvalue regions of the driven system enabling induced synchronization is studied for all the systems. The enhancement of synchronization through transient uncoupling observed in coupled second-order non-autonomous chaotic systems is reported in the literature for the first time.
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