Effect of an amplitude modulated force on vibrational resonance, chaos, and multistability in a modified Van der Pol-Duffing oscillator

Volume 16, Issue 2, pp 111--122 http://dx.doi.org/10.22436/jnsa.016.02.04

Publication Date: August 05, 2023 Submission Date: March 29, 2023 Revision Date: May 14, 2023 Accteptance Date: July 20, 2023

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Authors

C. H. Miwadinou - Laboratoire de Mecanique des Fluides, de la Dynamique Nonlineaire et de la Modelisation des Systemes Biologiques (LMFDNMSB), Institut de Mathematiques et de Sciences Physiques, Porto-Novo, Benin. - Departement de Physique, ENS-Natitingou, Universite Nationale des Sciences, Technologies, Ingenierie et Mathematiques (UNSTIM), Abomey, Benin. L. A. Hinvi - Laboratoire de Mecanique des Fluides, de la Dynamique Nonlineaire et de la Modelisation des Systemes Biologiques (LMFDNMSB), Institut de Mathematiques et de Sciences Physiques, Porto-Novo, Benin. - Departement de Genie Mecanique et Productique GMP, INSTI-Lokossa, Universite Nationale des Sciences, Technologies, Ingenierie et Mathematiques (UNSTIM), Abomey, Benin. C. Ainamon - Laboratoire de Mecanique des Fluides, de la Dynamique Nonlineaire et de la Modelisation des Systemes Biologiques (LMFDNMSB), Institut de Mathematiques et de Sciences Physiques, Porto-Novo, Benin. A. V. Monwanou - Laboratoire de Mecanique des Fluides, de la Dynamique Nonlineaire et de la Modelisation des Systemes Biologiques (LMFDNMSB), Institut de Mathematiques et de Sciences Physiques, Porto-Novo, Benin.

Abstract

This paper deals with the effects of an amplitude modulated (AM) excitation on the nonlinear dynamics of reactions between four molecules. The computation of the fixed points of the autonomous nonlinear chemical system has been made in detail using Cardan's method. Routes to chaos have been investigated through bifurcations structures, Lyapunov exponent and phase portraits. The effects of the control force on chaotic motions have been strongly analyzed and the control efficiency is found in the cases \(g=0\) (unmodulated case), \(g\neq 0\) with \(\Omega=n\omega\); \(n\) a natural number and \(\frac{\Omega}{\omega}\neq\frac{p}{q}\); \(p\) and \(q\) are simple positive integers. Vibrational Resonance (VR), hysteresis and coexistence of several attractors have been studied in details based on the relationship between the frequencies of the AM force. Results of analytical investigations are validated and complemented by numerical simulations.

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Keywords

• Modified Van der Pol oscillator
• Cardan's method
• chaos
• coexisting attractors
• vibrational resonance

MSC

•  34H10
•  92E20
•  37N99
•  34C28

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