In this section, we shall describe the partial contraction approach for studying global synchronization of coupled chaotic systems. This approach was given by [45]. method to analyze networks of coupled identical nonlinear oscillators, and study applications to synchronization. Specifically, we use nonlinear partial contraction theory to derive exact and global results on synchronization. The method can be applied to coupled networks of various structures and arbitrary size. For oscillators with positive-definite diffusion coupling, it can be shown that synchronization always occur globally for strong enough coupling strengths, and an explicit upper bound on the corresponding threshold can be computed through eigenvalue analysis.
Basically, a nonlinear time-varying dynamic system will be called contracting if initial conditions or temporary disturbances are forgotten exponentially fast, i.e., if trajec-tories of the perturbed system return to their nominal behavior with an exponential convergence rate. The concept of partial contraction allows one to extend the applica-tions of contraction analysis to include convergence to behaviors or to specific properties (such as equality of state components, or convergence to a manifold) rather than tra-jectories. We briefly summarize the basic definitions and main results of Contraction Theory here. Consider a nonlinear system
dx
dt = f(x, t)
where x∈ Rn is the state vector and f is an n× 1 vector function. Assuming f(x, t) is continuously differentiable, we have
d
dt(δxTδx) = 2δxTδdx
dt = 2δxT ∂f
∂xδx≤ 2λmaxδxTδx (2.26) where δx is a virtual displacement between two neighboring solution trajectories, and λmax(x, t) is the largest eigenvalue of the symmetric part of the Jacobian J = ∂x∂f. Hence, if λmax(x, t) is uniformly strictly negative, any infinitesimal length kδxk converges exponentially to zero. By path integration at fixed time, this implies in turn that all solutions of system (2.26) converge exponentially to a single trajectory, independently
of the initial conditions. Note that differential analysis yields global results.
We now introduce the concept of partial contraction, which forms the basis of the contraction analysis. It derives from the very simple but very general result which follows.
Theorem 2.5.1. Consider a nonlinear system of the form
˙x = f(x, x, t) and assume that the auxiliary system
˙y = f(y, x, t)
is contracting with respect to y. If a particular solution of the auxiliary y-system verifies a smooth specific property, then all trajectories of the original x-system verify this property exponentially. The original system is said to be partially contracting.
Let us now move to networked systems under a very general coupling structure. Con-sider a coupled system containing m identical nodes
dxi
dt = f(xi, t) +X
j∈Ni
Kji(xj− xi), i = 1,· · · , m, (2.27) where Ni denotes the set of indices of the active links of elements i. It is equivalent to
dxi where K0 is chosen to be a constant symmetric positive definite matrix (we will discuss its function later). Again, construct an auxiliary system
dyi According to Theorem 2.5.1, if the auxiliary system in (2.29) is contracting, then all system trajectories will verify the independent property x1 =· · · = xm exponentially.
Theorem 2.5.2. Regardless of initial conditions, all the elements within a generally coupled system in (2.27) will reach synchrony or group agreement exponentially if
1) the network is connected, 2) λmax(Jis) is upper bounded, 3) the couplings are strong enough.
where Jis =
∂f
∂y(y, t)
s, and Fs = 12(F + FT).
Chapter 3
Global Synchronization via Matrix Measures Approach
This chapter contains the main results of the first part of the thesis. In particular, we use matrix measures approach to study global synchronization of coupled chaotic systems. Our coupling rules are allowed to be asymmetric and/or some competitive (gij < 0, i 6= j) couplings between cells xi and xj as long as the coupled system is bounded dissipative. In addition, the partial-state coupling in our approach is allowed to have the form satisfying (3.14). Moreover, by merely checking the structure of the vector field of the single oscillator, we shall be able to determine if the system is globally synchronized. We also obtain a rigorous lower bound on the coupling strength for the global synchronization of all oscillators with coupling configuration satisfying (3.3a), and (3.3b). Part of the results in this direction is bases on the paper [27]. To conclude this section, we define the global synchronization as in the following.
Definition 3.0.3. (i) System (2.15) is said to be globally synchronized if for any given initial values x0 there exists a d = dx0 such that system (2.15) is synchronized for the initial conditions x0. Here dx0 is a constant depending on x0. (ii) System (2.15) is said to be uniformly, globally synchronized if there exists a d = d1 such that system (2.15) is synchronized for all initial values x0.
3.1 Preliminaries
Chaotic synchronization is a fundamental phenomenon in physical systems with dissi-pation. In this section, we introduce the concept of the bounded dissipation to coupled system (2.15). Then, we use this concept of the matrix measure theory to achieve the behavior of global synchronization in coupled system (2.15). Hence, we give the definition of bounded dissipation as follows.
Definition 3.1.1. (i) A system of n ordinary differential equations is called bounded dissipative provided that for any r > 0 and for any initial conditions x0 in Bn(r), there exists a time t∗ ≥ t0 and αr such that kx(t)k ≤ αr for all t ≥ t∗. (ii) If, in addition, αr is independent of r, then the system is said to be uniformly bounded dissipative with respect to αr.
To prove global synchronization of coupled chaotic systems, one needs to assume bounded dissipation of the system, which plays the role of an a priori estimate. Without such an a priori estimate, as in the case of R¨ossler system, the global synchronization is much more difficult to obtain. Only local synchronization was reported numerically in literature (see e.g., [4]). We remark that in certain cases of the R¨ossler system, the trajectory of each oscillator grows unbounded yet approaches each other (see e.g., [4]).
An interesting question in this direction is how bounded dissipation of the coupled system is related to the uncoupled dynamics and its connectivity topology. Not much general theorems have been provided so far. In the case that G is diffusively coupled with periodic boundary conditions or zero-flux and D satisfies (3.3c), it was shown in [5] that bounded dissipation of the single oscillator implies that of the coupled chaotic oscillators. Moreover, the absorbing domain of the coupled system is a topological product of the absorbing domain of each individual system. Moreover, it often requires to construct an approximate Lyapunov function to prove the bounded dissipation of the system. The following proposition gives the type of Lyapunov functions that would ensure the bounded dissipation of the system.
Proposition 3.1.2. Let a system of n ordinary differential equations be given. Let V be a continuous real-valued function V : Rn→ R+ so that V is strictly decreasing along
the solution of the system on Rn− Γ, where Γ is homeomorphic to an open ball in Rn. Suppose
kxk→∞lim V (x) =∞. (3.1)
Then the system is bounded dissipative.
Proof. For any x0 ∈ Rn, we first prove that x(t) must enter Γ at a certain time.
Otherwise, the values of V at the points of the ω-limit set of x(t) must be the same, a contradiction. The contradiction comes from the facts that the ω-limit set is closed and invariant and V is strictly decreasing along the solution trajectory, which stays in Rn− Γ. We then find a ball Bn(r) so that Bn(r)⊃ Γ. Let k1 = maxx∈ ¯Bn(r)V (x), and Bn(αr) be a ball satisfying V (x) > k2 whenever x∈ Rn−Bn(αr), where k2 > k1. Then we conclude that if x0 ∈ Bn(r), x(t) stays in Bn(αr) for all time t. We just complete the proof of the proposition.
In our derivation of synchronization of system(3.1), we need the concept of matrix measure. For the completeness and ease of references, we also recall the following definition of the matrix measure and its properties (see e.g., [44]).
Theorem 3.1.3. (see e.g., 3.5.32 of [44]) Consider the differential equation ˙x(t) = A(t)x(t) + v(t), t ≥ 0, where x(t) ∈ Rn, A(t) ∈ Rn×n, and A(t), v(t) are piecewise-continuous. Let k · ki be a norm on Rn, and k · ki, µi denote, respectively, the corre-sponding induced norm and matrix measure on Rn×n. Then whenever t≥ t0 ≥ 0, we have