Nonlinear Dynamics in a Chemical Reaction under an Amplitude-Modulated Excitation: Hysteresis, Vibrational Resonance, Multistability, and Chaos

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 the Cardan’s method. Hopf bifurcation has been also successfully checked. Routes to chaos have been investigated through bifurcations structures, Lyapunov exponent, phase portraits, and Poincaré section. The effects of the control force on chaotic motions have been strongly analyzed, and the control efficiency is found in the cases <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>g</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mtext> </mml:mtext></mml:math>(unmodulated case) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>g</mml:mi><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>=</mml:mo><mml:mi>ω</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>Ω</mml:mi><mml:mo>/</mml:mo><mml:mi>w</mml:mi><mml:mo>≠</mml:mo><mml:mrow><mml:mi>p</mml:mi><mml:mo>/</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:math>; <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi>p</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mi>q</mml:mi></mml:math> are simple positive integers. Vibrational resonance (VR), hysteresis, and coexistence of several attractors have been studied in detail based on the relationship between the frequencies of the AM force. Results of analytical investigations are validated and complemented by numerical simulations.