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We propose the use of a simple, cheap, and easy technique for the study of dynamic and synchronization of the coupled systems: effects of the magnetic coupling on the dynamics and of synchronization of two Colpitts oscillators (wireless interaction). We derive a smooth mathematical model to describe the dynamic system. The stability of the equilibrium states is investigated. The coupled system exhibits spectral characteristics such as chaos and hyperchaos in some parameter ranges of the coupling. The numerical exploration of the dynamics system reveals various bifurcations scenarios including period-doubling and interior crisis transitions to chaos. Moreover, various interesting dynamical phenomena such as transient chaos, coexistence of solution, and multistability (hysteresis) are observed when the magnetic coupling factor varies. Theoretical reasons for such phenomena are provided and experimentally confirmed with practical measurements in a wireless transfer.

In electronics and in nonlinear dynamics, oscillators involving inductors are always modeled without any coupling from external magnetic field. However, in certain experimental condition, non-physically coupled systems may interact.

In this paper, we study the effect of the magnetic coupling on the behavior of a Colpitts coupled oscillator. Coupled chaotic systems have become a topic of active research in many scientific areas such as chemistry, biology, optics, and communications. In particular, the synchronization of coupled chaotic systems has received a lot of attention since the early 1990s [

The classical Colpitts oscillator is widely used in electronic devices and communication system as a source of sinusoidal waveform with low harmonic content [

Hyperchaos is characterized as chaotic dynamics with more than one positive Lyapunov exponent (LE). Hyperchaos has attracted increasing attention from the scientific and engineering communities [

In a complex system several attractors may coexist for a given set of system parameters. This coexistence is termed as multistability and has been found in almost all research areas of natural science, such as mechanics, electronics, biology, environmental science, and neuroscience [

Generally speaking, chaos synchronization can be referred to as behavior in which two or more coupled systems exhibit identical chaotic oscillations. For such dynamical systems, the loss of synchronization between the subsystems is strictly related to the Lyapunov exponents (LEs) of the global system. When the subsystems lose synchrony, a transition from chaotic to hyperchaotic behavior takes place; that is, at least two LEs become positive [

In this paper, a new method is introduced. It has been experimentally and has been numerically characterized, leading to a simple, cheap, and easy technique for the study of dynamic and synchronization of Colpitts coupled oscillator. When compared with other circuit techniques applied to the chaotic oscillator [

The idea underlying our approach is also applied to chaotic Colpitts oscillator. It demonstrates how it is possible to control and synchronize a system in a simple and effective way, with high frequency oscillations. This coupling gives a possibility of obtaining a rich and complex dynamic system. To the best of our knowledge, the magnetic coupling of two chaotic Colpitts oscillators has never been investigated.

Our aims in this work are the following: (a) extend the study of magnetic coupling to higher frequencies chaotic systems (they are more useful in practical secure communication systems), (b) investigate the conditions leading to practical synchronization of the coupled Colpitts circuits, and (c) present an electronic implementation of the magnetically coupled systems. The rest of the paper is organized as follows: In Section

The Colpitts oscillator shown in Figure

The electrical circuit of the system consisting of two mutually coupled Colpitts oscillators (a) and the circuit diagram of a bipolar junction transistor (BJT) (b). The circuit elements are 2N3904;

Let us consider the two magnetic core coils of Figure

In our model (Figure

For the numerical investigation, an appropriate rescaling is introduced with the corresponding change of variables:

We consider the normalized system (

We obtain the equilibrium points

The equilibrium point

The stability of the equilibrium point

Using the Routh-Hurwitz criteria on system (

If

If

Representation of the eigenvalues, solution of the characteristic equation (

The critical value can be obtained by making recourse to numerical methods. We have one critical value

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 graph of Lyapunov exponent’s spectra [

For an equilibrium point,

For a limit cycle (periodic orbits),

For 2-torus (quasiperiodic orbits),

For 3-torus,

For chaotic orbits,

For hyperchaotic orbits,

The evolution process of the system is analyzed precisely by the means of the Lyapunov exponents spectrum, bifurcation diagrams, and phase portraits when varying the parameter

Bifurcation diagram of magnetically coupled Colpitts systems (

Lyapunov exponents spectrum of magnetically coupled Colpitts systems (

Qualitative comparison of numerical phase forms

Generally, multistable systems are characterized by a high degree of complexity in dynamical behavior due to the interaction among the coexisting attractors. First of all, the dynamics of a multistable system are extremely sensitive to initial conditions. Due to the coexistence of different attractors and complex fractal basin boundary structures very small perturbations of the initial state may influence the final attractor. Second, the qualitative behavior of the system often changes under the variation of the parameters system. Similarly, attractors exist only in small intervals of the parameters system. A slight change in a control parameter may cause a rapid change in the number and type of coexisting attractors. Third, multistable systems are extremely sensitive to noise. Noise may cause a popping process between various attractors. As an example of the system which possesses multistability, we consider the system described by (

Bifurcation diagrams for (blue) increasing and (red) decreasing of magnetic coupling.

For instance, In Figure

Coexistence of two different attractors, 3-period and hyperchaos, for

The appearance of chaos and hyperchaos on finite time scales is known as transient chaos and transient hyperchaos. In our system, the appearance of periodic or chaos or hyperchaos motion strongly depends not only on the circuit parameters but also on the initial conditions. When the magnetic coupling of the stimulus is selected as

Transition from chaotic to period doubling: time-domain waveform of variable

The transient hyperchaos phenomenon exists when

Transition from hyperchaos to period doubling: time-domain waveform of variable

Although the coupling scheme implemented here is bidirectional, we need to define a master (drive) and a slave (response) system. We 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:

It is clear that the synchronization problem is replaced by the equivalent problem of stabilizing (

The synchronization problem for the magnetically coupled Colpitts system is to achieve the asymptote of the zero solution of the error system (

Consider a Lyapunov function candidate in the following form:

If the following condition holds

Since

In this section, we consider two oscillators

A three-parameter diagram showing regions of synchronization (blue) in the (

Three parameters of system

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

The graphs of Figure

Graphs of time variation of the synchronization errors

Figure

Bifurcation diagram for the in-phase synchronization, respectively,

The complex behavior of the system (Figure

Figure

Experimental circuit.

Some examples are reported in Figure

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

The aim of this paper was to introduce a new method which is a simple, cheap, and easy technique for the study of dynamic and synchronization of Colpitts coupled oscillator. This method has gone through experiment and has been numerically characterized. The influence of the magnetic field via a magnetic coupling between the two oscillators has been presented. The stability of the fixed points has been investigated and the analytical formulas describing the stability of the system have been established. The bifurcation analysis of the phenomena leading to the creation of multistability of the system has been studied. Numerical simulations and electronic circuit implementation of the system show that the complex dynamics (transient chaos and transient hyperchaos) of the circuit are heavily dependent on the initial state of this system and on the magnetic coupling factor

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