Multistability and Four-Scroll Chaos in a Pair of Coupled Second-Order Damped Oscillators with Hyperbolic Sine Function: Theoretical Study and Circuit Simulation

• Published in: Circuits, systems, and signal processing systems, and signal processing, 2024-04, Vol.43 (4), p.2016-2050
• Main Authors: Sriram, Sridevi, Kana Kemgang, Lucas, Rakhmatullaeva, M. F., Rajagopal, Karthikeyan, Kengne, Jacques
• Format: Article
• Language: English
• Subjects: Bifurcation theory; Circuit design; Circuits and Systems; Coupling coefficients; Design; Electrical Engineering; Electronic circuits; Electronic components; Electronics and Microelectronics; Engineering; Extreme values; Hopf bifurcation; Hyperbolic functions; Instrumentation; Liapunov exponents; Linear functions; Nonlinearity; Numerical integration; Operational amplifiers; Oscillators; Signal,Image and Speech Processing; Trigonometric functions
• ISSN: 0278-081X; 1531-5878
• DOI: 10.1007/s00034-023-02573-2

It is known that the coupling of nonlinear oscillators can result in extremely rich and complex dynamics not found in an isolated oscillator. This work introduces a novel four-dimensional autonomous system designed by coupling two second-order damped oscillators with hyperbolic sine nonlinearity in such a way that each oscillator is perturbed by a signal proportional to the amplitude of the other one. The model presents both odd and rotational symmetry. The stability and nature of the equilibrium points are studied by resorting to the Routh–Hurtwitz criterion and the Hopf bifurcation theorem. A direct numerical integration of the evolution equation reveals rich and striking features such as multiple Hopf bifurcations, four-scroll chaotic attractors, and the coexistence of multiple modes of oscillations (i.e. homogeneous and heterogeneous multistability) when adjusting the coupling coefficients and the dissipation as well. These striking features are visualized by utilizing phase space trajectory plots, basins of attraction, plots of maximum Lyapunov exponents and bifurcation diagrams. A simple analogue electronic circuit emulating the coupled oscillators system is designed by using only basic electronic components consisting of operational amplifiers, resistors, capacitors and two pairs of diodes (implementing the hyperbolic sine nonlinearities). PSpice simulations and the microcontroller-based implementation are in accordance with the predictions of the theoretical investigations. One of main conclusions of the present study is that the coupling of second-order damped oscillators with hyperbolic sine function (only) can be seen as an alternative method to design multiscroll chaos generators. The proposed approach then differs from previous ones employing piecewise-linear functions, sine functions, and hysteresis functions or coupled third-order oscillators where the electronic design of the nonlinearity is relatively complex.