A Novel Disturbance Rejection Zeroing Neurodynamic Approach for Robust Synchronization of Chaotic Systems

Robust synchronization of chaotic systems becomes a hot topic in scientific and engineering fields because of the ubiquitous existence of time-variant external disturbances in complex application scenarios. In contrast with existing studies that the resultant synchronization error has a supremum or even diverges under the influence of time-variant external disturbances, this paper proposes a disturbance rejection zeroing neurodynamic (DRZN) approach and its related controller for the robust synchronization of chaotic systems against time-variant external disturbances. The controller designed by the proposed DRZN approach distinctively features the rejection of external disturbances with the generated synchronization error being convergence toward zero. Theoretical analyses guarantee that the DRZN approach and its related controller inherently possess robustness. Moreover, numerical studies including three examples substantiate the effectiveness of the proposed DRZN approach and its related controller for the synchronization of chaotic systems against the time-variant external disturbances. Comparisons with existing approaches, e.g., the conventional zeroing neurodynamic (CZN) approach and the linear-active control (LAC) approach, show the superiority of the proposed DRZN approach. Extensive tests further verify that the proposed DRZN approach possesses the outstanding anti-disturbance performance, and thus is suitable for the practical applications with time-variant external disturbances.


I. INTRODUCTION
Lorenz [1] creatively introduced the study on chaotic attractor half a century ago. After such a pioneering work, a lot of research on the chaotic control has been developed and investigated [2]- [8]. As an interesting and attractive phenomenon in chaotic control, the chaotic synchronization has become a heated research topic, which has attracted attentions of the researchers in recent years [9]- [13]. The synchronization of chaotic systems is a procedure that two (either identical or nonidentical) chaotic systems adjust a provided property of their motion to a frequent behavior by forcing or coupling [14]. In other words, starting with random initial The associate editor coordinating the review of this article and approving it for publication was Nishant Unnikrishnan.
conditions, all states of the slave (or termed response) chaotic system are forced to asymptotically track all states of the master (or termed drive) chaotic system. The synchronization of chaotic systems has numerous practical applications [15], [16], such as secure communications [17], [18], finance systems [19], electronic systems [20], ecological systems [21], and many different engineering systems [22]. For instances, Mobayen and Ma [13] proposed a new nonlinear feedback approach for the outstanding performance synchronization of chaotic systems under the influence of external disturbances, parametric uncertainties, Lipschitz nonlinearities as well as time delays. Yang and Zhang [17] introduced an effective global chaotic synchronization strategy of the identical systems, and then successfully applied to the secure communication. Naderi and Kheiri [18] detailedly investigated the exponential synchronization of the chaotic system without linear term, and then successfully applied to the secure communication. In [20], numerical, analog together with digital-circuit models were detailedly investigated with a specific chaotic system. In addition, a novel synaptic weight-update-learning rule of Hermite neural network was proposed in [22], and it was applied to the cryptography via combining Hermite polynomials. Moveover, Chen et al. [23] novelly investigated the hybrid synchronization behavior in an array of coupled chaotic systems with ring connection.
Up to now, a large number of approaches and techniques for the synchronization of chaotic systems have been creatively proposed and effectively employed, such as the sliding-mode-control approach [2], the neurodynamic approach [24]- [29], the active-control approach [14], [30], and the adaptive-backstepping control approach [31]. For instances, Ahmad et al. [14] studied and investigated a new global chaotic synchronization problem for identical chaotic systems as well as nonidentical chaotic systems by novelly utilizing a linear-active control (LAC) approach. In addition, Li et al. [24] presented a relatively simple controller generated by the conventional zeroing neurodynamic (CZN) approach for the synchronization of chaotic systems considering the influence of parameter perturbation, model uncertainty as well as external disturbance. Lin et al. [31] proposed an effective systematic approach for modeling and neural adaptive-backstepping control of an uncertain chaotic system by employing only input-and-output data from the underlying-dynamical systems. In the work by Song and Huang [32], the stabilization as well as synchronization of chaotic systems considering the time-variant delays via intermittent control were detailedly investigated. Li et al. [33] novelly developed an effective unified approach for impulsive lag-synchronization of chaotic systems in consideration of time-delay by employing the stability theory of impulsivedelayed-differential equations.
Among effective methodologies for the synchronization of chaotic systems, the neurodynamic approach emerges to be a prior alternative for researchers and engineers, due to the advantages of parallelism, distributed storage, as well as adaptive self-learning capability [34]- [44]. As typical kinds of recurrent neural networks (RNN) [45]- [48], a large number of neural network models designed by the CZN approach, are developed as feasible schemes for the time-variant engineering problems solving (including the time-variant synchronization of chaotic systems) [49]- [52]. For instances, Zhang et al. [50] detailedly investigated the tracking control problems of Lorenz, Chen and Lu chaotic systems via combining the CZN approach and gradient neurodynamic approach for developing an effective controller. In addition, a simple stabilization control approach of hyper-chaotic Lu system with one control input was proposed in [51] by employing the neurodynamic approach. Jin et al. [52] novelly proposed an effective controller design approach for the tracking control of a modified Lorenz nonlinear system in consideration of singularities (or termed division by zero issue) handling.
Although the exhibiting approach, such as the CZN approach [24] and the LAC approach [14], have found extensive applications in different scientific and engineering problems including the synchronization of chaotic systems. However, the research on robustness of the CZN approach for applying to synchronization of chaotic systems still remains deficient. Specifically, in complex application scenarios, there unavoidably exist different forms of external disturbances for chaotic systems such as offset errors, interactions with the environment, electromagnetism interferences in circuit systems, as well as bounded-random noises during signals transmission, etc. [53]. Sometimes those time-variant external disturbances might have negative impacts on both the stability and accuracy of the related controller or model for the synchronization of chaotic systems, and even worse, may cause the failure of the synchronization process [54]. Therefore, a robust approach is urgently needed for the synchronization of chaotic systems to reject the time-variant external disturbances in practice. This is the involved scientific problem and also the motivation that we propose and investigate a novel disturbance rejection approach and its related controller in this work.
As a preliminary attempt, the researchers in reference [24] proposed a relatively simple controller on the basis of the CZN approach to reduce the negative impacts derived from time-variant external disturbances. It was theoretically proven that, with bounded-and-random disturbances, the resultant synchronization error has a supremum by using such a controller [24]. However, the error accumulation phenomenon may occur if such a controller is applied in synchronization of chaotic systems with multiple sub-systems or long synchronization duration finally leading to a high computation burden and low synchronization accuracy. To accelerate the synchronization process of chaotic systems, another work in reference [55] introduced a super-exponential-zeroing neurodynamic approach together with a controller without considering the time-variant external disturbances. It could be theoretically proven that the resultant synchronization error would have a supremum by using the related controller in reference [55] under the influence of disturbances. In addition, Ahmad et al. investigated a new global chaotic synchronization problem by novelly utilizing an LAC approach in [14]. However, the impacts of time-variant external disturbances on the accuracy and stability of chaotic systems are not considered in the work [14].
To handle the difficulties and limitations discussed above, unlike the research based on the CZN approach [24] and LAC approach [14] or focusing on the rate of convergence [55], this paper proposes a novel disturbance rejection zeroing neurodynamic (DRZN) approach and its related controller for the robust synchronization of chaotic systems against the time-variant external disturbances. The controller designed by the proposed DRZN approach distinctively features the rejection of time-variant external disturbances with the synthesized synchronization error being convergence toward zero. Detailed theoretical analyses on the anti-disturbance property and the convergence performance are provided and compared with existing approaches to guarantee the inherent robustness and effectiveness of the proposed DRZN approach and its related controller. Numerical studies including three synchronization examples, comparisons with existing approaches, and extensive tests verify the efficacy and superiority of the proposed DRZN approach and its related controller in practical applications. To the best of the authors' knowledge, such a DRZN approach and its related controller with the outstanding anti-disturbance property for the synchronization of chaotic systems have not been reported in other existing studies. Therefore, this work makes new progresses in both theory and practice for the robust synchronization of chaotic systems. Moreover, the zeroing neurodynamic architecture for designing the controller of chaotic systems is presented in Fig. 1 for better understanding the main principle. As shown in Fig. 1, the whole control system via the proposed DRZN approach is a typical closed-loop control system. The control system uses the initial states of the master and slave chaotic systems as the input information, and the control outputs are used as the system feedback for the slave chaotic system. Both the final state outputs of the master and slave chaotic systems can be integrated to the plant and implemented as the effector for the whole control system.
The rest of the paper is structured as follows. In Section II, the problem formulation of the synchronization between two chaotic systems under the influence of time-variant external disturbances is presented as preliminaries. In Section III, the DRZN approach and its related controller are proposed with theoretical analyses. Section IV shows numerical studies including three synchronization examples, comprehensive comparisons and extensive tests. Section V concludes the paper. The main contributions of the paper are highlighted as below.
• In contrast with existing works that the resultant synchronization error has a supremum or even diverges under the influence of external disturbances, this paper proposes a novel DRZN approach and its related controller for the synchronization of chaotic systems against the time-variant external disturbances.
• The controller generated by the proposed DRZN approach distinctively features the rejection of external disturbances with the synthesized synchronization error being convergence toward zero. It is a breakthrough in the robustness research of neurodynamic approach as well as the synchronization of chaotic systems.
• Theoretical analyses on the anti-disturbance property and the convergence performance are presented and compared to guarantee the inherent robustness and effectiveness of the proposed DRZN approach and its related controller.
• Numerical studies including three synchronization examples, comparisons with existing approaches, as well as extensive tests verify the efficacy and superiority of the proposed DRZN approach and its related controller under the influence of time-variant external disturbances in practical applications.

II. PROBLEM FORMULATION
The problem formulation of the synchronization between two chaotic systems under the influence of time-variant external disturbances are presented in this section. Consider a master chaotic system with a general form as below: where x m (t) = [x m1 (t), x m2 (t), · · · , x mn (t)] T ∈ R n is the state vector of the master chaotic system, and f m (·) : R n → R n is the nonlinear-mapping vector of a specific master chaotic system. Then, the slave chaotic system with a general form is depicted iṅ where vector x s (t) = [x s1 (t), x s2 (t), · · · , x sn (t)] T ∈ R n is the state vector of the slave chaotic system, and mapping f s (·) : R n → R n is the nonlinear-mapping vector of a specific slave chaotic system. Besides, vector u(t) = [u 1 (t), u 2 (t), · · · , u n (t)] T ∈ R n is the control-input vector sent to the slave chaotic system for synchronization.
Note that the exhibiting approaches have found extensive applications in the synchronization of chaotic systems. However, with unexpected impacts, such as the electromagnetism interferences in communication systems and continuous-andrandom noises during the signal transmission, which could be deemed as time-variant external disturbances for the related chaotic systems, a more practical form of master chaotic system can be depicted in the following form to describe the involved scientific problem: n, max being a constant, and T d being a synchronization duration, of the master chaotic system.
If the initial-state vectors x m (0) and x s (0) of the master and slave chaotic systems are different from each other, the state trajectories of such two systems may differ very much. The objective in the paper for synchronization between the master chaotic system (3) with external disturbances (t) and the slave chaotic system (2) is to design a control-input vector u(t) so that the slave system is forced to track the master system with the synthesized state error e(t) = x m (t) − x s (t) converging toward zero.
Remark 1: Due to the complexity of the synchronization process of chaotic systems, the external disturbances may be unavoidable in real-world applications. In addition, most of specific chaotic systems could be formulated as the master chaotic system (3) with a general form of nonlinear-differential equations, such as Lu chaotic systems [50], Chen chaotic systems [51], and Lorenz chaotic systems [52], which covers most common chaotic systems [24].

III. DESIGN APPROACHES AND THEORETICAL ANALYSES
In this section, we propose the DRZN approach for designing the controller of chaotic systems. For further investigation and better comparison, the controller designed by the CZN approach is also provided. In addition, theoretical analyses are presented to guarantee the effectiveness and robustness of the proposed DRZN approach and the related controller.

A. DESIGN APPROACHES
To achieve the synchronization between the slave chaotic system (2) and the master chaotic system (3) under the influence of external disturbances, a controller is designed by the DRZN approach with detailed process shown as below.
Firstly, to monitor the synchronization process of chaotic systems (2) and (3), a vector-valued error function to measure the difference between the states of the master and slave chaotic systems is defined as below: To make each element e i (t) (with i = 1, 2, · · · , n) of the synthesized error (4) converge towards zero with respective to time t, the DRZN approach is employed with its dynamic equation described as below: where design parameters κ ∈ R + and ν ∈ R + are chose for the stability as well as convergence of the neurodynamic model, and vector (t) denotes the time-variant external disturbances that need to be rejected. In addition, ϒ(·) : R n → R n is an alternative activation-function vector mapping. Note that the elements of the activation-function vector mapping can be a monotonically-increasing odd function which could be adjusted to improve the convergence performance of the neurodynamic model. Without losing of generality and also for simplicity, a linear activation-function ϒ(e(t)) = e(t) is utilized and investigated in the paper.
By substituting chaotic systems (2) and (3) into dynamic equation (5), the corresponding neurodynamic model for the synchronization of chaotic systems can be depicted as follows: Due to the fact that there unavoidably exist unexpected time-variant external disturbances in complex application scenarios, the synchronization of chaotic systems would be a knotty time-variant problem. As readily founded in neurodynamic model (6), the proposed DRZN approach can effectively handle the disturbance rejection issue in a relatively simple manner by fully exploiting both the time-derivative and integral information of the involved chaotic systems. According to neurodynamic model (6), the related controller with explicit control-input vector u(t) is thus designed as follow: (7) Note that the related controller (7) designed by the proposed DRZN approach does not require any information VOLUME 7, 2019 FIGURE 2. Neuron-connection architecture of neurodynamic model (6) for designing the controller of chaotic systems with x si denoting the i th neuron.
of external disturbances which is thus applicable in practical applications. For better understanding of practitioners, the neuron-connection architecture of neurodynamic model (6) for designing the controller of chaotic systems via the proposed DRZN approach is presented in Fig. 2. As we can see from the figure, the neurodynamic model (6) is a typical kind of Hopfield-type RNN [56], which can be developed and implemented easily on analog circuits such as very-large-scale-integration [57].
To lay a basis for further investigation and comparison, the controller designed by the CZN approach for the synchronization of chaotic systems is also provided as below [24]: (8) where design parameter γ ∈ R + is set for the stability and convergence of the above controller, and (·) : R n → R n is an alternative activation-function vector mapping for the CZN approach. For better comparison, design parameters are set to be γ = κ, and activation-function vector mappings ϒ(·) and (·) are both set to be linear activation function.

B. THEORETICAL ANALYSES
In this subsection, to guarantee the effectiveness and robustness of the proposed DRZN approach as well as the related controller (7) for the synchronization of chaotic systems under the influence of time-variant external disturbances, the theoretical analyses are presented. Then, the corresponding theoretical results of the CZN approach are also presented for better comparison.
Definition 1: For the synchronization of chaotic systems (2) and (3), starting with a random initial state x s (0), a vector-valued error function e(t) at time t ≥ 0 synthesized by a control system is said to be convergent toward zero if it satisfies lim t→∞ sup{ e(t) E } = 0, where symbol sup{·} denotes the supremum of a sequence, and symbol · E denotes the Euclidean norm of a vector.

Definition 2:
For the synchronization of chaotic systems (2) and (3), starting with a random initial state x s (0), a state trajectory x s (t) of slave chaotic system (2) at time t ≥ 0 synthesized by a control system is said to be convergent toward the state x m (t) of master chaotic system (3) with external disturbances if it satisfies Theorem 1: For the synchronization of chaotic systems (2) and (3), starting with a random initial state x s (0), the vectorvalued error function e(t) at time t ≥ 0 synthesized by control system equipped with controller (7) converges toward zero.
Proof: Review the neurodynamic model (6) designed by the proposed DRZN approach as Within the region of convergence and e(t) = x m (t) − x s (t), the Laplace transformation [58] of the ith subsystem of neurodynamic model (9) is depicted as where operator L(·) denotes the Laplace transformation with i = 1, 2, · · · , n with activation-function vector mapping being a linear activation function, i.e., ϒ i (e i (t)) = e i (t). From (10), we have where +∞ 0 i (t)exp(−st)dt is the Laplace transformation of each element of time-variant external disturbances, i.e., i (t). Note that equation (11) could be rewritten as the following form: Note that in the field of signals and systems, the neurodynamic model with time-variant disturbances can be skillfully characterized and analyzed in the transform domain by algebraic manipulations. According to the final-value theorem [58], for bounded external disturbances during the synchronization process, i.e., | i (t)| ≤ max for t ∈ [0, T d ], and i (t) = 0 for t > T d with i = 0, 1, · · · , n, we have With each element of error function bing | lim t→∞ e i (t)| = lim t→∞ e i (t) = 0 (i = 1, 2, · · · , n), we finally obtain the following result: According to Definition 1, we finally have the result that the vector-valued error function e(t) synthesized by the control system equipped with controller (7) converges toward zero. The proof is thus completed.
Corollary 1: For the synchronization of chaotic systems (2) and (3), starting with a random initial state x s (0), the state trajectory x s (t) of slave chaotic system (2) at time t ≥ 0 synthesized by a control system equipped with controller (7) converges toward the state x m (t) of master chaotic system (3) with external disturbances.
Proof: It can be generalized from Definition 2 and the proof of Theorem 1.
For better comparison, the corresponding theoretical results of the controller (8) designed by CZN approach for the synchronization of chaotic systems (2) and (3) are also presented as the following lemma [24].
Lemma 1: For the synchronization of chaotic systems (2) and (3), starting with a random initial state x s (0), the vector-valued error function e(t) at time t ≥ 0 synthesized by control system equipped with controller (8) converges with a steady-state error having a supremum, i.e., lim t→∞ sup{ e(t) E } = (t) E /γ . Proof: Review the neurodynamic model (6) designed by the CZN approach as follows: According the chaotic systems (2) and (3), with e(t) = x m (t)−x s (t), one can obtain the following dynamic equation: Solving the above dynamic equation, one can readily obtain: where α is a constant. Therefore, one can finally have: Therefore, it has the result that the vector-valued error function e(t) at time t ≥ 0 synthesized by control system equipped with controller (8) converges with a steady-state error having a supremum (t) E /γ . The proof is thus completed.

IV. NUMERICAL STUDIES
In this section, numerical studies including three synchronization examples, comparisons with existing approaches, and extensive tests are performed to verify the effectiveness, robustness as well as superiority of the proposed DRZN approach and the related controller (7) for the synchronization of chaotic systems with time-variant disturbance rejection.

A. SYNCHRONIZATION EXAMPLES
In the examples, we successively consider the synchronization of two identical Lu chaotic systems, synchronization of two identical autonomous chaotic systems, and synchronization of two nonidentical chaotic systems. Without losing generality, the synchronization duration is selected to be T d = 10 s. In addition, design parameters are selected to be κ = 3 and ν = 30. The initial value of each state of both the master chaotic systems and slave chaotic systems is generated randomly between 0 and 1.

1) SYNCHRONIZATION OF TWO IDENTICAL LU CHAOTIC SYSTEMS
Firstly, consider the following Lu chaotic system [24]: where a = 36, b = 3 and c = 20. For the synchronization of two identical Lu chaotic systems, one can have the master chaotic system with external disturbances is described aṡ and the slave chaotic system with control-input vector is described aṡ The corresponding numerical results of synchronization of two identical Lu chaotic systems (17) and (18) equipped with controller (7) via the proposed DRZN approach are presented in Fig. 3 and Fig. 4. Specifically, Fig. 3(a) shows VOLUME 7, 2019 FIGURE 3. Synchronization and disturbance rejection performance between two identical Lu chaotic systems (17) and (18) equipped with controller (7) using the proposed DRZN approach.  (17) and (18) equipped with controller (7) using the proposed DRZN approach.
real-time synchronization of such two identical Lu chaotic systems (17) and (18) in 3-dimensional space. With initial value of each state randomly generated in [0, 1], the slave Lu chaotic systems (18) quickly synchronizes toward the master Lu chaotic systems (17). In addition, Fig. 3(b) through Fig. 3(d) respectively illustrate each state, i.e., x s1 , x s2 and x s3 of the slave system, which coincides well with each state, i.e., x m1 , x m2 and x m3 , of the master system even under the influence of external disturbances. As detailedly shown in Fig. 4, the absolute values of synchronization errors of all states are relatively small (or to say, ignorable), and quickly converge toward zero. Moreover, the supremum of each error, i.e., sup{|x mi − x si |} (with i = 1, 2 and 3), shows convergence tendency, which is consistent with the theoretical result provided in Theorem 1, i.e., with the error function being convergence toward zero. The above results illustrates the great synchronization and disturbance rejection performance of the proposed DRZN approach as well as the related controller (7) for the synchronization of two identical Lu chaotic systems.

2) SYNCHRONIZATION OF TWO IDENTICAL AUTONOMOUS CHAOTIC SYSTEMS
Consider the following new autonomous chaotic system proposed in [14]: where p = 40, q = 5, r = 30 and s ∈ [0, 10]. For the synchronization of two identical autonomous chaotic systems with the above form, we have the master chaotic system with external disturbances is described aṡ (20) and the slave chaotic system with control-input vector is described aṡ The corresponding numerical results of synchronization of two identical autonomous chaotic systems (20) and (21) equipped with controller (7) using the proposed DRZN approach are illustrated in Fig. 5 and Fig. 6. Specifically, Fig. 5(a) shows synchronization of such two identical autonomous chaotic systems (20) and (21) in 3-dimensional space. With initial value of each state randomly generated in [0, 1], the slave autonomous chaotic systems (21) also quickly synchronizes toward the master autonomous chaotic systems (20). In addition, Fig. 5(b) through Fig. 5(d) respectively show each state, i.e., x s1 , x s2 and x s3 of the slave system, coincides well with each state, i.e., x m1 , x m2 and x m3 , of the master system even under the influence of external disturbances. As presented in Fig. 6, the absolute values of synchronization errors of all states are also relatively small (or say, ignorable), and quickly converge toward zero. Moreover, the supremum of each error, i.e., sup{|x mi −x si |} (with i = 1, 2 and 3), shows convergence tendency, which is consistent with the theoretical result presented in Theorem 1, i.e., with the error function being convergence toward zero.

3) SYNCHRONIZATION OF TWO NONIDENTICAL CHAOTIC SYSTEMS
In the example, we further consider and achieve the synchronization of two nonidentical chaotic systems, i.e., with the maser chaotic system being the Lu chaotic system (17) and the slave chaotic system being the autonomous chaotic system (21).
The corresponding numerical results of synchronization of two nonidentical chaotic systems (17) and (21) equipped with controller (7) using the proposed DRZN approach are illustrated in Fig. 7 and Fig. 8. Specifically, the real-time synchronization of such two nonidentical chaotic systems (17) and (21) is shown in Fig. 7(a) in 3-dimensional space. With initial value of each state randomly generated in [0, 1], the slave autonomous chaotic system (21) still quickly synchronizes toward the master Lu chaotic system (17). In addition, Fig. 7(b) through Fig. 7(d) respectively show each state, i.e., x s1 , x s2 and x s3 of the slave autonomous chaotic VOLUME 7, 2019 FIGURE 6. Absolute errors between two identical autonomous chaotic systems (20) and (21) equipped with controller (7) using the proposed DRZN approach.

FIGURE 7.
Synchronization and disturbance rejection performance between two nonidentical chaotic systems (17) and (21) equipped with controller (7) using the proposed DRZN approach. system (21), almost overlaps each state, i.e., x m1 , x m2 and x m3 , of the master Lu chaotic system (17) even under the influence of external disturbances. As detailedly shown in Fig. 8, the absolute values of synchronization errors of all states are also relatively small and can be ignorable, and quickly converge toward zero. Moreover, the supremum of each error, i.e., sup{|x mi − x si |} (with i = 1, 2 and 3), shows convergence tendency, which is also consistent with the theoretical result provided in Theorem 1, i.e., with the error function being convergence toward zero, for the case of synchronization of two nonidentical chaotic systems.
Remark 2: Note that this work investigates the robust synchronization of multiple dimensional chaotic systems against time-variant external disturbances. Specifically, the dimension of the involved chaotic systems is set to be n.
Without loss of generality, the dimension n is chosen as 3 in the above synchronization examples of numerical studies. Actually, the dimension of the involved chaotic systems can be chosen as 2 for low dimensional chaotic systems. As shown in the above synchronization examples, the absolute values of synchronization errors of all states are relatively small (or to say, ignorable) with high enough synchronization accuracy, and quickly converge toward zero within shot time.

B. COMPARISONS WITH OTHER APPROACHES
In the subsection, to verify the robustness and superiority of the proposed DRZN approach, we conduct and show the numerical comparisons by using the CZN approach as well as the LAC approach for the synchronization of two identical FIGURE 8. Absolute errors between two nonidentical chaotic systems (17) and (21) equipped with controller (7) using the proposed DRZN approach. autonomous chaotic systems (20) and (21) under the influence of the same external disturbances.
The CZN approach was derived from Zhang et al. [24], and it has been proven to effectively solve the time-variant problems including the synchronization of chaotic systems without external disturbances. To be specifically, for synchronization of chaotic systems, the related controller designed by the CZN approach is depicted as (8). For better comparison, design parameter is set to be γ = 3. Other numerical conditions are set the same as those in Section IV-A.2. The corresponding numerical results of synchronization performance between two identical autonomous chaotic systems (20) and (21) equipped with controller (8) using the CZN approach are presented in Fig. 9 and Fig. 10. Specifically, with 3dimensional trajectories illustrated in Fig. 9(a), the slave system (21) can not show an acceptable synchronization performance. With initial value of each state randomly generated in [0, 1], the slave autonomous chaotic systems (21) synchronizes toward the master autonomous chaotic systems (20) with a bounded error. In addition, Fig. 9(b) through Fig. 9(d) shows each state, i.e., x s1 , x s2 and x s3 of the slave system, can not coincide well with each state, i.e., x m1 , x m2 and x m3 , respectively, of the master system under the same influence of external disturbances. As detailedly shown in Fig. 10, all the absolute values of synchronization errors of all states are also non-ignorable, and have an error bound but not VOLUME 7, 2019 FIGURE 10. Absolute errors between two identical autonomous chaotic systems (20) and (21) equipped with controller (8) using the CZN approach. converge toward zero. Moreover, the supremum of each error, i.e., sup{|x mi − x si |} (with i = 1, 2 and 3), does not show convergence tendency but keeps within an error bound, which is also consistent with the theoretical result presented in Lemma 1, i.e., with the error function having a supremum, i.e., (t) E /γ . Compared with the CZN approach, the above results verify the robustness and superiority of the proposed DRZN approach as well as the related controller (7) for the synchronization and disturbance rejection of chaotic systems.
Recently, as a novel alternative method for the synchronization of chaotic systems, the novel LAC approach [14] is proposed for the design of controller. Specifically, the related controller for the synchronization of two identical chaotic systems (20) and (20) generated by the LAC approach is provided as below: For better comparison, other numerical conditions are set the same as those in Section IV-A.2. The simulated results of synchronization between two identical chaotic systems (20) and (21) equipped with controller (22) using the LAC approach are presented in Fig. 11. As one can find in the figure, all the absolute values of synchronization errors of all states are also non-ignorable with the errors |x m1 (t) − x s1 (t)| and |x m2 (t) − x s2 (t)| being divergent and |x m3 (t) − x s3 (t)| having an error bound. Therefore, the time-variant external disturbances would have a negative impact on the accuracy and stability of chaotic systems equipped with controller (22), and even worse, may destroy the corresponding synchronization process.
Further, to further highlight the advantages of the proposed DRZN approach and its related controller (7) compared with the related works in recent years, comprehensive comparisons among different approaches for the synchronization of chaotic systems with external disturbances are summarized in Table 1. As seen from the comparisons, the DRZN proposed in this paper shows the anti-disturbance performance and high synchronization accuracy, which is substantiated via three examples in Section IV-A. Compared with other existing approaches in recent years, the vector-valued error e(t) synthesized by control system equipped with the proposed controller converges toward zero, which is better than other approaches (e.g., the CZN and LAC approaches [14], [24], as well as the other recent approaches in [50]- [52], [55], [59]) with a bounded error or divergence tendency.

C. EXTENSIVE TESTS
To further investigate the anti-disturbance performance of the proposed DRZN approach and the related controller (7), extensive tests of chaotic systems with longer synchronization duration (i.e., 50 s) as well as under different forms of time-variant disturbances are conducted successively. The corresponding numerical results are shown in Fig. 12 and Fig. 13. Specifically, the absolute values of synchronization errors illustrate a persistent convergence tendency for a long   (17) and (18) for 50 s duration equipped with controller (7) using the proposed DRZN approach. FIGURE 13. Absolute errors between two identical autonomous chaotic systems (20) and (21) for 50 s duration equipped with controller (7) using the proposed DRZN approach.  (17) and (18) under different forms of disturbances equipped with controller (7) using the proposed DRZN approach.  (20) and (21) under different forms of disturbances equipped with controller (7) using the proposed DRZN approach.
synchronization duration (see both Fig. 12 and Fig. 13). It means that the synchronization errors can decrease to sufficiently small as time tends to appropriately large enough.
In order to evaluate the synchronization performance in finite time and also to monitor the convergence process during the whole synchronization under different forms of VOLUME 7, 2019 external disturbances, the maximal absolute value of synchronization error, i.e., max{e(t)}, synthesized by the proposed DRZN approach and its related controller (7) at different time instants, i.e., t = 10 s, 20 s, 30 s, 40 s, 50 s, is presented in Table 2 and Table 3 for synchronization of Lu and autonomous chaotic systems, respectively. As both shown in Table 2 and Table 3, all the maximal absolute values of synchronization errors show convergence properties with different forms of time-variant disturbances (i.e., the constant form, sine form, exponential-decay form, and bounded random form). The above graphical results are also consistent with theoretical results. In other words, control system using the DRZN approach and its related controller (7) can converge toward the solution rapidly. All the numerical results of extensive tests verify that the proposed DRZN approach possesses the outstanding and inherent anti-disturbance performance, and thus is suitable for practical applications in the complex environment with time-variant external disturbances.

V. CONCLUSION AND FUTURE WORKS
In this paper, a DRZN approach and its related controller (7) have been proposed for the synchronization of chaotic systems against the time-variant external disturbances. Differing from existing works that the resultant synchronization error has a supremum or even diverge under the influence of external disturbances, the proposed DRZN approach can effectively suppress the external disturbances with the synthesized synchronization errors being convergence toward zero. It has been proven that the proposed DRZN approach and its related controller (7) inherently possess robustness. Moreover, numerical studies via three different examples have substantiated the effectiveness of the proposed DRZN approach as well as its related controller (7) for the synchronization of chaotic systems against the time-variant external disturbances. Comparisons with existing approaches, e.g., the CZN approach and LAC approach, have shown the superiority of the proposed DRZN approach. Meanwhile, extensive tests have further shown that the proposed DRZN approach possesses the outstanding and inherent anti-disturbance performance, and thus is suitable for the practical applications in complex application scenarios with time-variant external disturbances.
Future works and improvements lie in the following facts: i) investigation on different dimensional and different types of chaotic systems under the influence of time-variant external disturbances via the proposed DRZN approach and its related controller; ii) investigation on the finite-time convergence property of the proposed DRZN approach under the influence of time-variant external disturbances; iii) extension and implementation of synchronization of chaotic systems with internal system uncertainty with complete theoretical analyses on robustness of the proposed DRZN approach; and iv) development of the proposed DRZN approach and its related controller on an electrical systems to verify the physical realizability. As a final remark of this paper, to the best of authors' knowledge, this is the first work in the framework of zeroing neurodynamic which is able to elegantly handle the synchronization of chaotic systems under the influence of time-variant external disturbances with zero-oriented convergence performance.