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 16th 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 x0 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