A Complete Analytical Study on the Dynamics of Simple Chaotic Systems

TL;DR

This paper provides a comprehensive analytical investigation of bifurcations and chaos in certain second-order dissipative systems, confirming numerical observations with explicit solutions and experimental data.

Contribution

It presents the first explicit analytical solutions for bifurcations and chaos in these systems, including basins of attraction and phase portraits.

Findings

Analytical solutions confirm chaotic phenomena like period-doubling and Feigenbaum remerging.

Bifurcation diagrams derived analytically match numerical results.

Experimental attractors support the analytical findings.

Abstract

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, phase-portraits 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.