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**Abstract:**
This work studies the active control of chemical oscillations governed by a forced modified Van der Pol-Duffing oscillator. We considered the dynamics of nonlinear chemical systems subjected to an external sinusoidal excitation. The approximative solution to the first order of the modified Van der Pol-Duffing oscillator is found using the Lindstedt’s perturbation method. The harmonic balance method is used to find the amplitudes of the oscillatory states of the system under control. The effects of the constraint parameter and the control parameter** **of the model on the amplitude of oscillations are presented. The effects of the active control on the behaviors of the model are analyzed and it appears that with the appropriate selection of the coupling parameter, the chaotic behavior of the model has given way to periodic movements. Numerical simulations are used to validate and complete the analytical results obtained.

Since the discovery of chaos in the 20th century, the chaos has been recognized as a very interesting behavior in nonlinear dynamical systems [

This work takes into account all nonlinear chemical systems as a kinetic example which can be described by the following equations [

When we assume that the sink of the product is a first order reaction and we base ourselves that the laws of mass action and conservation, we get after some mathematical transformations that the self-oscillations in some nonlinear chemical systems can be modelised by the following single second order differential equation [

where

It has been noticed the presence of chaotic behaviors of this model [

where y is the control force and λ the control gain parameters, ω the free frequency of the linear oscillator and η the damping coefficient. Accordingly, the equation of the nonlinear chemical system under such a control scheme becomes:

Without the external sinusoidal excitation force, Equation (7) is solved using the Lindstedt’s perturbation method [

where the frequency ω is given by

and

With the external sinusoidal excitation force, assuming that the fundamental component of the solutions has the period of the external excitation, we express the solution ζ as follows:

To determine the amplitude of the vibration of the chemical system under control, we use the harmonic balance method [

with

Then, inserting the Equation (15) in the first equation of the system of Equations (10) gives us:

When we introduce the expression (14) in Equation (16) and equalize the coefficient of the terms in sinus and cosine, we obtain after a few algebraic manipulations the following differential equation satisfied by the amplitude A_{c} of the oscillatory states

It is noted that control is effective when amplitude A_{c} of vibration of the system under control is lower than amplitude A_{nc} of oscillations of the uncontrolled system obtained for λ = 0. We determine the domain of the parameter of control λ for which control is effective by solving Equation (17) with the Newton-Raphson algorithm. The amplitude A_{c} obtained from this resolution is plotted on _{c} is also observed when the amplitude of the external excitation E is varied. The existence of the phenomenon of hysteresis and of the jump amplitude of the harmonic oscillations has been noticed in the system under control. We notice the disappearance of these phenomena with the increase of the control parameter λ (see

The conditions for suppressing chaotic oscillations or instabilities in the modified and forced Van der Pol-Duffing oscillator coupled to the considered linear oscillator are investigated by doing the numerical simulation of the Equation (10).

In this paper we have investigated the active control of chaotic oscillations in certain nonlinear chemical dynamics. We have considered chemical dynamics modeled by a modified Van der Pol-Duffing oscillator subjected to external periodic

excitation. The model has been described and the corresponding equation presented. The Lindstedt’s perturbation method is used to find an analytic solution of the modified Van der Pol-Duffing oscillator. The amplitudes of the oscillatory states of the system under control have been found using the harmonic balance method. In the dynamics of the model under control, we noticed the appearance of the phenomena of hysteresis and jump amplitude, which is effectively controlled by the control parameter λ and the constraint parameter β. Active control is used to limit the presence of unwanted behaviors in the system. We have found the appropriate coupling parameter for which the linear oscillator effectively reduces the amplitude of the modified Van der Pol-Duffing oscillator. For a set of fixed values of the model parameters it was noted that the chaotic behavior of this oscillator has given way to periodic movements under the effect of active control.

The author thanks IMSP-UAC, DAAD for financial support and the anonymous referees whose useful criticisms, comments and suggestions have helped strengthen the content and the quality of the paper.

The authors declare no conflicts of interest regarding the publication of this paper.

Olabodé, D.L., Miwadinou, C.H., Monwanou, A.V., Lamboni, B. and Orou, J.B.C. (2019) Active Control of Chaotic Oscillations in Nonlinear Chemical Dynamics. Journal of Applied Mathematics and Physics, 7, 547-558. https://doi.org/10.4236/jamp.2019.73040