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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

Nonlinear dynamics is a multidisciplinary ﬁeld that covers not only mathematics but also engineering, physics, chemistry, and biology. A monograph, authored by Strogatz [

In this work, we consider the autonomous system of reactions between four molecules as in [

_{3}), and nitrogen monoxide and dihydrogen (NO + H_{2}), which all use the platinum surface Pt as catalyst, are well-known models of vacancies [

In the mean-ﬁeld approach, the kinetic equations for the evolution of the relative concentrations of the molecules (

It is easily shown that a condition of conservation is fulfilled:

The complex dynamic behaviors of this system under the action of a sinusoidal force have been studied in-depth [

Using the time rescaling

We consider the system (

The resolution of the system formed by these two equations comes down to that of the following equation:

The ﬁxed points of the autonomous system are

To solve equation (

So, we have

By taking the Cardan technique [

Here,

Equation (

On the contrary, by combining the equations of the system (

Thus, we have

In total, the autonomous system admits exactly four ﬁxed points

The study of the stability of each ﬁxed point is made by searching the Jacobian of the autonomous system associated. The Jacobian of the autonomous system associated with each ﬁxed point

For

If

In this case, the equilibrium point

Assuming

If

To support this theorem, we represent the real part of the eigenvalues associated with the equilibrium point

Real part of the eigenvalues versus

Phase portraits of the autonomous system for parameter values corresponding to Hopf bifurcation at fixed point

Before studying the influence of the AM excitation on the dynamics of the system, let us analyze the effect of each parameter

Bifurcation diagram and its corresponding Lyapunov exponent of nonlinear chemical reaction with

Chaotic phase portrait and its Poincaré section of nonlinear chemical reaction with parameters of Figure

Phase space and its corresponding Poincaré section of nonlinear chemical reaction with

Effect of

In a nonlinear dynamical system driven by a biharmonic force consisting of a low-frequency

We compute

For _{max} (see Figure _{max} decreases when the frequency

(a) Variation in numerically computed

Effect of frequency

(a) Effect of amplitude

Figure

In this section, we analyze the effect of

Bifurcation diagram and its corresponding Lyapunov exponent of nonlinear chemical reaction with

When we compare the dynamics of the system when

Poincaré section of nonlinear chemical reaction with

Effect of

The chemical oscillations become very varied, and we note periodic, multiperiodic, quasiperiodic, and chaotic behaviors with the presence and persistence of hysteresis and the coexistence of multiple attractors when

More precisely,

Here, the effect of the AM force is analyzed in the case

Effect of parameter

Effect of

For this, we take

Effect of parameter

Effect of parameter

In this work, we have studied the influence of an amplitude-modulated force on the chemical oscillations of reactions between four molecules. We have looked for fixed points and their natures when the system is autonomous. It appears that the autonomous system admits exactly four fixed points. The nature of the nontrivial fixed point depends of parameters

No data were used in this study.

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