The research concerning identical synchronization (IS) led to a new concept called generalized synchronization (GS), which is broader in its scope and includes the IS as a special case [2], [65]. While IS is concerned with coupled identical systems, GS focuses mainly on the coupling between systems with different dynamical properties. As a consequence, if GS occurs, the relationship between the state variables of the drive and the response system is not as simple as y = x anymore. Some of the concepts of IS can be transfered directly to GS. However, as a more general concept GS features many new aspects of synchronization.
In the simplest case GS involves a synchronization manifold that is a non-linear functional relationship y = Ψ(x) between drive and response states.
Depending on the systems the function Ψ(·) can either be only slightly non-linear or it can be very complex in its shape. It can either be smooth or even non-differentiable. However, in some cases the relationship between drive and response states is multi-valued, making it impossible to define any function Ψ(·) at all. Taking these difficulties into account, it is by no means a trivial task to identify GS for two unidirectional coupled systems. Two main detec-tion methods have been established, which differ in their approach. One is based on the detection of a functional relationship between drive and response system. The other method investigates the behavior of replica systems and is related to conditional Lyapunov exponents. Since the definition of GS depends on the detection method, the two approaches lead to concepts of GS that are not quite congruent [56], [54].
4.3.1 Definition I
Consider two unidirectionally coupled dynamical systems
˙
x = F(x),
˙
y = G(y,h(x)), (4.21)
with the state vectors x ∈ Rm and y ∈ Rn and the vector fields F and G. The first system is driving the second one with the coupling signal h(x).
For this coupling the definition of GS follows Definition(4.1) of IS except for the characterization of the synchronization manifold M. While for IS the manifold is simply a plane through the originy =x, the manifold for GS can be arbitrarily shaped
M={(x,y)∈Rm×Rn|y=Ψ(x)}, (4.22) with the Ψ : Rm → Rn representing the functional relationship between the state variables x and y. IS is included in the concept of GS as a special case where Ψ(·) =id(·).
A
x
y response
x state space
y
B
M
drive
Figure 4.6: Generalized synchronization occurs if two (possibly) nonidentical dynamical systems are coupled in an appropriate way. In the mutual state space the dynamics is then restricted to the synchronization manifold M, which can be an arbitrarily shaped hyperplane. The attractor Aof the coupled system lies inside M and is surrounded by a basin of attractionB.
Definition 4.2 In the unidirectional coupling scheme the drive system and the response system
˙
x = F(x),
˙
y = G(y,h(x)), (4.23)
with x∈ Rm,y∈ Rn, F : Rm → Rm, and h: Rm → Rk, have the property of generalized synchronization iff the two following conditions are met:
(i) There exists an attracting synchronization set
M={(x,y)∈Rm×Rn|y=Ψ(x)}, (4.24) containing the attractor A of the combined system in the mutual state space
A ⊂ M ⊂Rm×Rm. (4.25) (ii) There exists a basin of attraction B around the attractor A ⊂ B such
that
∀(x0,y0)∈Rm×Rn :
(x(t= 0),y(t= 0)) = (x0,y0)∈ B ⇒ lim
t→∞ky(t)−Ψ(x(t))k= 0. (4.26) The definition above was used for example in [42] and [35] and explicitly formulated in [56]. Note that it makes no statement about the properties of Ψ(·). Depending on the investigated drive and response systems the function may be smooth or even non-differentiable at all [63], [35]. For detecting GS in the above sense nearest neighbors statistics were suggested in [65] and [59].
These methods use the fact that for synchronized systems points in a small
response replica
response response
drive drive
response copy
Figure 4.7: A replica system (also auxiliary system) is simply an exact copy of the response system. During an application the original system and the replica are driven by the same driving signal.
ε-region in the state space of the drive system are mapped byΨ(·) to points in the state space of the response system which lie also very near to each other. In case that no synchronization is established, these points lie arbitrarily scattered in the state space of the response system.
4.3.2 Definition II
Another method for detecting GS is the application of a so called auxiliary or replica system as described in [1] (see Fig. 4.7). The replica system is an exact copy of the response system
˙
y′ =G(y′,h(x)). (4.27) The purpose of the replica is to test whether the response system retains any degrees of freedom or whether its dynamics is totally determined by the driving signal. For the former case the replica system y′ and the original response system y behave differently even when driven by the same signal h(x). In the latter case, when GS is established, both systems asymptotically behave in the same way. That means in spite of possibly different initial conditions their trajectories assimilate in the course of time
t→∞lim ky(t)−y′(t)k= 0. (4.28) This behavior can be interpreted as the response system forgetting its initial state. Following [56] the second definition for GS can be written as
Definition 4.3 In the unidirectional coupling scheme the drive system and the response system
˙
x = F(x),
˙
y = G(y,h(x)), (4.29) with x∈ Rm,y ∈Rn, F : Rm → Rm, and h: Rm → Rk, have the property of generalized synchronization iff the following condition is met:
(i) There exists an open basin of attraction B ⊂Rm×Rn such that
∀(x0,y0),(x0,y′0)∈ B ⇒ lim
t→∞ky(t)−y′(t)k= 0. (4.30) The second definition of GS is more general than the first one because it includes also the case where there is no functional relationship between the states of the drive and the response system due to multivaluedness.