Synchronization of Hyperchaos With Time Delay Using Impulse Control

Secure communication using hyperchaos has a better potential performance, but hyperchaotic impulse circuits synchronization is a challenging task. In this paper, an impulsive control method is proposed for the synchronization of two hyperchaotic Chen circuits with linear or nonlinear delays. Based on the Lyapunov theorem and some analysis techniques, the sufficient conditions for the synchronization of chaotic system with time delay are obtained. The upper bound of the impulse interval is derived to assure the synchronization error system to be asymptotically stable. Simulation and circuit experiment validated the proposed method for correctness and feasibility.

Time delay is a common phenomenon in practical commonplace, such as, communication [22], biology [23] and other fields. In nonlinear dynamics, time delay could generate an infinite dimensional hyperchaos [24], with infinite dimensional structure, promising more application benefits. Time delay could be different forms, including linear time delay or nonlinear time delay, constant or time varying time delay, The associate editor coordinating the review of this manuscript and approving it for publication was Jun Hu . simple or mixed time delay [20], [21]. Chen system with linear or nonlinear time delay is an example demonstrating multiple kinds of attractors, including the single scroll attractor [25], [26], the double scroll attractors [24], [27] and the composite multi-scroll attractors [22], which possesses multiple positive Lyapunov exponents [25], more complex dynamics, and better application potential. Therefore the combination of the impulse control and Chen system can be applied to encryption, secure communication, spacecraft guidance, etc.. However, the synchronization of the hyperchaotic systems is a challenge task.
Lots of research efforts have been dedicated to impulse control systems. For secure communication [28], [29], the impulse control benefits from significantly decreasing the energy cost and increasing information security. In neural networks [30], the impulse controller is widely applied, because some neurons cannot endure continuous control. Some researchers describe the impulse differential equations by considering the process with the ''Jump'' state at the beginning [12]. Liu et al. employs Lyapunov-Razumikhin theorem and provides the uniform asymptotic stability for the impulse differential equations [31], but they only consider one-dimensional system with time delay [31], [32]. The proof of the global exponential stability for impulse synchronization of network is also presented in [33], which does not consider time delay. In [34], the theoretical results on asymptotic stability of impulsive systems with time-varying delay are derived, which are quite useful to allow the existence of impulsive perturbations in the chaotic system. However, there is few investigation about impulse synchronization considering the systems with linear/nonlinear multiple time delays.
To deal with such problem, in this paper, we have derived the sufficient conditions to assure the synchronization error system are asymptotically stable, which provides the base for the corresponding controller design.
The contribution of this paper is to implement the synchronization between two infinite dimensional hyperchaos systems given by impulse delay differential equations by both simulation and circuit experiment. On the one hand, the uniform asymptotic stability is investigated based on the stability theory of Lyapunov for the hyperchaotic system with linear or nonlinear time delays. Furthermore, the result is extended to other systems with multiple types of time delay.
The rest of the paper is organized as follows. Section 2 gives the theoretical analysis of the stability of impulse synchronization error. Section 3 gives the simulation and experimental results of hyperchaotic system synchronization of the single-scroll attractor in Chen system with linear time delay using the proposed method. Conclusions are given in Section 4.

II. IMPULSE SYNCHRONIZATION OF TIME DELAY INDUCED HYPERCHAOTIC ATTRACTOR A. PRELIMINARY OF IMPULSE DELAY-DIFFERENTIAL EQUATIONS
In general, an impulsive differential equation is given by where x(t) ∈ R n is state vector, ϕ is the initial states of x(t), f : R + × S (ρ) → R n , I k : S (ρ) → R n , is continuous function vector on (t k−1 , t k ] × S (ρ), the impulse time t k satisfy 0 < t 0 < t 1 < t 2 < . . . and lim k→∞ t k = ∞. The abbreviated notation right-hand and left-hand limits for time approaching to t k in forward and reverse direction, respectively. Assuming that, for all k, f (t, 0) ≡ 0 and I k (0) = 0, equation (1) has a trivial solution. The following definitions and lemmas are introduced [31]: Definition 2.1 [31]: [31]: Assume that r ∈ N , D ⊂ R and F ⊂ R, PC (D, F) denotes a piecewise continuous function from D to F, namely, if φ ∈ PC (D, F), when t ∈ D, t = t k , φ is a continuous function, except t = t k , φ is a discontinuous function, but left side continuous. Denote PC r (D, F) as r-order piecewise global differentiable function from D to F, namely then, the trivial solution of system (1) is uniformly asymptotically stable.
B. HYPER-CHAOS SYNCHRONIZATION USING IMPULSE CONTROL For the drive system: where A ∈ R n×n represents the state matrix, B represents the delay gain matrix, ψ (x) represents the nonlinear function, τ represents the delay time. A response system using impulse control is considered as follows: where x (t) ∈ R n represents the state variables, x ∈ R n represents growth of states under the impulse control at time t k , C k ∈ R n×n represents impulse control matrix. The matrices A, B are the same as equation (2). From (2) and (3), the synchronization error system is given by: where e(t) represents e(t, t − τ ), and it denotes In this paper, we will give the theorem for error system (4), by which we can deduce the impulse interval to keep the VOLUME 8, 2020 asymptotically stability of Eq.(4). The uniform asymptotic stability theorem of error system (4) is given as follows.
Theorem 2.1: Considering the error system (4), if there exists a constant l 1 and symmetric and positive defined matrix P, such that: , where λ max (·) is the maximum eigenvalue of the matrix, λ min (·) is the minimum eigenvalue of the matrix, and ε is the impulse interval. Then, error system (4) is uniform asymptotically stable.
Proof of Theorem 1: Select Lyapunov function candidate as: For t = t k (6), as shown at the bottom of this page. here Therefore, (7) as shown at the bottom of this page. For t = t k (8), as shown at the bottom of this page. where assume c (s) = s, p (t) = W According to condition (4) of Lemma 1, we have: According to condition (2) of Theorem 1, if the following conditions are established: . Then, we conclude that the error system (4) is asymptotically stability.

End of proof. Corollary 2.1 (Theorem 1 Can Be Extended to the Delay Differential Equations With Multiple Linear Time Delays):
It is clear that conditions of Theorem 1 is independent of τ . We present Theorem 2 about the system with nonlinear time delays.
Proof of Theorem 2.2: Lyapunov function candidate is selected as The time derivation of (12) at t = t k is the same as Eq. (8).
End of Proof

A. HYPER-CHAOTIC SINGLE-SCROLL ATTRACTOR IN CHEN SYSTEM WITH TIME-DELAY
Chen system with linear time delay feedback is given as follows [27]: With different parameters, Chen system with time delay demonstrates four kinds of attractors, such as single scroll attractor(in Fig.1(a)), double scroll attractor (in Fig.1(b)), multi-scroll attractor(in Fig.1(c)) and D-scroll attractor(in Fig.1(d)). The parameters configuration are presented as shown in Table 1. The idea of the impulse  synchronization for Chen system with time delay will gain better insight into the image encryption and secure communication [22].

1) SYNCHRONIZATION OF CHEN SYSTEMS WITH LINEAR TIME DELAY USING IMPULSE CONTROL
Chen system with linear time delay feedback control can be given as:ẋ where The response system with impulse control is given as follows: where ψ x (t) = Simulation waveform of synchronization of the Chen system with linear time delay using impulse control.
The error system is given by: where are the same as that in Eq. (18). = 0.005611. In order to meet the conditions in Theorem 2.1, the impulse interval is selected as ε = 0.005. The initial states of the two systems are specified as (x(0), x (0)) = ((0.1, 0.1, −0.1), (0.2, 0.2, 5)). The simulation results are given in Fig. 2, the impulse control is activated from t = 5s. The subplot (a) gives the curves of states x of drive system, x of response system and the state error e 1 in black line with circle, blue line with star and red line with dot, respectively. From Fig. 2, we know that the synchronization is achieved after the impulse controller is active at t = 5s.

2) SYNCHRONIZATION OF CHEN SYSTEMS WITH NONLINEAR TIME DELAY USING IMPULSE CONTROL
Chen system with nonlinear time delay feedback control can be given as:       are (x(0), x (0)) = ((−0.1, −5, −0.1), (0.1, 1, 0.1)). The simulation results are shown in Fig. 3, the impulse control starts from t = 30s. The subplot (a), (b)and (c) give the state errors e 1 , e 2 and e 3 , respectively. From Fig. 3, we know that the synchronization of the chaos systems with nonlinear time delays is achieved with the impulse controller.

C. CIRCUIT EXPERIMENTAL RESULTS
For the circuit schematic diagram of impulse synchronization, we use multipath selectors (CD4066) and operation amplifiers (LF347N) as shown in Fig. 4.
In Fig. 4, signal x, y and z are the state variables of the drive system, and x , y and z are the state variables of the response system. Let e = (e 1 , e 2 , e 3 ) T be the synchronization error. The impulse control outputs are u 1 , u 2 and u 3 . The circuit parameters are summerized in Table 2. The 555 timer provides impulse signal with 10% duty ratio and 0.005s impulse interval. The block (a) in Fig. 4 is regard to the impulse gain C in Eq. (17), and the block (b) in Fig. 4 is used for signal power amplification. The switches in the Fig. 4 are employed as the multipath selector, on the moment that the switches are on-stated by a high level of 555 timer.
The circuit simulation results (using PSIM software) of the synchronization between the drive system and the response system are given in Fig. 5. The upper, middle and bottom subplot in Fig. 5 are the corresponding synchronization error curves e 1 , e 2 , e 3 , respectively. Here, the impulse control is activated after t = 80 second. We can see, from Fig. 5, that the synchronization errors tend to zero after the controller is activated.
In the following, the circuit experiment is built to observe the synchronization between the two hyperchaotic systems. The experimental results are shown in Fig. 6.
The experiment results are presented as shown in Fig. 6. Fig. 6 (a), (b) and (c) illustrate the states of the two systems and the corresponding synchronization errors after the controller is put into effect. From Fig 6, we learn that the synchronization is verified by the experiment.

IV. CONCLUSION
To conclusion, based on Lyapunov stability theory, we proposed two theorems for impulse synchronization of the hyperchaotic systems with linear and nonlinear time delays, respectively. The proposed method has been applied to the Chen system with time delay to show its correctness and effectiveness of the theory from both the simulation and experiment results. The proposed theorems could be extended to paradigmatic systems with linear and nonlinear time delays. This method gives bright prospects to explore some realistic applications.