The notion of synchronization is well known from the viewpoint of classical mechanics since early 16^th century. Since then, many other examples have been reported in the literature. However, the possibility of synchronizing chaotic systems is not so intuitive, since these systems are very sensitive to small perturbations on the initial conditions and, therefore, close orbits of the system quickly become un-correlated. Surprisingly, in 1990 it was shown that certain subsystems of chaotic systems can be synchronized by linking them with common signals [1] . In particular, the author reported the synchronization of two identical (i.e., two copies of the same system with the same parameter values) chaotic systems. They also show that, as the differences between those system parameters increase, synchronization is lost. Subsequent works showed that synchronization of non-identical chaotic systems is also possible. Many fundamental characteristics can be found in a chaotic system, such as excessive sensitivity to initial conditions, broad spectrum of Fourier transform, and fractal properties of the motion in phase space. Due to its powerful applications, both control and synchronization problems have extensively been studied in the past decades for chaotic systems such as Lorenz system [2] -[5] , Chua’s system [6] , Rossler system [7] , Chen system [8] , Lu system etc. [9] [10] . In fact, the state trajectories of chaotic systems evolve in a strange attractor. In addition, there are several control methods for chaotic systems which have extensively been studied in the literature, such as linear state error [11] , impulsive control [12] , adaptive control [13] , fuzzy model [14] , sliding mode control design [15] [16] , Robust chaos suppression [17] , etc.

However, some noises or disturbances always exist in the physical systems that may cause systems instability and thereby destroying the stability performance. Therefore, the problem that how to reduce the effect of the noise or disturbances in chaotic systems becomes an important issue.

This paper can be summarized as follows: In the next section, we introduce notion of chaos synchronization and propose some theorems for chaos synchronization based upon the analysis of nonlinear dynamical systems. In Section 3, we explain proposed theorems in terms of chaotic Lorenz system. Numerical simulations are given to verify proposed theorems in Section 4.

Given two systems

with The problem consists of choosing an appropriate controller in such a way as to have

Equations (1), (2) and (3) can describe both the problems of controlling and synchronizing a chaotic system. If the reference (2) is chosen to a chaotic system, identical to (1), starting from different initial condition, (3) describes a synchronization problem while if the reference model evolves along a periodic orbit a chaos control problem is described.

To explain the synchronization of (1) and (2), the error equation is formed

An orthogonal projection operator is found so that (4) could be rewritten as

where is the projection of on the complementary space of Im(B), which is assumed to be linear, and, are the projection on Im(B) of and respectively. Our aim is to find such that (1) synchronizes with (2) and the error equation stabilize and become regular.

M. D. Bernardo [18] proposed a modified adaptive approach in which for a given matrix such that is a Hurwitz matrix, that is all its eigenvalues are in the left half of the complex plane, we solve the Lyapunov equation

Theorem 1. [18] Let be the positive definite solution of (6) and let. The con- troller guarantees that for every initial condition

: 1), 2).

In Theorem 1, is adaptively estimated according to the law where is a continuous function such that,. We can decompose as .

Assumption 1: The projection of on the complementary space of Im(B) is linear, that is for some linear matrix L:

Under assumption 1, (4) becomes

We wish, now, to choose an appropriate function in order to solve the problem stated in (1). If we recall one of the main properties of feedback K, namely, the feedback linearization, we can try to achieve the control by linearizing the systems involved via a combined feedback plus feedforward action. Under assumption 1 we can trivially prove that:

Theorem 2. [19] If (L, B) is stabilizable then with

where is such that all eigenvalues of (L - BK) are in the left hand side of the complex plane will ensure (3).

It is difficult to provide the controller with a perfect knowledge of the function g but instead assume only that its projection l is bounded by a known continuous function in the following sense:

Assumption 2:, With continuous.

To exploit the fact that the system is a chaotic system evolving either on a strange attractor or on a periodic orbit/equilibrium point, to deduce that its solution must be bounded for all t. hence, for some known M, we assume

It then follows by assumption 2 that there exists such that

The idea is to exploit this property of the reference model, in order to achieve the control. In so doing so we consider a controller of the form

Hence we still have a linear term −ke and a feedback linearization term, but the feed forward, responsible of the compensation of has disappeared. What we have now is a discontinuous term depending upon the known bound W, which will dominate the nonlinearity of the reference model and therefore should guarantee the desired goal. In fact, the hypothesis of Theorem 2 and (10) yield the following.

Theorem 3. [19] Let be the positive definite matrix solution of the Lyapunov equation with.Define the set valued mapping such that

With B(1) denoting a closed ball of radius one in, and embed the feedback controlled error system (8) in the differential equation inclusion

Then the origin is asymptotical stable for the error system (8).

Proof: Note that is upper semi continuous on, with non-empty, convex and compact value. Therefore, for each, the system (8) with has a maximal solution. Let. Obviously:, , and for all

Let be a maximal solution, then, for almost all t and we may deduce that the error decays asymptotically to zero. Thus the proof of the Theorem 3 is complete.

Assumption 3: In order to find a sufficient synchronization criterion the following assumption on the drive system is needed. This assumption is in the light of the drive system being free and chaotic and based on a well-known fact that chaotic attractors are bounded in phase space

For any bounded initial state x₀ within the defined domain of the drive system, there exist some finite real constants such that

Keeping in view these facts we obtain the following theorem followed by a lemma which seems to be very significant to develop the subject of chaos theory.

Lemma 1: If system (1) involves r chaos terms in its dynamics, then control vector can be chosen such that

,

,

and,

Theorem 4. System (1) will synchronize with response (2) if control gain matrix B is chosen such that error dynamics of drive-response given by

is asymptotically stable provided the choice of positive Lyapunov function for the stabilization of (14) imply the derivative is negative definite.

Proof: Using lemma 3.1 Equation (14) can be re-written as

Now choosing the positive definite Lyapunov function

Equation (15) in matrix notation can be written as

where

is positive definite for certain values of and.

Now the derivative is given by

On solving Equation (17) using (14) we find a polynomial of the following form

where, and are functions of t only.

By excluding less important terms we get desired negative definite polynomial i.e.

where

is negative definite for certain values of and. Proof of the Theorem 4 is complete.

The Lorenz system described by the following system of non linear differential equations

For the parameter values has a chaotic attractor portrayed in Figure 1.

Now, consider the Lorenz chaotic system

as a drive system and the response system given by

where are unknown feedbacks controlling functions.

Now using, , and, , , error system becomes

Now we can re-state Theorem4 for Lorenz system as follows:

Theorem 5. System (20) will synchronize with response (21) if control gain matrix B is chosen such that error dynamics of drive-response given by (22) is asymptotically stable provided the choice of positive Lyapunov function for the stabilization of (22) imply the derivative is negative definite.

Proof: Using lemma 1 we take control functions as, ,.

We choose Lyapunov function

where. Differentiating (23) with respect to t, we obtain

Equation (24) is negative definite if the matrix

is negative definite. The following conditions must hold

1)

2)

3).

Proof of the Theorem 4 is complete.

In this section we verify the control laws presented in the previous sections via numerical simulations. Keeping

, , , , fixed and choosing, , according to proposed theorem one can obtain a number of values for which Lorenz system synchronized. For example if we choose a particular approximation for which, , and, , ,

and, , , for the chaotic drive and response Lorenz systems and, , , , , , for the synchronized drive and response Lorenz systems then we can see an excellent agreement with the proposed theorems. Figures 2(a)-(d) are the trajectories of chaotic Lorenz system. Figures 2(e)-(h) are the synchronization between two identical Lorenz systems.

We analyze chaotic dynamics of nonlinear systems and study chaos suppression. We extend our study up to chaos synchronization by discussing Pecora-Carroll and other methods available in literature. We proposed a theorem in general and another one for a particular case of Lorenz system. Numerical simulations are given to

verify the proposed criterion. In a particular approximation for which, , and

, , and, , , for the chaotic drive and response Lorenz systems and, , , , , , for the synchronized drive and response Lorenz systems then we show an excellent agreement with the proposed theorems.

AyubKhan,PrempalSingh, (2015) Chaos Synchronization in Lorenz System. Applied Mathematics,06,1864-1872. doi: 10.4236/am.2015.611164