Abstract: This study addresses the synchronization and adaptive synchronization problem of a hyperchaotic dynamical system with unknown system parameter. This technique is applied to achieve synchronization for hyperchaotic Lu system. Lyapunove direct method of stability is used to prove the asymptotic stability of solutions for the error dynamical system. Numerical simulations results are used to demonstrate the effectiveness of the proposed control strategy.
INTRODUCTION
In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under specific conditions exhibit dynamics that are sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, the behavior of chaotic systems appears to be random, because of an exponential growth of errors in the initial conditions. This happens even though these systems are deterministic in the sense that their future dynamics are well defined by their initial conditions and there are no random elements involved. This behavior is known as deterministic chaos, or simply chaos.
Chaotic behavior has been observed in the laboratory in a variety of systems including electrical circuits, lasers, oscillating chemical reactions, fluid dynamics and mechanical and magneto-mechanical devices. Observations of chaotic behaviour in nature include the dynamics of satellites in the solar system, the time evolution of the magnetic field of celestial bodies, population growth in ecology, the dynamics of the action potentials in neurons and molecular vibrations. Everyday examples of chaotic systems include weather and climate (Sneyers, 1998). There is some controversy over the existence of chaotic dynamics in the plate tectonics and in economics (Serletis and Gogas, 1997, 1999, 2000).
In recent years, researches on chaos control and synchronization have attracted increasing attention due to its potential applications to physics, chemical reactors, control theories, biological networks, artificial neural networks and secure communication (Ott et al., 1990; Pyragas, 1992; Tao et al., 2005; Wang and Tian, 2004).
Chaos synchronization has been observed in various fields. Fujisaka and Yamada (1983) showed criterion of chaos synchronization using Lyapunov exponents. Since Pecora and Carroll (1990) proposed a synthesis method for synchronized chaotic systems, many methods have been proposed and its applications in chaos communication provide very fascinating studies (Pecora and Carroll, 1990).
Synchronization of chaos is a phenomenon that may occur when two, or more, chaotic oscillators are coupled, or when a chaotic oscillator drives another chaotic oscillator. Because of the butterfly effect, which causes the exponential divergence of the trajectories of two identical chaotic system started with nearly the same initial conditions, having two chaotic system evolving in synchrony might appear quite surprising. However, synchronization of coupled or driven chaotic oscillators is a phenomenon well established experimentally and reasonably understood theoretically.
It has been found that chaos synchronization is quite a rich phenomenon that may present a variety of forms. When two chaotic oscillators are considered, these include: identical synchronization, generalized synchronization, phase synchronization, anticipated and lag synchronization and amplitude envelope synchronization. All these forms of synchronization share the property of asymptotic stability. This means that once the synchronized state has been reached, the effect of a small perturbation that destroys synchronization is rapidly damped and synchronization is recovered again. Mathematically, asymptotic stability is characterized by a positive Lyapunov exponent of the system composed of the two oscillators, which becomes negative when chaotic synchronization is achieved.
Hyperchaotic systems have received much attention in recent years, particularly the hyperchaotic Rossler attractors and its variation, which are obtained by introducing a quadratic term to a linear system (Rossler, 1979a; Liao and Huang, 1999), or by using piecewise-linear systems (Matsumat et al., 1986; Tsubone and Saito, 1998). Owing to their strong resistance to dynamics reconstruction, hyperchaotic systems are more suitable for some special engineering applications such as chaos-based encryption and secure communication.
Hyperchaotic systems is usually classified as a chaotic system with more than one positive Lyapunov exponent, indicating that the chaotic dynamics of the system are expanded in more than one direction giving rise to a more complex attractor. In recent years, hyperchaos has been studied with increasing interests, in the fields of secure communication (Udaltsov et al., 2003), multimode lasers (Shahverdiey et al., 2004), nonlinear circuits (Barbara and Silvano, 2002), biological networks (Neiman et al., 1999), coupled map lattices (Zhan and Yang, 2000) and so on.
Since the discovery of the hyperchaotic Rossler (1979b) system, many hyperchaotic systems have been developed such as the hyperchaotic MCK circuit (Matsumot et al., 1986), the hyperchaotic Chen system (Li et al., 2005; Yan, 2005), hyperchaotic Lü system (Elabbasy et al., 2006), etc.
First we need to recall some concepts and terms from synchronization theory.
Consider the systems of differential equation:
| (1) |
and
| (2) |
where, x ∈ Rn, y ∈Rn, f, g: Rn → Rn are assumed to be analytic function
Let x(t, x0) and y(t, y0) be solutions to (1) and (2), respectively. The solutions x(t, x0) and y(t, y0) are said that are synchronized if
SYSTEM DESCRIPTION
In this study we study the synchronization of the hyperchaotic Lü system (Elabbasy et al., 2006)
| (3) |
where, a, b, c and r are four unknown uncertain parameters. This new system exhibits a chaotic attractor at the parameter values a = 15, b = 5, c = 10 and r = 1 (Fig. 1).
The divergence of the flow Eq. 3 is given by:
where,
Hence the system is dissipative when: c<a+b+c
The system has three equilibrium points:
To study the stability of E0 the associated Jacobian J0 is
The characteristic polynomial of the matrix J0 is given by
| (4) |
The eigenvalues are λ1 = -a, λ2 = c, λ3 = -b and λ4 = -r. Then the equilibrium point E0 is stable if c < 0 other with the equilibrium is unstable.
To study the stability of E+ the associated Jacobian J+ is
The characteristic polynomial of the matrix J+ is given by
| Fig. 1a: | Chaotic attractor of hyperchaotic Lü system at a = 15, b = 5, c = 10 and r = 1 in x, y, z subspace |
| Fig. 1b: | Time responses for the variable w(t) of the hyperchaotic Lü system |
| (5) |
where:
A set of necessary and sufficient conditions for all the roots of Eq. 5 to have negative real parts is given by the well-known Routh-Hurwitz criterion in the following form
i.e.,
However, the above values of c1, c4 and c3 guaranteed that c1c2-c3<0. Hence the equilibrium point E+ is unstable.
To study the stability of E_ the associated Jacobian J_ is
The characteristic polynomial of the matrix J_ is given by
| (6) |
where:
As above, one can see that E_ is also unstable since c1c2-c3 will be negative.
SYNCHRONIZATION OF HYPERCHAOTIC Lü SYSTEM
We study the synchronization problem of the familiar hyperchaotic Lü system using the method proposed by Pecora and Carroll (1990) and Carrol and Pecora (1991), where, a stable subsystem of a chaotic system is synchronized with a separate chaotic subsystem under suitable conditions. This method has been further extended to cascading chaos synchronization with multiple stable subsystem.
The drive system is:
| (7) |
Here, since the (x1, z1, w1) subsystem is stable for all values of a, b and r, in which the conditional lyapunov exponents are negative. Then we will use y1 to be drive the (x2, y2, z2) subsystem of the response system:
| (8) |
and the difference system for:
| (9) |
then the error dynamical system is given by:
| (10) |
The solution of system Eq. 10 is given by:
| (11) |
where, α1, α2 and α3 are constants of integration.
Then
| (12) |
and then the response system with y-derive configuration does synchronize.
Numerical Results
We have verified that when applying the synchronization method of Pecora
and Carroll (1990) of the hyperchaotic Lü system using only y(t) as the
drive the stability condition can be satisfied while a = 15, b = 5, c = 10 and
r = 1. By using Fourth-order Runge-Kutta method with time step size 0.001. The
initial states of the drive system are
| Fig. 2: | Solutions of the drive and response systems with Pecora and Carroll method, (a) signals x1 and x2, (b) signals z1 and z2 and © signals w1 and w2 |
| Fig. 2d: | Behaviour of the trajectories ex, ey and ez of the error system tends to zero as t tends to 3 |
ADAPTIVE IDENTICAL SYNCHRONIZATION
In order to observe the adaptive synchronization behaviour in hyperchaotic Lü system, we have two identical hyperchaotic Lü systems where the drive system with four state variables denoted by the subscript 1 drives the response system having identical equations denoted by the subscript 2. However, the initial condition of the drive system is different from that of the response system, therefore two hyperchaotic Lü systems are described, respectively, by the following equations:
| (13) |
and
| (14) |
We have introduced four control inputs, u1(t), u2(t), u3(t) and u4(t) in Eq. 14, u1(t), u2(t), u3(t) and u4(t), are to be determined for the purpose of synchronizing the two identical hyperchaotic Lü systems with the same but unknown parameters a, b, c and r in spite of the differences in initial conditions.
Remark 1: The hyperchaotic Lü system is dissipative system and has a bounded, zero volume, globally attracting set. Therefore, the state trajectories x1(t), y1(t), z1(t) and w1(t) are globally bounded for all t = 0 and continuously differentiable with respect to time. Consequently, there exist three positive constants s1, s2, s3 and s4 such that:
hold for all t = 0.
Let us define the state errors between the response system that is to be controlled and the controlling drive system as:
Then the error dynamical system can be written as:
| (16) |
Then the synchronization problem is now replaced by the equivalent problem of stabilizing the system Eq. 16 using a suitable choice of the control laws u1(t), u2(t), u3(t) and u4(t). Let us now discuss the following one case of control input u2(t).
The state variable y1 of the drive system is coupled to the second equation of the response system and an external control with the state y2 as the feedback variable is also introduced into the second Eq. in 16. Therefore, the feedback control law is described as:
| (17) |
where,
| (18) |
Then the resulting error dynamical system can be expressed by:
| (19) |
Consider a Lyapunov function as follows:
| (20) |
where,
| (21) |
If we choose
Where, s2 and s3 are defined in remark 1. If
Numerical Experiment
Fourth-order Runge-Kutta method is used to solve differential equations.
A time step size 0.001 is employed. The three parameters are chosen as a = 15,
b = 5, c = 10 and r = 1 in all simulations so that the hyperchaotic Lü
system exhibits a chaotic behaviour if no control is applied. The initial states
of the drive system are x1(0) = -20, y1(0) = 5, z1(0)
= 0 and w1(0) = 15 and of the response system are x2(0)
= 10, y2(0) = -5, z2(0) = 5 and w2(0) = 10.
Then ex(0) = 30, ey(0) = -10, ez(0) = 5 and
ew(0) = -5. In this case, we assume that the drive system and the
response system are two identical hyperchaotic Lü system with different
initial conditions. The evolutions of state synchronization errors and the history
of the estimated feedback gain using the feedback control law (17) associated
with the adaptation algorithm (18). These numerical results demonstrate the
systems have been asymptotically synchronized using the proposed adaptive schemes
(Fig. 3).
| Fig. 3a: | Behaviour of the trajectory ex of the error system tends to zero as t tends to 2 when the parameter values are a = 15, b = 5, c = 10 and r = 1 |
| Fig. 3b: | Behaviour of the trajectory ez of the error system tends to zero as t tends to 2 when the parameter values are a = 15, b = 5, c = 10 and r = 1 |
| Fig. 3c: | Behaviour of the trajectory ez of the error system tends to zero as t tends to 2 when the parameter values are a = 15, b = 5, c = 10 and r = 1 |
| Fig. 3d: | Behaviour of the trajectory ew of the error system tends to zero as t tends to 8 when the parameter values are a = 15, b = 5, c = 10 and r = 1 |
CONCLUSION
In this study synchronization and adaptive synchronization using uncertain parameters of the hyperchaotic Lü system is demonstrated. The Pecora and Carroll method has been applied to achieve the synchronization of the hyperchaotic Lü system. All results are proved by using Lyapunov direct method. The proposed scheme is efficient in achieving simple synchronization in our example and can be applied to similar chaotic systems.