Energy computation and multistability in a class of second-order chaotic systems with simple nonlinearities: numerical, experimental and analytical results

Abstract

The present work is aimed at the design, implementation and analysis of generic new second-order non-autonomous chaotic systems with simple nonlinear functions that have algebraically simple representations in mathematical form. These systems involve simple second-order non-autonomous ordinary differential equations with different simple nonlinearities like $G(x) (= x^2, |x|, max(x)$ and $min(x))$. The present study deals with numerical simulation, experimental analog circuit simulation results to demonstrate the ordered and chaotic phenomena being exhibited by these systems. Of most interest the striking phenomenon of multistability is uncovered for certain nonlinearities. Coexisting attractors are harnessed through the offset boosting approach. Also, the Hamilton energy is established through the Helmholtz theorem. It is found that both methods can be exploited as a non-bifurcation approach in characterizing multistability in the model. Further, analytical solutions developed for systems with certain nonlinearities are presented and the chaotic behavior of the systems studied using the solutions validating the numerical and experimental results are reported.

Emergence of dynamical hysteresis in a second-order non-autonomous chaotic circuit

“…Offset boosting is an easy and reliable method to identify any coexisting dynamics in the phase space [22][23][24]. Offset boosted systems are obtained by introducing a constant vector into the state variables of an investigated system.…”

“…The MLC circuit and its variants have wide applications in secure communication, image encryption, electronic noise generation, chaotic logic gates, and reservoir computing [17][18][19][20][21]. Furthermore, analytical solutions for a class of second-order chaotic systems have been reported recently [22].…”