In this section we study the synchronization between two coupled nonlinear dynamical systems. The presentation is adapted from [Wu and Chua 1994].
Synchronization is closely related to notions of stability and controllability in systems theory. To this end, let us first recall the following definitions of stability and asymptotic stability.
Definition 2
The system dx/dt = f(x,t) or x(t+1) = f(x(t),t) is uniform stable if for all ε > 0 there exists δ(ε) > 0 such that for all t ≥ t0
||x(t,x0,t0)- x(t,x1,t0)|| < ε whenever ||x0-x1||< δ(ε).
Definition 3
The system dx/dt = f(x,t) or x(t+1) = f(x(t),t) is uniform asymptotically stable if it is uniform stable and there exists δ > 0 such that for every ε > 0 there exists T(ε) ≥ 0 such that given any two initial conditions x(t0), y(t0) with ||x(t0)-y(t0)|| < δ, the corresponding trajectories satisfies ||x(t)-y(t)|| ≤ ε for all t ≥ t0+T(ε).
Definition 3 implies that ||x(t)-y(t)|| → 0 as t→∞. These definitions of stability are taken from [Yoshizawa 1966], but there are many other definitions of stability such as Lyapunov stability, absolute stability [Vidyasagar 1993] etc. with some of them requiring a boundedness condition on the trajectories as well. Note that Definitions 2 and 3 do not specify whether the trajectories are bounded or not. As in Definition 1 in the introduction, these definitions are global and refers to global stability, but local version can be easily defined as well. Definition 3 along with a boundedness condition on the trajectories coincides with the notion of unique steady state in circuit theory [Chua and Green 1976]; the condition that the asymptotic behavior of a circuit does not depend on the initial conditions (e.g. the initial voltages and the initial currents through the capacitors and inductors respectively).
The right hand side f of the system dx/dt = f(x,t) depends only on the state x and time t. We introduce a dependence of f on external input η and can also define stability with respect to the external input η.
Definition 4 ([Wu and Chua, 1994])
The system dx/dt = f(x, η(t), t) or x(t+1) = f(x(t),η(t),t) is uniform asymptotically stable and i-uniformly with respect to all η(t) if it is uniform asymptotically stable for all fixed η(t) and the constants δ and T(ε) in Definitions 2 and 3 can be chosen to be independent of η(t).
Definition 5
A function f:Rn → Rn is increasing if for all x, y ∈ Rn, (x-y)T(f(x)-f(y)) ≥ 0.
A function f:Rn → Rn is strictly increasing if for all x ≠ y ∈ Rn, (x-y)T(f(x)-f(y)) > 0.
A function f:Rn → Rn is uniformly increasing if there exists γ > 0 such that for all x , y ∈ Rn, (x-y)T(f(x)-f(y)) ≥ γ||x-y||2.
A function f is (uniformly strictly) decreasing if -f is (uniformly strictly) increasing.
The following result shows when a function is increasing or strictly increasing.
Theorem 1 ([Chua and Green 1976])
A C1 function f:Rn → Rn is
Many classical results in absolute stability theory such as passivity theory [Vidyasagar 1993] and Luré-Postnikov theory [Curran et. al 1997] give conditions under which a system is asymptotically stable. The following results show asymptotic stability using increasing functions.
Theorem 2 ([Chua and Green 1976, Wu and Chua 1994])
The system dz/dt = -g(Az,u(t)) is uniform asymptotically stable i-uniformly with respect to all continuous u(t) if
We are interested in synchronization in a network of n coupled dynamical systems. The simplest case occurs when there are only two identical systems, i.e. n=2. We will look at the case of n > 2 in later sections. Starting with two identical uncoupled systems dx1/dt = fa(x1,t) and dx2/dt = fa(x2,t) we connect their dynamics by introducing coupling between them. If the state equations of the coupled systems can be decomposed as:
dx1/dt = f(x1,x1,x2,t)
dx2/dt = f(x2,x1,x2,t)
then the following synchronization theorem follows directly from the definition of asymptotic stability (Definition 3).
Theorem 3 ([Wu and Chua 1994, Kocarev 1995])
The two coupled systems
dx1/dt = f(x1,x1,x2,t)
dx2/dt = f(x2,x1,x2,t)
synchronizes if dz/dt = f(z,u(t),v(t),t) is uniform asymptotically stable and i-uniformly with respect to u(t) and v(t).
This result reduces the synchronization problem into a problem of stability for a related system. At the synchronized state, x2(t) = x1(t) and thus dx1/dt = f(x1,x1,x1,t). Therefore if the consistency condition
f(x1,x1,x1,t) = fa(x1,t)
is satisfied for all x1, t, then when synchronization is reached, each system will follow the dynamics of the uncoupled system dx1/dt = fa(x1,t).
We say that dx1/dt = f(x1,x1,x2,t) is in the same functional form as dx2/dt = f(x2,x1,x2,t) because if we consider the second and third argument of f as inputs signals v(t) and u(t) respectively, the two systems can be written as dx1/dt = f(x1,u(t),v(t),t) and dx1/dt = f(x1,u(t),v(t),t), where u(t) = x1(t) and v(t) = x2(t). The discrete-time version of this result is:
Theorem 4
The two coupled discrete-time systems
x1(t+1) = f(x1(t),x1(t),x2(t),t)
x2(t+1) = f(x2(t),x1(t),x2(t),t)
synchronizes if z(t+1) = f(z(t),u(t),v(t),t) is uniform asymptotically stable and i-uniformly with respect to u(t) and v(t).
Several configurations that follows as corollaries of Theorems 3 and 4 are of particular interest:
We consider two systems which are driven by the exact same excitation signal η(t).
dx/dt = f(x,η(t))
dy/dt = f(y,η(t))
If f(y,η(t)) is uniform asymptotically stable and i-uniformly with respect to η(t), then the coupled system is synchronized. An application of this result is the homogeneous driving or subsystem decomposition approach of Pecora and Carroll [Pecora and Carroll, 1990]:
The system
dv/dt = f(v,u) \__ driving system
du/dt = g(v,u) /
dw/dt = g(v,w) → driven system
synchronizes if the sense that w → u if du/dt = g(η(t),u) is uniform asymptotically stable i-uniform with respect to η(t).
Theorem 5
The two coupled systems:
dx/dt = f(x,g(x,y,t),t)
dy/dt = f(y,g(x,y,t),t) (AC)
synchronizes the system dz/dt = f(z,η(t),t) is uniform asymptotically stable i-uniform with respect to η(t).
The term g(x,y,t) can be thought of the coupling between two systems where each system receives the exact same coupling. At first glance this is different from reciprocal coupling where one system receives the coupling g(x,y,t) while the other system receives the coupling term g(y,x,t), but as we see below reciprocal coupling can be cast into this form. Some special cases of this coupling configurations are:
Let the master system be decomposed into two subsystems as:
dx1/dt = g1(x1,y1,t) ← first subsystem
dy1/dt = g2(x1,y1,t) ← second subsystem
Consider a slave system with the same state equations as the second subsystem and driven by the state of the first subsystem x1 with state equations:
dy2/dt = g2(x1,y2,t)
This slave system synchronizes to the master subsystem, i.e. ||y1-y2||→ 0 as t→∞ if dy1/dt = g2(v(t),y1,t) is asymptotically stable for all functions v(∙).
Consider two coupled systems:
dx/dt = g(x,t)
dy/dt = g(y,t) + K(x-y)
where K is a square matrix. The coupled systems synchronize if the system dz/dt = g(z,t) - Kz + Kη(t) is uniform asymptotically stable i-uniform with respect to η(t).
We can consider -Kz as a stabilizing linear feedback that when added to dz/dt = g(z,t) turns it into a stable system.
Consider the coupled systems
dx/dt = f(x,c(x,s(t)),t) ← transmitter
dy/dt = f(y,c(x,s(t)),t) ← receiver
In this configuration, s(t) is an information signal, c(•) is an coding function that transforms the information into a signal c(x,s(t)) that is tranmitted from the transmitter to the receiver. At the receiver there is a decoding function
d(•,•) that is continuous in the first variable such that d(x,c(x,s(t))) = s(t) for all x. An example pair of functions c and d is c(x,s) = x+s and d(x,y) = y-x.
If dz/dt = f(z,η(t),t) is uniform asymptotically stable i-uniform with respect to η(t), then coupled systems synchronizes and the decoded signal d(y,c(x,s(t)) → s(t) as t→∞.
Consider the coupled systems
dx/dt = h(x,t) - K1(x,t) + K1(y,t)
dy/dt = h(y,t) - K2(y,t) + K2(x,t) (SC)
where K1 and K2 are nonlinear mappings from Rm+1 to Rm. By setting g(x,y,t) = K2(x,t)+K1(y,t) and f(x,z,t) = h(x,t) - K1(x,t) - K2(x,t) + z, we show that the system in Eq. (SC) can be cast into the form of Eq. (AC). Applying Theorem 5 this shows that the system in Eq. (SC) synchronizes if dz/dt = h(z,t) - K1(z,t) - K2(z,t) + η(t) is uniform asymptotically stable i-uniform with respect to η(t). One can view the term - K1(z,t) - K2(z,t) as stabilizing feedback that makes dz/dt = h(z,t) stable.
When K1 = K1= K, this reduces to the reciprocal coupling case:
dx/dt = h(x,t) - (K(x,t) - K(y,t))
dy/dt = h(y,t) - (K(y,t) - K(x,t))
and this coupled system synchronizes if dz/dt = h(z,t) - 2K(z,t) + η(t) is uniform asymptotically stable i-uniform with respect to η(t).
Next section: synchronization in a network of nonlinear dynamical systems.
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Copyright 2006-2009 by Chai Wah Wu.