Generalized Analytical Solutions and Synchronization Dynamics of Coupled Simple Nonlinear Electronic Circuits

TL;DR

This paper derives generalized analytical solutions for coupled nonlinear circuits and explores their synchronization dynamics using bifurcation analysis, Lyapunov exponents, and stability functions, providing new insights into chaotic system synchronization.

Contribution

It presents the first explicit analytical solutions for coupled nonlinear circuits and analyzes their synchronization behavior through bifurcation and stability methods.

Findings

Synchronization occurs via eigenvalue bifurcation in piecewise linear regions.

Stable synchronized states confirmed by master stability function.

Conditional Lyapunov exponents and Kaplan-Yorke dimension validate synchronization.

Abstract

In this paper we present a generalized analytical solution to the generalized state equations of coupled second-order non-autonomous circuit systems. The analytical solutions thus obtained are used to study the synchronization dynamics of two different types of circuit systems, differing only by their constituting nonlinear element. The synchronization dynamics of the coupled systems are studied through two-parameter bifurcation diagrams, phase portraits and time-series plots obtained from the explicit analytical solutions. The mechanism of synchronization is realized through the bifurcation of the eigenvalues as functions of the control parameter in each of the coupled piecewise linear regions of the drive and response systems. The stability of the synchronized state for coupled identical chaotic states are studeid through the {\emph{master stability function}}. Further, {\emph{conditional Lyapunov exponents}} and {\emph{Kaplan-Yorke dimension}} are obtained to confirm the synchronized states of both coupled identical and non-identical chaotic states. The synchronization dynamics of coupled chaotic systems studied through two-parameter bifurcation diagrams obtained from explicit analytical solutions is reported in the literature for the first time.