Communications in Nonlinear Science and Numerical Simulation
Synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control
Highlights
► Exponential synchronization of non-autonomous systems with time-varying delay via delayed feedback control is derived. ► Exponential synchronization of non-autonomous systems with time-varying delay via delayed intermittent control is derived. ► Exponential synchronization conditions are formulated in terms of solutions of certain Riccati differential equations.
Introduction
The problem of chaos synchronization has attracted a wide range of research activity in recent years. A chaotic system has complex dynamical behaviors that possess some special features, such as being extremely sensitive to tiny variations of initial conditions, having bounded trajectories in the phase space. The concept of chaos synchronization is making two or more chaotic systems oscillate in a synchronized manner. There are several schemes which can be used to achieve chaos synchronization of autonomous chaotic systems, for example linear feedback method [13], [28], active control [1], adaptive control [1], [22], [30], impulsive control [3], [23], back-stepping design [20] and time-delay feedback control [11], intermittent control [14], [15], [16], [31], [32], etc.
In fact, non-autonomous systems for modeling the behavior of many engineering systems, such as offshore platforms, earthquake dynamics, electronic circuits and so on have been widely explored [6], [8], [9], [17], [19]. In [5], Carroll and Pecora studied synchronization non-autonomous chaotic circuits by using a feedback device to correct the phase of the periodic forcing in the response system. In [2], Cai et al. presented synchronizing two identical non-autonomous chaotic systems coupled by sinusoidal state error feedback control are derived by the Lyapunov direct method and the Gerschgorin disc theorem. More flexible criteria has been further obtained by the property that similar matrices have the same eigenvalues. In [28], Suykens et al. studied H∞ synchronization criteria master–slave non-autonomous Lur’e synchronization schemes for the static linear state error feedback control and dynamical output error feedback control.
Intermittent control has been used for a variety of purposes such as manufacturing, transportation and communication. In [32], the authors introduced intermittent control to nonlinear dynamical systems. However, results using intermittent control to study synchronization are few. Recently, by using intermittent control, the authors of [14], [15] investigated the synchronization of coupled chaotic systems with or without delay by using intermittent state feedback, the authors of [31] discussed the synchronization problem for a class of complex delayed dynamical networks by pinning periodically intermittent control, the authors of [16] studied the synchronization of coupled chaotic systems with time delay in the presence of parameter mismatches by using intermittent linear state feedback control. However, to the best of our knowledge, few published papers deal with the problem of synchronization of non-autonomous chaotic systems with time-varying delay by using intermittent linear state delayed feedback control. So, our paper presents a new non-autonomous system and we also approach to establishing both delay and non-delay controller to the system.
It is well known that the existence of time delay in a system may cause instability and oscillations system such as chemical engineering systems, biological modeling, electrical networks and many others, [4], [10], [12], [21], [27]. Stability of time-delay systems has been studied for decades and many results on this subject have been reported, see, e.g. [24], [25], [26] and references therein. In [24], Niamsup et al. studied the exponential stability condition for a class of linear time-varying systems with nonlinear delayed perturbations was derived by using an improved LyapunovKrasovskii functional. The proposed exponential stability conditions are formulated in terms of the solution of Lyapunov differential equations. In [25] Phat and Ha presented H∞ control and exponential stability of nonlinear non-autonomous systems with time-varying delay. By using the improved Lyapunov–Krasovskii functional and the Razumikhin-type stability theorem, sufficient conditions for exponential stabilization have been presented the delay-dependent and formulated by solving the standard Riccati differential equations.
In this paper, we shall investigate synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control. By utilizing Lyapunov–Krasovskii functional and combination of Riccati differential equation approach. The sufficient conditions are obtained for the exponentially stable of the error system via solving Riccati differential equation. The designed controller ensures that the synchronization of nonlinear non-autonomous chaotic systems are proposed via delayed feedback control and intermittent linear state delayed feedback control. Finally, we will provide numerical examples to illustrate the effectiveness of these synchronization criteria.
Section snippets
Problem formulation
The following notation will be used in this paper: R+ denotes the set of all real non-negative numbers: Rn denotes the n-dimensional Euclidean space; L2([0, t], Rn) denotes the Hilbert space of all L2-integrable and Rn-valued functions on [0, t]; C([0, t], Rn) denotes the Banach space of all Rn-valued continuous functions mapping [ −h, 0]; 〈x, y〉 or xTy denotes the scalar product of two vectors x, y; ∥ · ∥ denotes the Euclidean vector norm of x; AT denotes the transpose of the vector/matrix A; A is
Linear delayed feedback control
In this section, we will given some sufficient conditions for the synchronization of system (2.4), (2.5), firstly, some propositions are introduced. Then, we present exponential stability criteria for the system (2.9) and thus the system (2.4) synchronize with the system (2.5). Given positive number β, h, α, γ, γ1, γ2, ϵi, i = 1, 2, 3, 4, we set
Numerical examples
In this section, we now provide an example to show the effectiveness of the result in Theorem 3.1, Theorem 3.2 Example 4.1 We assume that there are two non-autonomous chaotic systems with time-varying delay systems and the state feedback controller satisfying (H1) such that the master system (with the subscript m) and the slave system (with subscript s). The master and slave systems are given, respectively, byand
Conclusions
This paper has investigated synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control. We have obtained some sufficient conditions for the exponential stability of the error system via solving Riccati differential equation. The delay feedback controller H1 and H2 designed can guarantee exponential stability of the error system. The validity of the approach has been demonstrated by numerical examples.
Acknowledgment
The authors thank anonymous reviewers for their valuable comments and suggestions. The first author is supported by the Graduate School, Chiang Mai University and Thai Government Scholarships in the Area of Science and Technology (Ministry of Science and Technology). The second author is supported by the Center of Excellence in Mathematics, CHE, Thailand. The third author was supported by NSERC Canada.
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