Synchronization of Chaotic Systems: A Generalized Hamiltonian Systems Approach

Abstract

A reapproach to chaotic systems synchronization is presented from the perspective of passivity-based state observer design in the context of Generalized Hamiltonian systems including dissipation and destabilizing vector fields. The synchronization and lack of synchronization of several well-studied chaotic systems is reexplained in these terms.

Fault Detection and Isolation of Nonlinear Systems with Generalized Hamiltonian Representation

“…A special class of Generalized Hamiltonian representation is defined by Sira-Ramierez and Cruz-Hernandez [23],…”

Design and Applications of Continuous-Time Chaos Generators

“…This section presents the synchronization using Hamiltonian forms and an observer approach (Sira-Ramírez & Cruz-Hernández, 2001). Let's consider the dynamical system described by the master circuit in (13).…”

“…Definition: Two chaotic systems described by a set of states x 1 , x 2 ...x n (13) and ξ 1 , ξ 2 ...ξ n (14) will synchronize if the following limit fulfills (Shuh-Chuan et al, 2005;Sira-Ramírez & Cruz-Hernández, 2001):…”

“…If the energy flows in one direction between the systems, it is one-way coupling, known as master-slave configuration. This section is based on the work of (Sira-Ramírez & Cruz-Hernández, 2001). To synchronize two systems by applying Hamiltonian approach, their equations must be placed in the Generalized Hamiltonian Canonical form.…”

“…The vector K in (19) is the observer gain and it is adjusted according to the sufficiency conditions for synchronization (Sira-Ramírez & Cruz-Hernández, 2001). By selecting R io = 10kΩ, R fo = 3.9MΩ and R ko = 22Ω in Fig.…”