Eur. Phys. J. Plus (2022) 137: 789
https://doi.org/10.1140/epjp/s13360-022-03000-2

Regular Article

Coupled van der Pol and Duffing oscillators: emergence of antimonotonicity and coexisting multiple self-excited and hidden oscillations

¹ Center for Nonlinear Systems, Chennai Institute of Technology, Chennai, India
² Laboratoire d’Automatique et Informatique Appliquée (LAIA), IUT-FV Bandjoun, University of Dschang, Dschang, Cameroon

^b kengnemozart@yahoo.fr

Received: 24 February 2022
Accepted: 23 June 2022
Published online: 8 July 2022

Abstract

Due to its fundamental and technological relevance, the system formed by a van der Pol oscillator coupled to a Duffing oscillator is the subject of increasing research interest. In this contribution, we study the dynamics of a system composed of a van der Pol oscillator coupled to a Duffing oscillator with two asymmetric potential wells. In the considered coupling scheme, the van der Pol oscillator is disturbed by a signal proportional to the difference of position while the Duffing oscillator is driven by a signal proportional to the speed difference. We use analytical and numerical methods to shed light on the complex behaviors exhibited by the coupled system. We highlight striking phenomena such as hysteresis, parallel bifurcation branches, the coexistence of multiple (i.e., three, four or five) self-excited and hidden attractors, and the coexistence of symmetric and asymmetric bifurcation bubbles depending on the numerical values of initial conditions and parameters. An in-depth study of the coexistence of solutions is carried out by constructing basins of attraction. Sample experimental results captured from STM32F407ZE microcontroller-based digital implementation of the coupled system verify the striking dynamics features observed during the theoretical analysis. It should be noted that the coexistence of a hidden attractor with self-excited others is unique for the coupled oscillators system considered in the context of this work and thus deserves dissemination.