Chaos Synchronization in Lorenz System

In this paper, we analyze chaotic dynamics of nonlinear systems and study chaos synchronization of Lorenz system. We extend our study by discussing other methods available in literature. We propose a theorem followed by a lemma in general and another one for a particular case of Lorenz system. Numerical simulations are given to verify the proposed theorems.


Introduction
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.
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 In Theorem 1, ( ) k t is adaptively estimated according to the law ( ) ( ) ( ) The projection of f g − on the complementary space of Im(B) is linear, that is for some linear matrix L: Under assumption 1, (4) becomes

e t Le t B h x t l y t u t h B B B f l B B B g
We wish, now, to choose an appropriate function ( ) u t in order to solve the problem stated in (1).If we re- call 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: 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 γ in the following sense: 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 W R 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 ( ) , h x t , but the feed forward, re- sponsible of the compensation of ( ) , f y t has disappeared.What we have now is a discontinuous term de- pending 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.
With B(1) denoting a closed ball of radius one in m R , and embed the feedback controlled error system (8) in the differential equation inclusion , .

e t Le t B l y t u t u t D e E e t
Then the origin is asymptotical stable for the error system (8).
Proof: Note that ( ) . e 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 x 0 within the defined domain of the drive system, there exist some finite real constants i i x x t M i n t ≤ = ∀ ≥  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 ( ) ( ) and [ ] ii k R i r ∈ ∀ =  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

, , , e t Ae t f e M t u t u t Bu t
is asymptotically stable provided the choice of positive Lyapunov function ( ) T , V e t e Pe = for the stabilization of ( 14) imply the derivative ( ) T , V e t e Qe =  is negative definite.Proof: Using lemma 3.1 Equation ( 14) can be re-written as , V e t e t e t e t α δ ( ) 15) in matrix notation can be written as where Now the derivative ( )

V e t e t e t e t e t e t e t
On solving Equation ( 17) using ( 14) we find a polynomial of the following form where is negative definite for certain values of i β and ( ) Proof of the Theorem 4 is complete.

Synchronization of Two Lorenz Systems
The Lorenz system described by the following system of non linear differential equations as a drive system and the response system given by 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

has a chaotic attractor portrayed in Figure 1 .
Now, consider the Lorenz chaotic system ) is asymptotically stable provided the choice of positive Lyapunov function ( ) T , V e t e Pe = for the stabilization of (22) imply the derivative ( ) T , V e t e Qe =  is negative definite.Proof: Using lemma 1 we take control functions as