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We propose a new paradigm for the synchronization of two unconnected magnetic core coils based modified Van der Pol Duffing (MVDPD) oscillators circuits. The method is such that only magnetic field based coupling is sufficient to drive two identical chaotic circuits to a synchronized state as well as achieving the global stabilization of the system to its regular dynamics. The dynamics of the coupled system is investigated and Lyapunov stability theory is applied to prove that under some conditions the drive-response system can achieve practical synchronization. Numerical and PSpice simulations are given to demonstrate the effectiveness of the controller.

In 1990, the problems of chaos control [

In most of the methods mentioned above, the controllers and the design approach are often very complex and could be difficult to achieve in practice. Thus, designing simple and available control inputs that can achieve global and stable synchronization of coupled oscillators is generally significant and of practical interest in view of the foreseen applications of chaos synchronization in circuits and lasers. This is an open challenge that has remained unresolved. The synchronization in chaotic circuits coupled by mutual inductors has been extensively studied [

This paper has four aims.

We analyse the dynamics of modified Van der Pol Duffing oscillators coupled by tunable magnetic core coils transformer.

We use the mutual inductance between the windings (the coupling coefficient) to investigate the synchronization of the two coupled circuits.

We use the Lyapunov theory, to investigate the conditions leading to practical synchronization of the coupled circuits.

We present an electronic implementation of this methodology.

The rest of the paper is organized as follows. In Section

Figure

(a) The electrical model of the system consisting of two mutually coupled modified Van der Pol Duffing oscillators (MVDPD). (b) The practical realization of the electrical circuit corresponding to the cubic nonlinear negative resistance: the operational amplifier can be any compatible TL082; the signal diodes are IN 4148; the circuit elements are

Let us consider the two magnetic windings of (Figure

The flux due to the second winding in the first one is

The mutual inductance is given by

However, using the reciprocity theorem which combines Amperes’s law and Biot-Savart’s law, one may show that the mutual inductance is finally

In order to compare experimental and numerical results, an appropriate rescaling is introduced. With the corresponding change of variables:

As we mentioned above, one of the main advantages of this circuit for experimental applications is the richness of its dynamics. Here, the types of behaviors are identified using two indicators. The first indicator is the bifurcation diagram, the second being the largest 1D numerical Lyapunov exponent denoted by

Considering the effects of

Bifurcation diagram (a) and the graph of 1D largest numerical Lyapunov exponent (b).

Qualitative comparison of numerical phase portraits forms (

Let us consider here that the drive system is given by the following set of coupled differential equations:

According to our aim, we define the controlled response system as the following set of differential equations:

At this stage, we need to prove that the dynamics of the error system equation (

The time derivative of this function with respect to the system of errors equations (

In this case, (

If

Using the fact that

Thus, one obtains the following that presents the bounds of the Lyapunov function

Hence, one has

Relation (

This concludes the proof.

We now present numerical simulation results to verify the effectiveness of the controller. In all cases, we select the parameters

In Figure

Graphs of time variation of the synchronization errors

Bifurcation diagram for the in-phase synchronization.

The experimental simulation is a nice way to scan the parameter range in order to find the proper parameter values for a numerical simulation. Another advantage of such an implementation with respect to numerical simulation is that there is no need to wait for long transient times. This justifies the increasing interest devoted to this type of implementation for the analysis of nonlinear and chaotic physical systems. Our aim in this section is to consider and implement an appropriate analog real circuit for the investigation of the system described by (

Figure

In-phase synchronization of the magnetically coupled chaotic circuits (a)

In this paper, we have proposed a new paradigm for the synchronization of two unconnected magnetic core coils based modified Van der Pol Duffing (MVDPD) oscillators circuits. The method is such that only magnetic field based coupling is sufficient to drive two identical chaotic circuits to a synchronized state as well as achieving the global stabilization of the system to its regular dynamics. The dynamics of the coupled system was investigated and Lyapunov stability theory was applied to prove that under some conditions the drive-response system can achieve practical synchronization. Numerical and experiment simulations were given to demonstrate the effectiveness of the controller. In future works, it will be shown that collective dynamics of several magnetically coupled MVDPD systems could be controlled by moving a transformer’s core or varying distances between oscillators.

The authors declare that there is no conflict of interests regarding the publication of this paper.