Stability and Persistence of Synchronization in a System with a Diffusive-Time-Lag Coupling

We study synchronization in the framework of invariant manifold theory for systems with a time lag. Normal hyperbolicity and its persistence in infinite dimensional dynamical systems in Banach spaces is applied to give general results on synchronization and its stability.

Basic to the study of synchronization, two fundamental questions are of interest.The first is to do with the stability of the synchronization state of the system and the second is its robustness.Robustness of the synchronization state is its ability to be insensitive to small perturbations in the system that generates it.The two questions have been done for diffusively coupled systems without a time lag in the coupling.Normal hyperbolicity and the generalized Lyapunov exponents have been used to establish conditions for the stability and persistence of synchronization manifold for lattice dissipative systems each with a compact global attractor, see for example (Chow and Liu, 1997;Wasike, 2002;Wasike, 2003;Josica, K, 2000;Wasike and Rotich, 2007).
The subject of stability and persistence of synchronization in a system with a diffusive-time-lag coupling has received less attention.Grasman and Jansan (1979) have studied synchronization in oscillators coupled in only one variable with a time lag in which phase differences have been used to detect synchrony.Numerical results have also been used to determine the Lyapunov exponents in order to exhibit stability of the synchronization manifold, see for instance (Pyragas, 1998;Rossoni, 2005).To the best of my knowledge, no rigorous mathematical results in the framework of robustness of invariant manifolds for delay differential equations have been reported.This is how we approach this problem.We set the problem in the framework of dynamical systems and then consider the two aspects of synchronization based on invariant manifold theory for systems with a time lag.For the theory of invariant manifold for systems with a time lag, see for instance Halanay (1967), Kurzweil (1967).Normal hyperbolicity is exactly the right condition for persistence of invariant manifolds (see Bates et. al., 1998).In particular, we shall apply the theory of normally hyperbolic invariant manifolds for semiflows in Banach spaces as defined in Bates et al., (1998Bates et al., ( , 1999Bates et al., ( , 2000)).This approach will enable us to compare the rate of growth in the transversal direction to the synchronization manifold and that along the manifold.This is our road map.Most of the definitions and terminologies will be given in § 2 while in § 3 we present the main results on the robustness of a synchronization manifold where normal and tangential growth rates are compared.In § 4 we give an example of an all-to-all coupled system.§5 is the conclusion.

Let
) be the space of continuous functions from [ ] , 0 r − to endowed with the usual supremum topology.Consider the following system where are continuous functions defined on X to and , respectively. .
and assume all solutions are uniquely defined for , then is a -semigroup on X Suppose that the system in equation ( 1) is dissipative, then there exists a global attractor .

A
The following definition is motivated by the works of Halanay (1967) and also Kurzweil (1967).In many applications, we are interested in local synchronization.We are interested in the local attractivity of graph (H); that is, the graph of (H) is exponentially attracting in the neighbourhood of D,

Definition. System
where , k α are positive constants.

Definition. Suppose equation (1) is locally synchronized with map H. The synchronization is stable if for any
is locally synchronized with map and H H − <∈ see (Bates et .al ;1998 p. 119).

Main Results: Robustness of a synchronized manifold
Let us recall some of the invariant manifold theory for infinite dimensional systems in a Banach space.Let X represent a Banach space with norm ⋅ ⋅ In subspaces the same norm symbol is used.The notation ⋅ ⋅ will be reserved for the linear operator norm where means the domain of the operator L.
( ) For any , X ϕ ∈ there is a unique solution ( ) , z t ϕ to equation (1) that defines a semiflow on X ; that is, and for each is and 0, : where ( , z t ) ϕ is the solution of Equation ( 1) with ( )  such that for all and For this definition see Bates et al., (1998Bates et al., ( , 1999)).A few remarks on the above definition of normal hyperbolicity are useful.

Remark. Inequality (4) suggests that near contracts in the direction of ,
X , and at a rate greater than that on M. Furthermore (4) suggests that for an invariant splitting to persist we need both to be satisfied, see for instance Bates et al. (1998, pages 124 and 11, respectively).

Generalized Lyapunov-type numbers
For the purpose of calculation it will prove convenient to phrase inequalities ( 5) and ( 6) more quantitatively in terms of rates of growth.By the equivalence of norms, .
Thus inequality (6) reduces to ( ) Let us now compare the growth bounds of the two semigroup operators.The growth bound of the linear operator is a real number ( ) ≥ This is the same as see for instance Diekmann et al (1995, p. 470).We also know that the   ( ) By the variation-of-constants formula, the solution ( , w t ) ϕ to the first equation in equation ( 18) is given by

Conclusion
Time delay in the coupling does not always destabilize synchronization states of similar systems with delay 0. r = example, in coupled identical systems or any system symmetric with respect to D, the synchronization occurs for H = I.In this case, the diagonal in X that consists of the set of functions ( set for equation (1).
System (1), there exists a smooth connected invariant manifold M = graph that is, for each Consider the linearization of equation (1) along Letbe the linearized semiflow associated with equation (3).

X
The tangent bundle of X restricted to M splits into two continuous subbundles , is the tangent bundle of M and s X is transversal to c X .The superscripts c and s stand for "center", and "stable", respectively.(H2)This splitting is invariant under that is, for each ; fundamental matrix solution of the first equation in equation (18); that is, ( ) to the left of the complex plane(see for instanceBose, 1986).Equation (19) gives .
(1) is synchronized for y with respect to x, if there exists a map m C C r = −with m the dimension of y such that the graph of H denoted graph (H), is invariant and globally attracting.