Upper Semicontinuity of Attractors and Synchronization

Abstract

In this paper we prove that diffusively coupled abstract semilinear parabolic systems synchronize. We apply the abstract results obtained to a class of ordinary differential equations and to reaction diffusion problems. The technique consists of proving that the attractors for the coupled differential equations are upper semicontinuous with respect to the attractor of a limiting problem, explicitly exhibited, in the diagonal. ᮊ

Synchronization of Systems With Multiplicative Noise

“…The synchronization of coupled dissipative systems has been investigated mathematically in the case of autonomous systems by Afraimovich and Rodrigues 1 , Carvalho et al 7 and Rodrigues 14 , both for asymptotically stable equilibria and general attractors, such as chaotic attractors. Analogous results also hold for nonautonomous systems (Ref.…”

Chaotic synchronization in lattice of partial-state coupled Lorenz equations

“…Thus, three different coupling configurations in the x i -, y i -or z i -component of Lorenz equations are considered. (c) The invariant manifold method and the analysis techniques proposed by Carralho and Rodrigues [5] and He and Vaidya [15], respectively, cannot be directly used to prove the synchronization of partial-state coupled Lorenz equations and to determine the dependence of coupling coefficients on the lattice size. Thus, a mathematical proof of synchronized behavior should be treated differently.…”

The persistence of synchronization under environmental noise

“…It can be shown (see Afraimovich and Rodrigues (1998), and Carvalho et al (1998)) that this also has a unique equilibrium (x ν ,ȳ ν ), which is globally asymptotically stable. Moreover, (x ν ,ȳ ν ) → (z,z) as ν → ∞, wherez is the unique globally asymptotically stable equilibrium of the "averaged" system…”

“…Analogous results hold for more general autonomous attractors (cf. Afraimovich and Rodrigues (1998), Carvalho et al (1998)) as well as for nonautonomous systems with appropriately defined nonautonomous attractors (cf. Kloeden (2003)).…”