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Introduction

Chaos is an interesting nonlinear phenomenon and has been known for a rather long time in the mathematics, physics, engineering and other related fields [5]. Nevertheless, duo to its sensitive dependence on initial conditions, chaos was believed in a long time to be neither predictable nor controllable. However, a great many researchers from various fields have made insightful investigations on this subject, and many important and funda- mental results have been reported.

Chaos synchronization is a contemporary topic in nonlinear science with its applica- tions to diverse areas such as secure communications, biological chemical reactions and so on. In general, in order to synchronize two or more nonlinear dynamical systems, it is natural to address the proper control approaches. Up to now, many typical and useful control methods have been proposed such as adaptive control [9, 25], feedback control [6,17], pinning control [10,18], intermittent control [19,23] and impulsive control [7,13]. Recently, discontinuous control approaches such as impulsive, switched [20] and intermittent controls have attracted increasing attention since they are commonly used in transportation, manufacturing and communication engineering. For instance, in commu- nications, intermittent control scheme is usually used as a central means of transmitting in- formation between transmitter and receiver in order to realize synchronization. However, discontinuous control dynamical systems are governed by complex mathematical models displaying rather irregular dynamical behaviors with interesting challenges. Generally speaking, intermittent control strategy is composed of control time and the rest time in turn. And intermittent control is activated during the work time and is off during the rest time.

Compared with impulsive control, intermittent control is easier to be implemented because it may be more reasonable to realise control process in some time intervals other than some time instants in practice control application. The intermittent controller has been triumphant applied to stabilize and synchronize neural networks [26, 27], com- plex networks [10, 18, 19, 23], chaotic systems [9, 11] in recent decades. Nevertheless, periodic intermittent control may be inadequate in the practical application and may cause unreasonable and unnecessary results. For instance, the generation of wind power is emblematical aperiodically intermittent. Therefore, for the real applications and the theoretical analysis, it is momentous to consider the synchronization problem of chaotic systems under aperiodically intermittent control scheme.

As everyone knows, adaptive control scheme are designed under control objective because of the characteristics of considered system [1, 4, 15, 16, 22, 28]. The strong merit of adaptive control is that the control parameters can automatically adjust themselves according to some appropriate updating laws. Adaptive consensus control for a class of nonlinear multiagent time-delay systems using neural networks was investigated in [4]. In [1], adaptive synchronization of fractional-order memristor-based neural networks with time delay was studied. In [15], Li and Hu proposed Pinning adaptive and impulsive synchronization of fractional-order complex dynamical networks. Pinning synchroniza- tion of complex directed dynamical networks under decentralized adaptive strategy for aperiodically intermittent control was investigated in [28].

It is well known that due to the finite switching speeds of transmission and spreading, time delay is a very typical phenomenon in some fields and may lead to undesirable dy- namic behaviors such as oscillation and instability behavior. Therefore, it is extraordinary essential to consider the influence of time delay on the dynamical behavior for systems. Recently, a great many of results for delays dynamical systems have been obtained [2, 3, 8]. However, to the best of our knowledge, the synchronization problem for delayed dynamical systems under intermittent adaptive control, until now, receives few attentions. We will devote our efforts to this problem in this letter.

Motivated by the above discussion, synchronization of nonlinear dynamical systems with time-varying delays is investigated via a novel and generalized adaptive intermittent control protocol. The main contributions in this paper can be summarized as follows. The first one is that the time-varying delays is taken into account for the chaos systems, which may be more consistent with the real world case. When dealing with delays, it is difficult to construct a piecewise Lyapunov auxiliary function and use analytical techniques. The second one is that adaptive intermittent control strategy is adopted to synchronize nonlinear dynamical systems. The controller is an economic and realistic method for network and is an extension of periodic intermittent adaptive control strategy. The third one is that by constructing a piecewise Lyapunov auxiliary function and making use of piecewise analysis technique, some effective and novel criteria are obtained to ensure the global synchronization of delayed chaotic systems by means of the designed control protocols. At the end, two numerical examples with are provided to verify the effectiveness of the theoretical results proposed scheme.

The rest of this paper is organized as follows. In Section 2, model of nonlinear dy- namical systems with time-varying delays and preliminaries are given. Synchronization of the considered model under the aperiodically intermittent control and its adaptive strategy is investigated in Section 3. In Section 4 the effectiveness and feasibility of the developed methods are presented by two numerical examples. Finally, some conclusions are obtained in Section 5.

Notations. Let be the space of n-dimensional real column vectorsdenotes a vector norm defined bydenotes the set of all n × n real matrix. I_n is the identity matrix with n dimensions. For a square matrix A, A^T denotes its transpose, λ_max (A) and λ_min(A) are the maximum and minimum eigenvalues of matrix A. For a real matrix M write is positive (negative) definite, and is semipositive (seminegative) definite be the set of all natural numbers. Z⁺ be the set of all nonnegative integers.

Preliminaries

In this section a class of delayed nonlinear chaotic systems is considered. The differential equations are described by

where denotes the state vector, is a nonlinear vector function, the time-varying delay τ (t) may be unknown but is bounded by known constant, are three constant matrices, is a constant vector, which may be an external disturbance or the system bias

The initial condition of system (1) is given , where represents the set of all n-dimensional continuous functions defined on the interval

In the case that system (1) reaches complete synchronization, we have response chaotic system described as follows:

where u(t) is an appropriate control input designed in the following.

In order to obtain our main results, the following assumptions, definition and lemmas are necessary:

Assumption1. (See [21].) For the nonlinear function f (t), there exist positive constant l such that for any ,

Assumption 2. (See [21].) For the aperiodically intermittent control strategy, there exist two positive scalars such that, for

where

Assumption 3. Time-delay function is real-valued continu- ous function and satisfies

Definition 1. (See [21].) For the aperiodically intermittent control, define

Obviously, the aperiodically intermittent control becomes the continuous control. In the following, without loss of generality, we always suppose that .

Lemma 1. (See [18].) If Y and Z are real matrices with appropriate dimensions, then there exists a positive constant such that

Lemma 2. (See [18].) Suppose that function is continuous and nonnegative for

where are constants and m ∈ Z⁺. Suppose that for the aperiodically intermittent control, there exists a constant , where Ψ is defined in Definition 1.If

then

where q > 0 is the unique positive solution of the equation

Main results

Let be synchronization errors. In this section, two control schemes will be designed to synchronize nonlinear system (1) to the desired state (2). The main results are stated in the following.

Intermittent control with constant control gains

To achieve the synchronization, firstly, a generalized intermittent control u(t) with adap- tive constant control gain defined as follows

where is a constant

According to the control law (3), the error dynamical system is then governed as follows:

It is easy to see that the synchronization of the delayed dynamical systems (1) and (2) are achieved if the zero solution of the error dynamical system (4) is stable.

Theorem 1. Suppose that Assumptions 1–2 hold. Under controller (3), if there exist positive constants such that

(i)

(ii)

(iii)

(iv) where q > 0 is the unique positive solution of the equation

then the delayed dynamical systems (1) and (2) are globally asymptotically synchro- nized.

Proof. Define the Lyapunov function as

Then its derivative with respect to time . along with solutions of (4) can be calculated as follows.

When

According to condition (i), it can be obtained that

Similarly, for , using condition (ii), it can be derived that

Hence, it can be gotten that

By Lemma 2 and conditions (iii), (iv) it can be obtained that

Therefore, the asymptotical synchronization of the controlled system (2) is realized, and the proof of Theorem 1 is completed.

ntermittent control with adaptive control gains

For the controlled system (2), the feedback control gains may be chosen according to Theorem 1 to derived synchronization. However, feedback gains may be given much larger than those needed in practice. A better way is to use adaptive approach to tune the feedback gains. In the section an adaptive method is introduced to determine the intermittent control gains, and the global synchronization will be investigated based on the aperiodically intermittent adaptive control gains. For the sake of simplicity, the control input in the controlled system (2) is defined as follows:

Where

and

here are the positive constant control strengths.

According to the control law (5)–(7), the error dynamical system is then governed as follows:

where is a diagonal matrix.

It is easy to see that the synchronization of the delayed dynamical systems (1) and (2) is achieved if the zero solution of the error dynamical system (8) is stable.

Theorem 2. Suppose that Assumptions 1–3 hold. Under controller (5) satisfying (6) and (7), systems (1) and (2) are globally asymptotically synchronized.

Proof. Let λ is the largest eigenvalue of the matrix

Choose a positive number µ such that. Construct a piecewise function described by

for in which is a positive constant to be determined later. It follows from (9) that W (t) is continuous except for

where and denote the right limit and the left limit of W(t) at time respectively.

Define the Lyapunov function as

where

Evidently, U (t) is continuous for all , and V (t) is continuous except for

Then its derivative with respect to time t along with solutions of (8) can be calculated as follows.

When

where . It is easy to see that some suitable dˆcan be chosen such that

which shows that So

So,

Similarly, for

Hence, for

which leads to

In the following, it can be proved that

By virtue of (10), (11) and (12) it can be derived that

which implies that

and then

It shows that

therefore, by the theory of series,

In addition, for, in view of the nonnegativity of d_i(t) and it is easy to estimate that

In view of this, it can be gotten that

Evidently,when , by the above results we obtain

Therefore, the asymptotical synchronization of the controlled system (2) is realized, and the proof of Theorem 2 is completed.

If for all m , the aperiodically intermittent adaptive control (5) satisfying (6) and (7) becomes the following periodically intermittent adaptive control:

where

and

hereare the positive constant control strengths.

Then based on Theorem 2, the following Corollary 1 is immediately obtained.

Corollary 1.Suppose that Assumptions 1–3 hold. Under controller (13) satisfying (14) and (15), system (1) and the controlled delayed system (2) are synchronized.

If for all the aperiodically intermittent adaptive control becomes the following general continuous control:

Then, based on Theorem 2, the following Corollary 2 is immediately gotten.

Corollary 2. Suppose that Assumptions 1–3 hold. Under controller (16), system (1) and the controlled delayed dynamical system (2) are synchronized.

Suppose , we can rewrite nonlinear system model (1) as follows:

Correspondingly, the response chaotic system y(t) is represented by

where is the aperiodically intermittent adaptive control gain defined as follows:

where

and

Based on Theorem 2, we can get the following Corollary 3.

Corollary 3. Suppose that Assumptions 1 and 2 hold. Under controller (19) satisfy- ing (20) and (21), system (17) and the controlled delayed system (18) are synchronized.

Remark 1. Due to the finite information transmission and processing speeds among the units, time delays are usually encountered in dynamical networks and may result in undesirable dynamic behaviors such as oscillation behavior and network instability. Hence, time delays should be taken into account in realistic modeling of dynamical networks. Otherwise, because of the technical reasons, time delays is not considered in [9, 10, 12, 14, 28]. In this paper, by establishing piecewise auxiliary function, piece- wise analysis technique and constructing novel generalized adaptive intermittent control, the synchronization of delayed dynamical systems has been realized. When τ (t) = 0, Corollary 3 in the paper is equivalent to Theorem 4 in [9]. This is to say, results in [9] are the special case of our results. So, the model we choose in the paper is more close to the reality.

Remark 2. In this paper a generalized adaptive intermittent control strategy, which con- tains the traditional periodically intermittent control and the aperiodic case, is introduced. Especially, when where are positive constants, ,, the generalized adaptive intermittent control becomes the adaptive periodic one which have been studied in [10, 23]. When , the adaptive intermittent control is reduced to the continuous-time adaptive control, which have been studied in [1, 4, 15, 25]. This is to say, our results in the paper are less conservative and more practically applicable.

Remark 3. Evidently, it follows from adaptive intermittent strategy (5)–(7) that the adaptive gains (t) for i = 1, 2, 3 are increasing in each work time according to the update law (7) and identically equal to zero in the rest time. When the synchronization is realized, the values of (t) are converge to some positive constants in each work time. The adaptive gains will be shown in the numerical examples in the four section.

Remark 4. In [24] the authors utilize the method of adaptive intermittent control and the theory of Lyapunov stability to realize the synchronization of chaotic systems with time-varying delay. The synchronization of chaotic system is obtained by constructing a conventional Lyapunov function in [24]. In the paper, by using a piecevise function described by

the synchronization of delayed chaotic system is gotten.

Numerical simulations

In this section, two numerical examples are given to present the effectiveness of our results achieved in this paper.

Example 1. Consider the following 3-D oscillator model with variable delay as follows:

In the following, we consider the synchronization between (22) and the following re- sponse system:

where and τ (t) are defined in system (22), the controller U(t) is an intermittent protocol defined in (3).

The dynamic property of (22) with the initial values = (0.2, 0.6, 0.2)^Twith can be emerged, which is revealed in Fig. 1, and the state is chaotic attractor in this case. Furthermore, we choose. Hereinafter, we will choose suitable control parameters such that (22) and (23) achieves the global synchronization.

By computation it can be obtained that . Moreover, the different dynamic properties of (23) with the initial values = ( 0.2, 0.6, 0.2)^T with are presented in Figs. 2–7. The intermittent control exists on time span

.

So,. By simple computing it is easy to verify that = 1/11. Then let a1 = 4, and it can be obtained that q = 3.3042 is the unique positive solution of the equation then we can derive that conditions (i)–(iv) of Theorem 1 are satisfied. From Theorem 1 systems (22) and (23) can be global asymptotical synchronized. Figures 2–4 show the synchronization of dynamics, and Figs. 5–7 show the errors of dynamics.

Example 2. Consider the 3-D neural networks model with variable delay as follows:

In the following, we investigate the synchronization of system (24) and the response system (25) described by

where and τ (t) are defined in system (24), the controller U(t) is an intermittent adaptive protocol defined in (5)–(7).

The dynamic property of (24) with the initial values = (0.8, 0.6, 0.2)T with can be emerged, which is revealed in Fig. 8, and the state is chaotic attractor in this case. Moreover, the different dynamic properties of (25) with the initial values = (0.1, 0.45, 0.45)T with are presented in Figs. 9-12 .The intermittent control exists on time span

So,. From Theorem 2 the networks (24) and (25) can be global asymptotical synchronized under the adaptive intermittent control rules (5)–(7). Figure 9 shows the error of dynamics, and Figs. 10–12 show the synchronization of dynamics. Time evolution of adaptive aperiodic intermittent control gain (t) with (0) = 1.1 and= 1, 2, 3 are given in Figs. 13–15.

Conclusion

The paper deal with synchronization of delayed chaotic systems by means of a novel generalized intermittent and its adaptive strategy. This is to say, this article solves the open question mentioned in [9, 10, 28]. And it is noted that the control protocols in the paper are more general and practical than the traditional periodic intermittent control. Some novel global synchronization criteria have been derived based on the method of piecewise auxiliary function and piecewise analysis technique via the designed control protocols designed in the paper. Finally, two numerical examples are provided to demonstrate the feasibility of the proposed theoretical results.

Acknowledgments

This work was supported by the Research Fund Project of Zhoukou Normal University (grant No. ZKNUC2020006), the Scientific Research Project of Shaanxi Provincial Department of Education (grant No. 18JK0140),the Scientific Research Project of Shaanxi University of Technology in 2016 (grant No. SLGKY16-06) and theNational Natural Science Foundation of China (grants Nos. 62003380, 62003289, U1703262, 61963033).

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