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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.

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 [

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

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

where

M. D. Bernardo [

Theorem 1. [

In Theorem 1,

Assumption 1: The projection of

Under assumption 1, (4) becomes

We wish, now, to choose an appropriate function

Theorem 2. [

where

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

Assumption 2:

To exploit the fact that the system

It then follows by assumption 2 that there exists

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

Theorem 3. [

With B(1) denoting a closed ball of radius one in

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

Proof: Note that

Let

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

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

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

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

Now the derivative

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

where

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

where

is negative definite for certain values of

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

For the parameter values

Now, consider the Lorenz chaotic system

as a drive system and the response system given by

where

Now using

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

Proof: Using lemma 1 we take control functions as

We choose Lyapunov function

where

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

and

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

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