Synchronization and Antisynchronization of a Planar Oscillation of Satellite in an Elliptic Orbit via Active Control

Abstract

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 mathematica.

1. Introduction

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 [1–3]. Studies in this field are partly motivated by experimental realization in lasers, electronic circuits, plasma discharge, and chemical reactions [2–4]. Synchronization techniques have been improved in recent years, and many different methods are applied theoretically as well as experimentally to synchronize the chaotic systems. Some of them are nonlinear feedback method [5], adaptive control method [6], antisynchronization method [7], sliding mode control method [8], and chaos synchronization using active control, which is introduced in [9] is one of these methods. Notable among these methods, chaos synchronization using active control scheme has recently been widely accepted as an efficient technique to synchronize the chaotic systems. The reason is because it can be used to synchronize identical as well as nonidentical systems; a feature that gives it an advantage over other synchronizing methods. This method based on the Lyapunov stability theory and the Routh-Hurwitz criteria to active control in order to achieve stable synchronization has been applied to many practical systems such as the electronic circuits, which model a third-order “jerk” equation [10], Lorenz, Chen, and Lü system [11], geophysical systems [12], nonlinear equations of acoustic gravity waves (Lorenz-Stenflo system) [13], Van-der Pol-Duffing oscillator [14], forced damped pendulum [15], nuclear magnetic resonance modeled by the Bloch equations [16], parametrically excited oscillators [17], permanent magnet reluctance machine [18], inertial ratchets [19], RCL-shunted Josephson junction [20], and modified projective synchronization [21].

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.

2. Description of the Model

Elliptically orbiting planar oscillations of satellites in the solar system make an interesting study, and significant contributions to this end can be found [22–30], all of which have studied the influence of certain perturbative forces, such as solar radiation pressure, tidal force, and air resistance. In the present work, we consider the planar oscillation of a satellite in elliptic orbit with the spin axis fixed perpendicular to the orbital plane. Let the long axis of the satellite makes an angle 𝜃 with a reference axis that is fixed in inertial space, the long axis of the satellite makes an angle 𝜓 with satellites planet centre line [31], and the satellite to be a triaxial ellipsoid with principal moments of inertia 𝐴<𝐵<𝐶, where 𝐶 is the moment about the spin axis. The orbit is taken to be a fixed ellipse with semimajor axis 𝑎, eccentricity 𝑒, true anomaly 𝑓, and instantaneous radius 𝑟. The equation of motion of satellite planar oscillation in an elliptic orbit around the earth is 2𝐶𝜃̈−3(𝐵−𝐴)𝐺𝑚_𝑝𝑟³sin2𝜓=0. (1) Taking 3(𝐵−𝐴)/𝐶=𝜔²₀, 𝐺𝑚_𝑝=𝜇, 𝜓=𝑓−𝜃 and 𝑙/𝑟=1+𝑒cos𝑓, (1) can be written as 𝑑²𝜃𝑑𝑡²=𝜇𝜔²₀𝑙³1+𝑒cos𝑓³sin2𝑓−𝜃. (2) In order to reduce three variables 𝜓,𝜃, and 𝑓 we are taking 𝑓as independent variable.

Now, 𝑑𝜃=𝑑𝑡𝑑𝜃𝑑𝑓𝑑𝑓=𝑓̇𝑑𝑡𝑑𝜃=ℎ𝑑𝑓𝑟²𝑑𝜃,𝑑𝑓using𝑟²̇,𝑑𝑓=ℎ₂𝜃𝑑𝑡²=−2ℎ𝑟̇𝑟³𝑑𝜃+ℎ𝑑𝑓𝑟²𝑑²𝜃𝑑𝑓²𝑑𝑓,𝑑𝑑𝑡₂𝜃𝑑𝑡²=−2ℎ𝑟̇𝑟³𝑑𝜃+ℎ𝑑𝑓𝑟²⁴𝑑²𝜃𝑑𝑓²using𝑟²𝑓̇=ℎ. (3) Now, from 𝑙/𝑟=1+𝑒cos𝑓, we have ̇𝑓̇𝑟=−𝑙−𝑒sin𝑓1+𝑒cos𝑓²,𝑟̇=𝑒𝑙sin𝑓1+𝑒cos𝑓²ℎ𝑟². (4)

Using (3) and (4) in (2), we have ℎ²𝑟⁴𝑑²𝜃𝑑𝑓²−2𝑒𝑙ℎ²sin𝑓𝑟⁵1+𝑒cos𝑓²𝑑𝜃=𝑑𝑓𝜇𝜔²₀𝑟³ℎsin2𝑓−𝜃₂𝑟⁴𝑑²𝜃𝑑𝑓²−2𝑒ℎ²sin𝑓𝑟⁴1+𝑒cos𝑓𝑑𝜃=ℎ𝑑𝑓₂𝜔²₀𝑙𝑟³sin2𝑓−𝜃usingℎ²,𝑑=𝜇𝑙₂𝜃𝑑𝑓²=2𝑒sin𝑓1+𝑒cos𝑓𝑑𝜃𝑑𝑓+𝜔²₀sin2𝑓−𝜃.1+𝑒cos𝑓 (5)

Using the binomial expansion for (1+𝑒cos𝑓)⁻¹ and for very small value of 𝑒 ignoring the higher order terms in 𝑒, we have 𝑑²𝜃𝑑𝑓²=2𝑒sin𝑓𝑑𝜃+𝜔𝑑𝑓2²⁰1−𝑒cos𝑓sin2𝑓−𝜃. (6)

3. Synchronization via Active Control

For a system of two coupled chaotic oscillators, the master system (𝑥̇=𝑓(𝑥,𝑦)) and the slave system (𝑦̇=𝑔(𝑥,𝑦)), where 𝑥(𝑡) and 𝑦(𝑡) are the phase space (state variables), and 𝑓(𝑥,𝑦) and 𝑔(𝑥,𝑦) are the corresponding nonlinear functions, synchronization in a direct sense implies lim_𝑡→∞|𝑥(𝑡)−𝑦(𝑡)|→0. When this occurs the coupled systems are said to be completely synchronized. Chaos synchronization is related to the observer problem in control theory [32]. The 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 in the course of time.

In order to formulate the active controllers, we write the system (6) in two first-order differential equations as shown below.

Let 𝑑𝜃/𝑑𝑓=𝑥^′₁ and 𝑑²𝜃/𝑑𝑓²=𝑥^′₂, then we have 𝑥^′₁=𝑥₂,𝑥^′₂=2𝑒𝑥^′₁𝜔sin𝑓+²⁰21−𝑒cos𝑓sin2𝑓−𝑥₁. (7) Let us define another system 𝑦^′₁=𝑦₂+𝑢₁𝑓,𝑦^′₂=2𝑒𝑦^′₁𝜔sin𝑓+²⁰21−𝑒cos𝑓sin2𝑓−𝑦₁+𝑢₂𝑓, (8) where (7) and (8) are called the master system and the slave system, respectively, and in slave system, 𝑢₁(𝑓) and 𝑢₂(𝑓) are control functions to be determined. Let 𝑒₁(𝑓)=𝑦₁(𝑓)−𝑥₁(𝑓) and 𝑒₂(𝑓)=𝑦₂(𝑓)−𝑥₂(𝑓) be the synchronization errors such that lim_𝑓→∞𝑒_𝑖(𝑓)→0 for 𝑖=1,2. From (7) and (8), we have 𝑒^′₁𝑓=𝑒₂𝑓+𝑢₁𝑓,𝑒^′₂𝑓=2𝑒𝑒^′₁𝑓𝜔sin𝑓+²⁰2×1−𝑒cos𝑓sin2𝑓−𝑦₁−sin2𝑓−𝑥₁+𝑢₂𝑓. (9) In order to express (9) as only linear terms in 𝑒₁(𝑓) and 𝑒₂(𝑓), we redefine the control functions as follows: 𝑢₁𝑓=𝑣₁𝑓,𝑢₂𝑓=−2𝑒𝑒^′₁𝑓𝜔sin𝑓−²⁰2×1−𝑒cos𝑓sin2𝑓−𝑦₁−sin2𝑓−𝑥₁+𝑣₂𝑓. (10) From (9) and (10), we have 𝑒^′₁𝑓=𝑒₂𝑓+𝑣₁𝑓,𝑒^′₂𝑓=𝑣₂𝑓. (11) Equation (11) is the error dynamics, which can be interpreted as a control problem where the system to be controlled is a linear system with control inputs 𝑣₁(𝑓)=𝑣₁(𝑒₁(𝑓),𝑒₂(𝑓)) and 𝑣₂(𝑓)=𝑣₂(𝑒₁(𝑓),𝑒₂(𝑓)). As long as these feedbacks stabilize the system, lim_𝑓→∞𝑒_𝑖(𝑓)→0 for 𝑖=1,2. This simply implies that the two systems (7) and (8) evolving from different initial conditions are synchronized. As functions of 𝑒₁(𝑓) and 𝑒₂(𝑓), we choose 𝑣₁(𝑓) and 𝑣₂(𝑓) as follows: 𝑣₁𝑓𝑣₂𝑓𝑒=𝐷¹(𝑡)𝑒₂(𝑡), (12) where 𝐷=^𝑎𝑏_𝑐𝑑 is a 2×2 constant feedback matrix to be determined. Hence the error system (11) can be written as 𝑒^′₁𝑓𝑒^′₂𝑓𝑒=𝐶¹𝑓𝑒₂𝑓, (13) where 𝐶=^𝑎1+𝑏_𝑐𝑑 is the coefficient matrix.

According to the Lyapunov stability theory and the Routh-Hurwitz criteria, if 𝑐()𝑎+𝑑<0,1+𝑏−𝑎𝑑<0, (14) then the eigen values of the coefficient matrix of error system (11) must be real or complex with negative real parts and, hence, stable synchronized dynamics between systems (7) and (8) is guaranteed. Let 𝑎+𝑑=𝑐(1+𝑏)−𝑎𝑑=−𝐸, (15) where 𝐸>0 is a real number which is usually set equal to 1. There are several ways of choosing the constant elements 𝑎, 𝑏, 𝑐, 𝑑 of matrix 𝐷 in order to satisfy the Lyapunov stability theory and the Routh-Hurwitz criteria (14).

4. Numerical Simulation for Synchronization

For the constant elements of feedback matrix, choosing 𝑎=𝑑=−0.5 and for the parameters involved in system under investigation, 𝑒=0.15, 𝜔₀=0.3 with the initial conditions [𝑥₁(0),𝑦₁(0)]=[0,0], and [𝑥₂(0),𝑦₂(0)]=[1,1.5], we have simulated the system under consideration using mathematica. The results obtained show that the system under consideration achieved synchronization. Phase portraits, time series analysis, and error diagrams are the witness of achieving synchronization between master and slave system. Further, it also has been confirmed by the convergence of the synchronization quality defined by 𝑒𝑓=𝑒²₁𝑓+𝑒²₂𝑓. (16) Figures 1–6 confirms the convergence of the synchronization quality defined by (16).

Figure 1 

Phase plot of master system.

Figure 2 

Phase plot of slave system.

Figure 3 

Time series analysis of master system.

Figure 4 

Time series analysis of slave system.

Figure 5 

Time series analysis of 𝑒1(𝑓).

Figure 6 

Time series analysis of 𝑒2(𝑓).

5. Antisynchronization via Active Control

Antisynchronization of two coupled systems 𝑥̇=𝑓(𝑥,𝑦) (master system) and 𝑦̇=𝑔(𝑥,𝑦) (slave system) means lim_𝑡→∞|𝑥(𝑡)+𝑦(𝑡)|→0. This phenomenon has been investigated both experimentally and theoretically in many physical systems [18, 19, 32–37]. A recent study of the antisynchronization phenomenon in nonequilibrium systems suggests that it could be used as a technique for particle separation in a mixture of interacting particles [19].

In order to formulate the active controllers for Antisynchronization, we need to redefine the error functions as 𝑒₁(𝑓)=𝑦₁(𝑓)+𝑥₁(𝑓) and 𝑒₂(𝑓)=𝑦₂(𝑓)+𝑥₂(𝑓), where 𝑒₁(𝑓) and 𝑒₂(𝑓) are called the antisynchronization errors such that lim_𝑓→∞𝑒_𝑖(𝑓)(𝑖=1,2)→0. From (7) and (8), error dynamics can be written as 𝑒^′₁𝑓=𝑒₂𝑓+𝑢₁𝑓,𝑒^′₂𝑓=2𝑒𝑒^′₁𝑓𝜔sin𝑓+²⁰2×1−𝑒cos𝑓sin2𝑓−𝑦₁+sin2𝑓−𝑥₁+𝑢₂𝑓. (17) In order to express (17) as only linear terms in 𝑒₁(𝑓) and 𝑒₂(𝑓), we redefine the control functions as follows: 𝑢₁𝑓=𝑣₁𝑓,𝑢₂𝑓=−2𝑒𝑒^′₁𝑓𝜔sin𝑓−²⁰2×1−𝑒cos𝑓sin2𝑓−𝑦₁+sin2𝑓−𝑥₁+𝑣₂𝑓. (18) From (17) and (18), we have 𝑒^′₁𝑓=𝑒₂𝑓+𝑣₁𝑓,𝑒^′₂𝑓=𝑣₂𝑓. (19) Equation (19) is the error dynamics, which can be interpreted as a control problem where the system to be controlled is a linear system with control inputs 𝑣₁(𝑓)=𝑣₁(𝑒₁(𝑓),𝑒₂(𝑓)), and 𝑣₂(𝑓)=𝑣₂(𝑒₁(𝑓),𝑒₂(𝑓)). As long as these feedbacks stabilize the system, lim_𝑓→∞𝑒_𝑖(𝑓)→0 for 𝑖=1,2. This simply implies that the two systems (7) and (8) evolving from different initial conditions are synchronized. As functions of 𝑒₁(𝑓) and 𝑒₂(𝑓), we choose 𝑣₁(𝑓) and 𝑣₂(𝑓) as follows: 𝑣₁𝑓𝑣₂𝑓𝑒=𝐷¹𝑓𝑒₂𝑓, (20) where 𝐷=^𝑎𝑏_𝑐𝑑 is a 2×2 constant feedback matrix to be determined. Hence the error system (19) can be written as 𝑒^′₁𝑓𝑒^′₂𝑓𝑒=𝐶¹𝑓𝑒₂𝑓, (21) where 𝐶=^𝑎1+𝑏_𝑐𝑑, is the coefficient matrix.

According to the Lyapunov stability theory and the Routh-Hurwitz criteria, if 𝑐()𝑎+𝑑<0,1+𝑏−𝑎𝑑<0, (22) then the eigen values of the coefficient matrix of error system (19) must be real or complex with negative real parts, and, hence, stable synchronized dynamics between systems (7) and (8) is guaranteed. Let 𝑎+𝑑=𝑐(1+𝑏)−𝑎𝑑=−𝐸, (23) where 𝐸>0 is a real number which is usually set equal to 1. There are several ways of choosing the constant elements 𝑎, 𝑏, 𝑐, 𝑑 of matrix 𝐷 in order to satisfy the Lyapunov stability theory and the Routh-Hurwitz criteria (19).

6. Numerical Simulation for Antisynchronization

For the constant elements of feedback matrix, choosing 𝑎=𝑑=−0.5 and for the parameters involved in system under investigation, 𝑒=0.15, 𝜔₀=0.3 with the initial conditions [𝑥₁(0),𝑦₁(0)]=[0,0] and [𝑥₂(0),𝑦₂(0)]=[1,1.5], we have simulated the system under consideration using mathematica. The results obtained show that the system under consideration achieved synchronization. Phase portraits, time series analysis, and error diagrams are the witness of achieving synchronization between master and slave system. Further, it also has been confirmed by the convergence of the synchronization quality defined by 𝑒𝑓=𝑒²₁𝑓+𝑒²₂𝑓. (24) Figures 7–12 confirms the convergence of the synchronization quality defined by (24).

Figure 7 

Phase plot of slave system.

Figure 8 

Time series analysis of slave system.

Figure 9 

Time series analysis of 𝑒1(𝑓).

Figure 10 

Time series analysis of 𝑒2(𝑓).

Figure 11 

Convergence of errors in synchronization.

Figure 12 

Convergence of errors in antisynchronization.

7. Conclusion

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 mathematica. For the errors in synchronization and antisynchronization behavior of the system under study, we have observed that the rate of convergence of errors is faster in antisynchronization. Figures 1–12 are proof of it.