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We have investigated the synchronization and antisynchronization behaviour of two identical planar oscillation of a satellite in elliptic orbit evolving from different initial conditions using the active control technique based on the Lyapunov stability theory and the Routh-Hurwitz criteria. The designed controller, with our own choice of the coefficient matrix of the error dynamics that satisfy the Lyapunov stability theory and the Routh-Hurwitz criteria, is found to be effective in the stabilization of the error states at the origin, thereby, achieving synchronization and antisynchronization between the states variables of two nonlinear dynamical systems under consideration. The results are validated by numerical simulations using

In the last two decades, considerable research has been done in nonlinear dynamical systems and their various properties. One of the most important property of nonlinear dynamical systems is synchronization, which classically represents the entrainment of frequencies of oscillators due to weak interactions [

In this article, we have applied the active control technique based on the Lyapunov stability theory and the Routh-Hurwitz criteria to study the synchronization and antisynchronization behavior of two identical planar oscillation of a satellite in elliptic orbit evolving from different initial conditions. The system under consideration is chaotic for some values of parameter involved in the system. In synchronization, the two systems (master and slave) are synchronized and start with different initial conditions. The same problem may be treated as the design of control laws for full chaotic slave system using the known information of the master system so as to ensure that the controlled receiver synchronizes with the master system. Hence, the slave chaotic system completely traces the dynamics of the master system in the course of time. The aim of this study is to trace the chaotic dynamics of the planar oscillation of a satellite in elliptic orbit based on synchronization and antisynchronization. To the best of my knowledge nobody studied this before.

Elliptically orbiting planar oscillations of satellites in the solar system make an interesting study, and significant contributions to this end can be found [

Now,

Using (

Using the binomial expansion for

For a system of two coupled chaotic oscillators, the master system (

In order to formulate the active controllers, we write the system (

Let

According to the Lyapunov stability theory and the Routh-Hurwitz criteria, if

For the constant elements of feedback matrix, choosing

Phase plot of master system.

Phase plot of slave system.

Time series analysis of master system.

Time series analysis of slave system.

Time series analysis of

Time series analysis of

Antisynchronization of two coupled systems

In order to formulate the active controllers for Antisynchronization, we need to redefine the error functions as

According to the Lyapunov stability theory and the Routh-Hurwitz criteria, if

For the constant elements of feedback matrix, choosing

Phase plot of slave system.

Time series analysis of slave system.

Time series analysis of

Time series analysis of

Convergence of errors in synchronization.

Convergence of errors in antisynchronization.

In this paper, we have investigated the synchronization and antisynchronization behaviour of the two identical planar oscillation of a satellite in elliptic orbit evolving from different initial conditions via the active control technique based on the Lyapunov stability theory and the Routh-Hurwitz criteria. The results were validated by numerical simulations using