Fuzzy modeling and synchronization of hyperchaotic systems

doi:10.1016/j.chaos.2005.01.023

Chaos, Solitons & Fractals

Volume 26, Issue 3, November 2005, Pages 835-843

https://doi.org/10.1016/j.chaos.2005.01.023 Get rights and content

Abstract

This paper presents fuzzy model-based designs for synchronization of hyperchaotic systems. The T–S fuzzy models for hyperchaotic systems are exactly derived. Based on the T–S fuzzy hyperchaotic models, the fuzzy controllers for hyperchaotic synchronization are designed via the exact linearization techniques. Numerical examples are given to demonstrate the effectiveness of the proposed method.

Introduction

Since the early 1990s researchers have realized that chaotic systems can be synchronized [1], [2]. An increasing interest has been generated in chaotic or hyperchaotic synchronization and its practical applications in different fields, particularly in communications because chaotic systems are deterministic but extremely sensitive to initial conditions and have noise-like behavior [3], [4], [5], [6], [7]. In particular, synchronization schemes of hyperchaotic systems were investigated recently [8], [9]. It should be noted that the presence of more than one positive Lyapunov exponent clearly improves the security by generating more complex dynamics.

Various control methods have been developed in order to synchronize chaotic systems which include linear and nonlinear feedback control, time delay feedback control, adaptive control, and impulsive control. Most recently, a new approach to control and synchronize chaos via LMI-based fuzzy control system design is suggested in [10], [11], where the idea is to use the Takagi–Sugeno (T–S) fuzzy model to represent typical chaotic models and then apply some effective fuzzy techniques. LMI-based fuzzy synchronization of Chen’s system is considered in [12]. Synchronization of generalized Henon map by using adaptive fuzzy controller is considered in [13].

Following the idea of representing a chaotic system via a T–S fuzzy model [10], [11], [14], in this paper, we intend to represent a class of typical continuous-time hyperchaotic systems via T–S fuzzy models and develop a fuzzy controller design method for hyperchaotic synchronization.

The rest of this paper is organized as follows. In Section 2, the T–S fuzzy models will be presented for two well-known hyperchaotic systems: the fourth Rössler system and the Matsumoto–Chua–Kobayashi (MCK) circuit. Synchronization issues will be discussed in Section 3. Simulation results is given in Section 4. Finally, Concludes is drawn in Section 5.

Section snippets

Fuzzy modeling of hyperchaotic systems

The T–S fuzzy dynamic model originates from Takagi and Sugeno [15], which is described by fuzzy IF-THEN rules where the consequent parts represent local linear models. Consider a continuous-time nonlinear dynamic system as follows: $\dot{X} (t) = f (X (t)),$ where $X (t) \in R^{n}$ is the state vector, f(x) is nonlinear function with appropriate dimension. In the form of T–S model, the system (1) can be exactly represented in a region of interest as follows:

Rⁱ: IF z₁(t) is M_i1 and ⋯ and z_p(t) is M_ip

THEN $\dot{X} (t) = A_{i} X (t) + b_{i},$

Fuzzy synchronization of hyperchaotic systems

In this section, we will achieve the synchronization of hyperchaotic systems based on the T–S fuzzy model. Given the fuzzy hyperchaotic drive system (11), we consider the following fuzzy controlled system as the response system,

Rⁱ: IF $\tilde{s} (t)$ is M_i

THEN $\dot{\tilde{X}} (t) = A_{i} \tilde{X} (t) + b_{i} + Bu (t), i = 1, 2 .$ Here, B is input matrix, u(t) is input signal. The defuzzification process is given as $\dot{\tilde{X}} (t) = \sum_{i = 1}^{2} h_{i} (\tilde{s} (t)) {A_{i} \tilde{X} (t) + b_{i}} + Bu (t), h_{i} (\tilde{s} (t)) = M_{i} (\tilde{s} (t)) .$ The fuzzy drive system (11) and the fuzzy controlled system (12)

Simulation results

In this section, we will give some numerical simulations to verify the hyperchaotic synchronization based on T–S fuzzy hyperchaotic model and fuzzy controller design. In the following, all the LMIs are solved by MATLAB LMI toolbox and all the differential equation are solved by using the fourth–fifth order Runge–Kutta method (ODE45 in MATLAB).

Firstly, we consider the synchronization between hyperchaotic Rössler systems. The initial conditions for drive and response systems are X₀ = [0, 0, 0, 30]^T, $\tilde{X}$

Conclusions

In this paper, we represent the typical hyperchaotic systems via T–S fuzzy models. Based on the fuzzy hyperchaotic models, simpler fuzzy controllers have been designed for synchronizing hyperchaotic systems. The proposed approach does not require the computation of the Lyapunov exponents. Finally, numerical examples are provided to demonstrate the effectiveness of our method.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 60271019 and the Youth Science and Technology Foundation of UESTC under Grant JX04009.

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Fuzzy modeling and synchronization of hyperchaotic systems

Abstract

Introduction

Section snippets

Fuzzy modeling of hyperchaotic systems

Fuzzy synchronization of hyperchaotic systems

Simulation results

Conclusions

Acknowledgments

Phys Rep

Phys Lett A

Chaos, Solitons & Fractals

Synchronization in chaotic systems

Phys Rev Lett

Synchronizing chaotic circuits

IEEE Trans CAS I

Taming chaos. Part I. Synchronization

IEEE Trans CAS I

Fundamentals of synchronization in chaotic systems, concepts, and applications

Chaos

From chaos to order methodologies, perspectives and applications

Chaos in nonlinear oscillators: controlling and synchronization