Synchronization of chaotic system with uncertain variable parameters by linear coupling and pragmatical adaptive tracking

Abstract

We study the synchronization of general chaotic systems which satisfy the Lipschitz condition only, with uncertain variable parameters by linear coupling and pragmatical adaptive tracking. The uncertain parameters of a system vary with time due to aging, environment, and disturbances. A sufficient condition is given for the asymptotical stability of common zero solution of error dynamics and parameter update dynamics by the Ge–Yu–Chen pragmatical asymptotical stability theorem based on equal probability assumption. Numerical results are studied for a Lorenz system and a quantum cellular neural network oscillator to show the effectiveness of the proposed synchronization strategy.

Appendix: GYC pragmatical asymptotical stability theorem

The stability for many problems in real dynamical systems is actual asymptotical stability, although may not be mathematical asymptotical stability. The mathematical asymptotical stability demands that trajectories from all initial states in the neighborhood of zero solution must approach the origin as t→∞. If there are only a small part or even a few of the initial states from which the trajectories do not approach the origin as t→∞, the zero solution is not mathematically asymptotically stable. However, when the probability of occurrence of an event is zero, it means the event does not occur actually. If the probability of occurrence of the event that the trajectories from the initial states are that they do not approach zero when t→∞, i.e., these trajectories are not asymptotical stale for the zero solution is zero, the stability of the zero solution is actual asymptotical stability though it is not mathematical asymptotical stability. In order to analyze the asymptotical stability of the equilibrium point of such systems, the GYC pragmatical asymptotical stability theorem is used.

Let X and Y be two manifolds of dimensions m and n (m<n), respectively, and φ be a differentiable map from X to Y; then φ(X) is subset of Lebesque measure 0 of Y [22]. For an autonomous system

$$ \dot{x} = f(x_{1}, \ldots, x_{n}) $$

(67)

where x=[x ₁,…,x _n ]^T is a state vector, the function f=[f ₁,…,f _n ]^Tis defined on D⊂R ⁿ, an n-manifold.

Let x=0 be an equilibrium point for the system (67). Then

$$ f(0) = 0 $$

(68)

For a nonautonomous system,

$$ \dot{x} = f(x_{1}, \ldots, x_{n + 1}) $$

(69)

where x=[x ₁,…,x _n+1]^T, the function f=[f ₁,…,f _n ]^T is defined on D⊂R ⁿ×R ₊, here t=x _n+1⊂R ₊. The equilibrium point is

$$ f(0, x_{n + 1}) = 0 $$

(70)

Definition

The equilibrium point for the system is pragmatically asymptotically stable provided that with initial points on C which is a subset of Lebesque measure 0 of D, the behaviors of the corresponding trajectories cannot be determined, while with initial points on D−C, the corresponding trajectories behave as that agree with traditional asymptotical stability [19, 20].

Theorem

Let V=[x ₁,x ₂,…,x _n ]^T:D→R ₊ be positive definite and analytic on D, where x ₁,x ₂,…,x _n are all space coordinates such that the derivative of V through Eqs. (67) or (69), $\dot{V}$, is negative semidefinite of [x ₁,x ₂,…,x _n ]^T.

For the autonomous system, let X be the m-manifold consisting of the point set for which ∀x≠0, $\dot{V}(x) = 0$ and D is a n-manifold. If m+1<n, then the equilibrium point of the system is pragmatically asymptotically stable.

For the nonautonomous system, let X be the m+1-manifold consisting of the point set for which ∀x≠0, $\dot{V}(x_{1}, x_{2}, \ldots, x_{n}) = 0$ and D is an n+1-manifold. If m+1+1<n+1, i.e., m+1<n, then the equilibrium point of the system is pragmatically asymptotically stable. Therefore, for both the autonomous and nonautonomous system, the formula m+1<n is universal. So, the following proof is only for the autonomous system. The proof for the nonautonomous system is similar.

Proof

Since every point of X can be passed by a trajectory of Eq. (67), which is one-dimensional, the collection of these trajectories, C, is a (m+1)-manifold [16, 17].

If m+1<n, then the collection C is a subset of Lebesque measure 0 of D. By the above definition, the equilibrium point of the system is pragmatically asymptotically stable.

If an initial point is ergodicly chosen in D, the probability of that the initial point falls on the collection C is zero. Here, equal probability is assumed for every point chosen as an initial point in the neighborhood of the equilibrium point. Hence, the event that the initial point is chosen from collection C does not occur actually. Therefore, under the equal probability assumption, pragmatical asymptotical stability becomes actual asymptotical stability. When the initial point falls on D−C, $\dot{V}(x) < 0$, the corresponding trajectories behave as that agree with traditional asymptotical stability because by the existence and uniqueness of the solution of initial-value problem, these trajectories never meet C.

For Eq. (7), the Lyapunov function is a positive definite function of n variables, i.e., p error state variables and n−p=m differences between unknown and estimated parameters, while $\dot{V} = e^{T}Ce$ is a negative semidefinite function of n variables. Since the number of error state variables is always more than one, p>1, m+1<n is always satisfied, by pragmatical asymptotical stability theorem we have

$$ \lim_{t \to \infty} e = 0 $$

(71)

and the estimated parameters approach the uncertain parameters. The pragmatical adaptive control theorem is obtained. Therefore, the equilibrium point of the system is pragmatically asymptotically stable. Under the equal probability assumption, it is actually asymptotically stable for both error state variables and parameter variables. □