Adaptive, Observer-Based Synchronization of Different Chaotic Systems

Abstract

In this study, the problem of master–slave synchronization of two different chaotic systems is considered and solved under a novel set of assumptions. The mathematical model of each of them contains unknown, constant parameters. Only a single output of the master system is available, and only a single input of the slave system is a control input. The proposed, novel approach is based on the active cooperation of the adaptive observer of the master system and adaptive controller of the slave. The tuning function technique is included in the observer–controller design to avoid overparameterization. Complexity explosion and unacceptable increases in adaptive parameters are prevented by proper adaptive techniques application. Due to the selected observer type, the derivation is restricted to the defined class of master systems—output-nonlinear parametric (ONP) systems. Linear transformation of several popular chaotic systems (e.g., Arneodo, Arneodo–Coullet, Genesio–Tesi, Lur’e) into the ONP form is discussed. The stability of the whole, closed-loop system is derived using Lyapunov techniques and examples of implementation (synchronization of Arneodo and 3D jerk systems) are provided.

Chaos Synchronization of Two Györgyi–Field Systems for the Belousov–Zhabotinsky Chemical Reaction

“…At the beginning, the trajectories of bidimensional and three-dimensional attractor exhibit large open cycles that shrink until they become stabile closed cycles (Figures 10 12). The trajectories obtained in the GF model indicate the pure oscillatory behavior of B reaction (Figures 10-12), in which the amplitudes and frequencies of oscillations the de pend by time [32] and temperature [33]. For a high flow in the open reaction system, the concentrations of all intermediate species and parameter values listed in Table 2 presented oscillatory behaviors (Figures 6-8) in time (τ).…”

“…At the beginning, the trajectories of bidimensional and three-dimensional attractors exhibit large open cycles that shrink until they become stabile closed cycles (Figures 10-12). The trajectories obtained in the GF model indicate the pure oscillatory behavior of BZ reaction (Figures 10-12), in which the amplitudes and frequencies of oscillations the depend by time [32] and temperature [33].…”

A New DC Offset Boostable Chaotic System with Multistability, Coexisting Attractors and Its Adaptive Synchronization

“…In the past few decades, chaos synchronization has received great attention owing to its applications in designing secure communication systems. Various adaptive synchronization schemes have been developed in recent years such as sliding mode controller [60,61], backstepping neural network method [62,63], observer-based synchronization [64] and so on.…”

Observer-Based, Robust Position Tracking in Two-Mass Drive System

“…Therefore, an obvious technique to consider is to join an adaptive observer and an adaptive controller. This problem was investigated in [38] and it was observed that it requires the active cooperation of two adaptation mechanisms: the first acting in the observer and the other one in the controller. It is well known that adaptive observers for nonlinear systems require special conditions [39,40] and that the derivation of the stability and tuning is complicated.…”