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Synchronization Between Adaptively Coupled Systems With Discrete and Distributed Time-Delays Wei Lin, Member, IEEE, and Huanfei Ma, Student Member, IEEE
Abstract—This paper investigates complete synchronization of unidirectionally and adaptively coupled systems with discrete and distributed time delays. Instead of the conventional hypothesis of a uniform Lipschitz condition on the system’s vector fields, only a local Lipschitz condition is adopted. It is proved that the local complete synchronization can be achieved through a unidirectional and adaptive coupling, and that the global complete synchronization can be realized when the nonlinear degree of the vector fields is smaller than some derived critical value. The results are illustrated in some representative models with time delays. Also considered is complete synchronization with an exponential convergence rate on the adaptively coupled time-delayed systems with vector fields that are one-sided uniformly Lipschitz (systems of this type can admit time-varying discrete and distributed delays). All the results can be further generalized to various types of synchronization between bidirectionally coupled time-delayed systems or among delayed kinetic systems of a complex network. Index Terms—Adaptive coupling, control, locally Lipschitzian, synchronization, time-delays.
I. INTRODUCTION INCE Huygens’ observation about the synchrony of pendulum clocks, systematic investigations and potential applications of synchronization of dynamical systems have ranged from physics to chemistry, electronic engineering, computer science, and neuroscience [1]–[5]. Various techniques have been developed for complete synchronization, generalized synchronization, lag synchronization, and phase synchronization in synthetic and natural dynamical systems [3], [4]. Representative approaches include the conventional linear and nonlinear coupling techniques, impulsive control method, invariant manifold method, adaptive coupling technique, and noise-based coupling method [6]–[14].
S
Manuscript received December 23, 2008; revised May 22, 2009, May 30, 2009, and July 23, 2009. First published February 02, 2010; current version published April 02, 2010. This work was supported by the National Natural Science Foundation of China under Grants 10501008 and 60874121, by the Rising-Star Program Foundation of Shanghai, China under Grant 07QA14002, and by the National Basic Research Program of China under Grants 2006CB303102 and 2007CB814904. Recommended by Associate Editor R. D. Braatz. W. Lin is with the Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, China, the School of Mathematical Sciences and the Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China. He is also with the CAS-MPG Partner Institute for Computational Biology, Chinese Academy of Sciences, Shanghai 200031, China (e-mail:
[email protected]). H. Ma is with the School of Computer Science and the Center for Computational Systems Biology, Fudan University, Shanghai 200433, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2010.2041993
In spite of the extensive exploration of synchronization theory, some underlying mechanisms and fundamental questions are still pending for illustrations and answers. For instance, when the conventional adaptive coupling technique is implemented to realize complete synchronization between coupled dynamical systems, one of the essential conditions assumed, a priori, is that the system’s vector fields are uniformly Lipschitz [15]–[19]. However, as ubiquitous in many systems such as the well-known Lorenz-like system [20], [21], the Rössler system [22], and the Lotka-Volterra (biological population) system [23], not uniform Lipschitz but merely local Lipschitz condition can be satisfied. Synchronization in systems that violate the uniform Lipschitz condition cannot be surely realized through the adaptive coupling with arbitrary initial values [24]. Thus, it becomes important and useful to establish criteria for synchronization in adaptively coupled systems that are only locally Lipschitz. On the other hand, in many systems, the future states of the system depend not only on the present state but also on the past state at discrete instant or in continuous time intervals [25], [26]. Thus, models with time-delays become more accurate. Such time-delayed models are infinite-dimensional in nature and can contribute to instability and dynamic complexity [25]–[27]. Complete synchronization in adaptively coupled uniformly Lipschitzian systems with time delays has been theoretically and experimentally considered [28]–[30]. Nevertheless, very few results have been reported in the literature for synchronization in adaptively coupled time-delayed systems with only locally Lipschitzian vector fields and time-varying delays. This paper establishes criteria for complete synchronization between unidirectionally and adaptively coupled systems with discrete and distributed time delays. Instead of the conventional hypothesis of a uniform Lipschitz condition on the system’s vector fields, only a local Lipschitz condition is assumed. Also investigated are adaptively coupled systems with time-varying delays that have vector fields that satisfy a one-sided uniform Lipschitz condition. The remainder of the paper is organized as follows. Section II introduces the problem of adaptive synchronization in unidirectionally coupled systems with various types of time delays and definitions for complete synchronization and several Lipschitz conditions. Section III shows that complete synchronization between locally Lipschitzian systems with time delays can be achieved by application of unidirectional and adaptive coupling for restricted initial values for the coupling variables. Section IV provides a feasible method to relax the restriction on the selection of the initial values and then shows that complete
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synchronization can be obtained globally if the nonlinear degree of the given time-delayed system’s vector fields is smaller than some derived critical value. Section V establishes a delay-independent theory for complete synchronization with an exponential convergence rate between adaptively coupled time-varying delayed systems with vector fields that satisfy the one-sided uniform Lipschitz condition. This is followed by concluding remarks. II. PROBLEM FORMULATION AND PRELIMINARIES Consider the problem of characterizing the unidirectional and adaptive coupling technique for complete synchronization of a class of nonlinear dynamical systems with only locally Lipschitzian vector fields and various forms of time-delays. More specifically, consider the driving system described by the following functional differential equations: (1) in which with the initial condition . Here, is a maximal time delay condenotes the space of all continuous stant, -valued functions on the interval , the state vector , and for . Moreover, each component of the vector field is given in the following explicit form: (2) in which each and each
is a finite natural number, belongs to the
, family . Here, consists of all the Dirac delta functions , and with a compact point set support in the interval consists of all the Borel measurable bounded normalized by nonnegative functions on the interval . Each can typically be regarded as a weight or density function in application. Specifically, when is selected as the Dirac delta function with a compact , in which , the driving system (1) support , system (1) has has multiple discrete time-delays. If all no time delays. In addition, when is a continuous and for some indexes and , the nonnegative function on driving system (1) has at least a time delay in the distributive form. Furthermore, to avoid the trivial case of no time delays, it is assumed that there exist at least one pair of indexes satisfying for all . In the above settings, one can see that many of well-known dynamical systems with time-delays, such as the delayed Logistic model, the chaotic delayed Ikeda model, the artificial neural network model with discrete and distributed time-delays, and the predator-prey model with distributed delays, can be written in the form of (2). , the objective is Now, given a driving signal to design a feasible coupling technique to realize complete synchronization between two identical systems with different initial conditions. In practice, the coupled systems can be regarded function
as those interacting dynamical elements in a large network of agents, such as physical particles, biological neurons, ecological populations, genic oscillators, and even automatic machines and robots [2]–[5]. A feasible coupling design for realizing complete synchronization allows a full command of the intrinsic mechanism regulating the evolution of interacting real systems, an exact fabrication of mimic systems, and a remote control of the machines and nodes in networks with large scales. Typically, adaptive coupling for complete synchronization between driving and response systems requires such a feasible design, in which the response systems are
(3) The initial values for system (3) are taken as , , where . Starting from these initial values, the trajectory of (3) , or equivalently, by is denoted by . In system (3), the definitions of the state vectors and are analogous to those of the corresponding state vectors in system (1), respectively. Moreover, and each is a properly selected positive constant. will be further clarified in the following The role of each section. Next, the mathematical definition of complete synchronization between the unidirectionally and adaptively coupled systems (1) and (3) is introduced. Definition 2.1: The driving system (1) and the response system (3) are said to be locally completely synchronized, provided that, for some subset of and for and , there exists any initial values of such that a subset
whenever the initial value for in system (3) is taken from the set . If, furthermore, , then the coupled systems (1) and (3) are said to be globally completely synchronized. In addition, the following mathematical assumptions on the system’s vector fields are imposed. in (2) is supAssumption 2.2: Each scalar function posed to be locally Lipschitz, i.e., for each pair of compact sets , there exists a positive constant such that (4) where denotes the Euclidean norm in . in AsRemark 2.3: Note that, if the value of the constant sumption 2.2 is independent of the choice of the compact sets and in , then the function in (2) is said to be uniformly Lipschitz or globally Lipschitz. However, if the value is crucially related to the structure of and of the constant , then the function is not uniformly but only locally Lipschitz.
LIN AND MA: SYNCHRONIZATION BETWEEN ADAPTIVELY COUPLED SYSTEMS
In conventional driving and response systems, complete synchronization can be realized through a linear coupling that involves a constant coupling gain. However, the characteristic of the unidirectionally adaptive coupling in (3) allows the coupling to be self-adjustable depending on the distance begain tween driving and response signals. The larger the difference between these signals, the stronger the coupling gain. Based can be arbitrarily on this mechanism, the initial value for selected and global complete synchronization can be realized as long as the system’s vector fields in (1) and (3) are uniformly Lipschitz. Nevertheless, global complete synchronization cannot be definitely realized for some systems whose vector fields are only locally Lipschitz. This is because the local Lipschitz condition is extraordinarily weaker than the uniform Lipschitz condition. In general, a sufficiently smooth vector field merely satisfies the local Lipschitz condition rather than in the uniform Lipschitz condition. Thus, two questions naturally arise: “What kind of adaptively coupled systems can be locally completely synchronized? ” and “What kind of coupled systems can be further globally completely synchronized?” These questions will be answered in the following sections. To be more practical, the following assumption on the boundgenerated by system edness of the driving signal (1) is also imposed. of system (1) is Assumption 2.4: The driving signal and for any bounded, that is, for some , there exists a compact set such that the trajectory of (1) starting from the given initial data is for all , or always in the set , i.e., for all and . Noticeably, Assumption 2.4 allows the case that system (1) can have unbounded trajectories aside from the bounded trajec. However, the trajectory, which tories starting from the set is selected as the driving signal, should be bounded on the entire . time interval In addition, two measures for any pair of elements in the funcare introduced. tions set , Definition 2.5: For any two functions , define
which is the conventional supremum norm for continuous -valued functions on a bounded interval, and define
where each is a constant. endowed with Remark 2.6: The function set in Definition 2.5 becomes a Banach the first measure space. Each in the second measure is a tunable constant, and its value depends on specific requirements in applications. This will be illustrated in the next section. Here, the second measure has the property
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for all . Together with Assumption 2.4 and Definition 2.5, the following property describes the wandering region of a driving signal in the function space. It will be useful in the next section. Proposition 2.7: Suppose that Assumption 2.4 is fulfilled. , and Then, there exist three positive constants, , , a constant vector function with , and a compact ball
such that the driving signal for all . Proof: First of all, from Assumption 2.4, it follows that and a constant such there exist a constant vector that, for all
Thus, this implies that, for any , so that . Set
and
, , in which
Since each integral , where the strict inequality holds at least for some pair of indexes and , one , giving an estimation has
for all
for all
. Moreover
. This implies
for all . Hence, . On the other is closed, and that hand, it is not hard to verify that are uniformly bounded and equi-conall functions in tinuous. Consequently, the compactness of directly follows from the Ascoli-Arzelà theorem [31]. III. SYNCHRONIZATION IN ONLY LOCALLY LIPSCHITZIAN SYSTEMS WITH TIME-DELAYS THROUGH ADAPTIVE COUPLING This section is to derive an analytical criterion for realizing complete synchronization between the unidirectionally and
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adaptively coupled systems (1) and (3), whose vector fields are only locally Lipschitz. be the initial-value set for generating bounded Let driving signal of system (1). For any given and , define a positive constant as . , where is the In addition, let positive constant in (3). The following theorem shows what kind of trajectories produced by the coupled systems (1) and (3) are always bounded. Theorem 3.1: Suppose that both Assumptions 2.2 and 2.4 are satisfied. Then, there exists a constant vector such that, for all , the trajectory
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where is specified in the proof of Proposition and yields . 2.7. Letting This inequality thus implies that
for all . Hence, on is bounded. Now, in light of the classical theory of functional differential equations [25], [32], it is concluded that the right end point can be further extended . This will complete the proof of the theorem. to Thus, what is left to prove is to verify for all . This assertion can be shown by contradiction. If the assertion is not true, then there exists a positive time constant
such that for all . Through a similar argument as performed above, one can show that, for all
where , is obtained in Proposition 2.7, the initial value for coupling variable is selected from the functions set
(5) is an arbitrary constant. Proof: As assumed, the driving signal of system (1) exists and is bounded on the entire interval . However, the promaximal interval of existence of the trajectory ; duced by the coupled systems (1) and (3) may not be thereby we suppose this maximal interval of existence is with . It is to prove that for . If this proof is completed, we have all and
Hence, from Proposition 2.7, it follows that both trajectories and stay in the compact ball for all , where
Therefore, according to Assumption 2.2, there exist positive such that numbers (6) for all and . With these positive numbers, one can determine the constant vector componentwise by (7)
for all . This inequality implies that is . Notice the combounded on , and that as shown in Proposition 2.7. Thus, pactness of the set , such there exists a convergent sequence that and
where is a constant. Next, consider the differentiable function of (8) On the one hand
as . Hence, from Definition 2.5, Remark 2.6, and Propoand sufficiently large sition 2.7, it follows that, for any because the initial value over, from the above definition of lows that:
, as defined in (5). Moreand , it fol-
LIN AND MA: SYNCHRONIZATION BETWEEN ADAPTIVELY COUPLED SYSTEMS
On the other hand, the derivative of the function with respect to on the interval gives
where the first inequality is attributed to the inequality (6); the second inequality is due to the elementary Young’s inequality; the third inequality follows from the setting of and the Cauchy-Schwartz inequality for integrals with . In addition, by the property of the Dirac delta function, the above estimation is still valid even if . Consequently, from the last inequality in the above estiis nonmation, it follows that the function with respect to , so that increasing on the interval
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. This leads to a contradiction, eventually completing the proof of the theorem. Remark 3.2: Note that any functions, belonging to either the set or the set , are not necessarily uniformly bounded, because these sets are not compact in the infinite-di. However, the boundedness of mensional space generated by the coupled systems (1) each trajectory and (3) on the interval is attributed to the distributive . Moreover, writing form of the trajectory, keeping both inequalities strict in the construction of the sets and is necessary to the completion of the above proof. in (7) can be Remark 3.3: In some case, the choice of . optimized by a proper selection of the positive constants More specifically, when some of the density functions are the Dirac delta function with compact support , the corresponding local Lipschitz constants, which are separated from , can contribute to dithe term rectly. This separation allows the value of to be further relaxed. Theorem 3.4 shows complete synchronization between the adaptively coupled systems (1) and (3) whose trajectories are all bounded. Theorem 3.4: Suppose that both Assumptions 2.2 and 2.4 is the initial-value set for generating are satisfied, and that bounded driving signal of system (1). Then, the adaptively coupled systems (1) and (3) are locally completely synchronized, , i.e., for any given initial values and , the difference between the driving and response signals satisfies
where , is an arbitrary constant strictly larger is defined in (5). than one, and Proof: Clearly, the coupled systems (1) and (3) constitute a group of autonomous functional differential equations whose vector fields are completely continuous, that is, is continuous into bounded and takes closed bounded sets of sets of . From Theorem 3.1, it follows that the trajectory starting from , denoted by , is bounded and refor all . mains in Rewrite the function (8) in the form of . Then, it is continis not hard to verify that the functional , denoted by . Moreuous on the closure of over, analogously to the proof of Theorem 3.1, the derivative of this functional along the trajectory of the coupled systems (1) and (3) yields
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for any initial values implies
. Thus, this
Therefore, by virtue of the LaSalle invariance principle for autonomous functional differential equations [32], the bounded tends to the largest invariance set trajectory contained in . In particsatisfies , ular, each trajectory restricted in for all . Therefore, the limit fori.e., mula in the theorem is validated. In addition, the state vector , restricted in , becomes a constant -valued function and the converges as the time tends monotonous coupling variable to infinity. Remark 3.5: When the driving signal is selected as an unstable equilibrium of the system (1) on the , the above-described complete synentire interval chronization between systems (1) and (3) can be regarded as an adaptive control of system (1) with a control target . In such a case, by simply setting , , and in Theorems 3.1 and 3.4, all the above-established analytical results on complete synchronization can be adapted to realizing the adaptive control in those time-delayed systems whose vector fields are only locally Lipschitzian. Also, the driving signal can be selected as an unstable periodic orbit of system (1). To illustrate an application of the above analytical results, consider the time-delayed Lotka-Volterra (LV) model, which is frequently employed in biology, medicine, and economics [26]. Example 3.6: Consider a four-dimensional LV model with multiple discrete time-delays as follows:
(9) where
, with
, model (9) exhibits different dynamical evolufor , tions. More specifically, when , model (9) possesses a chaotic strange attractor in the bounded region , as shown in Fig. 1(a). As shown in Fig. 1(b), some state variables of model (9) display divergent dynamics, when the initial values are selected for , . The ocas: currence of this difference in dynamics is attributed to the only locally Lipschitzian characteristic of the model. of model (9) as a driving Take the chaotic trajectory signal. Then, the adaptive coupling results in the response system
(10) where , . The initial value for the response state vector is taken as: for , . Hence, when in the definition of . Note that the driving setting all always stays in the compact ball , where signal and . Thus, . Furthermore, setting allows the estimation of the local Lipschitz constant of the coupled models (9) and (10) restricted . Here, in the compact set . In light of Theorem 3.1 and Remark 3.3, it . Therefore, that the initial value is adequate to take for satisfies (11) becomes a condition for realizing locally complete synchronization between the coupled models (9) and (10). This condition is verified by the numerical simulation shown in Fig. 1(c), where for , . Note that the criterion (11) for the choice of is only a sufficient condition for complete synchronization between models (9) and (10). Any violation of criterion (11) typically leads to either success or failure of the complete synchronization, as seen in Figs. 1(d) and 1(e). The above analytical and numerical discussions of models (9) and (10) indicate that global complete synchronization in some dynamical systems with only locally Lipschitzian vector fields cannot be surely realized by the above unidirectional and adaptive coupling. IV. FROM LOCAL SYNCHRONIZATION TO GLOBAL SYNCHRONIZATION IN ONLY LOCALLY LIPSCHITZIAN TIME-DELAYED SYSTEMS
, and the discrete time-delays are taken as . Set . Within these settings, model (9) is not uniformly but only locally Lipschitz, which falls into the above-specified class of systems (1) and (2). Starting from different initial values
From the analytical results and Example 3.6 in the preceding section, one can find out the limitation inherently rooted in the adaptive coupling technique: not global complete synchronization but merely local complete synchronization can be achieved for adaptively coupled time-delayed systems whose vector fields are only locally Lipschitz. However, in real applications, global complete synchronization is always desirable. Thus, the objective of this section is to design an adaptive coupling technique conquering the limitation to some extent. Also, this
LIN AND MA: SYNCHRONIZATION BETWEEN ADAPTIVELY COUPLED SYSTEMS
section will answer the question: “Under what kind of condition can global complete synchronization be realized in coupled time-delayed systems whose vector fields are only locally Lipschitz?” First of all, Theorems 3.1 and 3.4 together indicate that, for realizing complete synchronization, the initial value should be . Componentwise writing out this selected from the set condition of the initial value gives
Here, can be an arbitrary continuous function on except for its value at zero. Proposition 4.1 uses the above inequality to derive a lower bound on the constants that ensures complete synchronization between the coupled systems (1) and (3). Proposition 4.1: Suppose that both Assumptions 2.2 and 2.4 is the initial-value set for generating are satisfied, and that bounded driving signals of system (1). For any given initial , and , values the adaptively coupled systems (1) and (3) can be completely satsynchronized provided that the constants isfy the following condition:
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bounded quantity independent of and is an arbitrarily given constant strictly larger than one. Then, there exists a critical , such that value (i) if , the adaptively coupled systems (1) and (3) can be globally completely synchronized; , the local complete synchronization between (ii) if systems (1) and (3) can be guaranteed only. Correspondingly, the initial value for the coupling variable can be any continuous function on except that is , where is defined selected from the set in (5). Theorem 4.4 can be verified by a direct investigation of the following simple but nontrivial example. Example 4.5: Consider the adaptive control, as a particular case of synchronization, for the following 1-D system with a time delay (12) is regarded as an nonlinear index, is the where time delay, and is an only locally Lipschitzian funcis monotonous for , any trajectory tion. Because of system (12) starting from an positive function will increase is rapidly within a finite time evolution. This implies that an unstable equilibrium of system (12). Select this unstable but as the control target (or as a driving bounded equilibrium signal). Then, an adaptively controlled system is described by
where , is an arbitrary constant strictly larger is defined in (7). than one, and Proof: From the condition of the theorem, it follows that:
This manifests not only that the set is enlarged, but also . Therefore, acthat any given initial value belongs to cording to Theorem 3.4, the complete synchronization between systems (1) and (3) with the above-specified initial value and constants is guaranteed. Remark 4.2: Proposition 4.1 shows that complete synchronization can be realized for any initial values. However, the constants in the adaptive coupling should be adjusted, depending on the choice of the initial values. Clearly, this case is not consistent with the concept of the global complete synchronization given in Definition 2.1. Remark 4.3: Evidently, the lower bound on in Proposition 4.1 is not always practically useful. By adopting this proposition, some or even all of become tremendously large when the initial values for the response system are far from the driving signal. Moreover, a larger value of is more likely to destroy the true dynamics of the model either in numerical simulation or in real applications. This is because accuracy limit unavoidably exists in various algorithms for simulating continuous systems. In what follows, regardless of the role of , the global complete synchronization is investigated and an answer to the question posed at the beginning of this section is provided. as defined in (7) Theorem 4.4: Suppose that the constant can be estimated as , where stands for any
(13) Here, for simplicity, the initial value for and are, respectively, taken as constant functions: and for . By virtue of Definition 2.5 and Theorem 3.1, one has
if pact set
, and that both and stay in the comwith . Because of , the local Lipschitz constant of restricted in is
where
. Notice that is a number taken from arbitrarily. Therefore, from (5), it follows that the sufficient condition for the successful control of system (13) (or ) becomes for the complete synchronization between and , where interval is settled as
Clearly, the left ending point of the interval tends towards negative infinity as . However, the realization of a global control (or a global complete synchronization) of model as . (13) requires a condition that , i.e., , Specifically, if the right ending point of tends towards positive infinity,
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0 0
Fig. 1. Different dynamics generated by model (9) with different initial values. Chaotic attractor projected in the x x x phase space (a) and divergent trajectories in the time-state plane (b). The variation of Err = y (t; ; ) x (t; ) versus the evolutional time shows complete synchronization with speedy convergence between the chaotic model (9) and the response model (10) (c). The variations also show, respectively, successful synchronization (d) and failed for t [ ; 0], and that in synchronization (e), when the criterion (11) is violated. The initial value for the coupling variable in (d) is taken as: (t) 0 (e) is: (t) [0; 0; 1; 0] for t [ ; 0]. Here and throughout the paper, the Euler scheme with a time-step size 0.01 is employed in numerical simulations.
0
2
20
so that the global control can be realized; nevertheless, if , i.e., , the control can only be achieved locally. The above argument clearly validates Theorem 4.4 and, in particular, illustrates the reason why the . critical value is taken as As for this concrete example, one can treat the critical case more accurately. Indeed, one can compute the right as follows: ending point of
Thus, this right ending point tends towards positive infinity as , provided that . This means that global control can be realized in the critical case if the constant is above 8. Seemingly, this restriction on is similar to the condition derived in Proposition 4.1. Actually, this restriction is more relaxed, allowing arbitrary selection of the initial values for the critical case. Furthermore, for this example, the increase of the time delay can play a positive role in somewhat expanding the right ending point of when . To show this, one only needs to show that the function
is increasing with respect to for some and . To this end, the at is derivative of
Thus,
20
gives (14)
Clearly, a relatively larger delay can lead to a broader choice of when the restriction (14) on is fulfilled. With different parameter settings, Fig. 2 numerically depicts different feasible regions of the initial values for successful adaptive control. Specifically, with the decrease of the nonlinear index , the feasible region becomes larger and larger, as shown by the jackstraw-like feasible regions in Figs. 2(a) and 2(b). The occurrence of the asymmetrical and fractal-like structure in Fig. 2(b) is attributed to the introduction of the complex for the number expression . Compared with Fig. 2(a), the feasible non-integer and region in Fig. 2(c) expands obviously when increases. This is consistent with the conclusion indicated by Proposition 4.1. Moreover, as seen in Fig. 2(d), the arm-like areas are melt into the enlarged feasible region when increases. This typically accords with the above analytical result that a relatively larger time delay can enlarge the feasible region of model (13) as satisfies (14). From Theorem 4.4 and Example 4.5, it follows that the nonlinear degree of the only locally Lipschitzian system determines the success of the global complete synchronization. When the degree is below some critical value derived above, the global complete synchronization can certainly be achieved through the adaptive coupling. Otherwise, only the local complete synchronization can be guaranteed. The larger the degree, the more limited the choice of the initial values for the coupling variable.
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Assumption 5.1: The vector field in system (15) is assumed to satisfy the one-sided uniform Lipschitz condition, i.e., there exists a positive constant , such that APRIL
(16)
Fig. 2. Shaded zones represent different feasible regions of the initial constant functions [; ] for the successful adaptive control of system (13) with different parameter settings. Here, the parameters are, respectively, taken as: (a) = 3, = 1, and r = 1; (b) = 2:5, = 1, and r = 1; (c) = 3, = 1, and r = 10; (d) = 3, = 5, and r = 1.
Remark 4.6: The constant in Theorem 3.1 is independent of if the functions in system (1) are uniformly Lipschitz. In such a case, it directly follows from Theorem 4.4 that global complete synchronization between the coupled time-delayed systems (1) and (3) can be surely achieved. Remark 4.7: Note that, for some divergent trajectory generated by the driving system (1), the adaptive coupling synchronization technique can extend its maximal interval of existence in a finite length to an interval of infinite length.
V. ADAPTIVE SYNCHRONIZATION IN ONE-SIDED UNIFORM LIPSCHITZ SYSTEMS WITH TIME-VARYING DISCRETE AND DISTRIBUTED DELAYS In real applications, the expression of the system’s vector fields may not be obtained in an explicit form like (2). Also, time-dependent density functions and time-varying delays are always imported in mathematical modeling of real systems. These reasons motivate us to consider a more general time-delayed model given by (15) as the driving system. Here, the vector field is allowed to be non-autonomous, the initial value is the same as that for system (1), and the trajectory of (15), unnecessarily bounded, exists on the entire interval . Furthermore, the following assumption is imposed on this vector field.
denotes the inner product of two vectors in . where The local Lipschitz condition provided in Assumption 2.2 is a relaxed condition. The one-sided uniform Lipschitz condition (16) is also relaxed in some cases. For example, condition (16) allows for time-dependent density functions and time-varying discrete delays. Noticeably, the local Lipschitz condition in Assumption 2.2 does not include these types of delays yet. Moreover, any uniformly Lipschitzian system (1) with (2) certainly satisfies condition (16), and even some systems that are only locally Lipschitz fulfill this condition. Naturally, condition (16) is applicable to many dynamical systems, such as the time-delayed neural network models with sigmoidal activation functions, the time-delayed Mackay-Glass biological models, and some particular forms of LV models with time-varying delays [26], [27]. Example 5.2: Let the vector field
where the density function satisfies . Obviously, this vector field is only locally Lipschitz. However, for any ,
Clearly, satisfies the one-sided uniform Lipschitz condition. Similarly, the adaptively coupled response system for the driving signal produced by system (15) is
(17) where all the initial values are the same as those for response system (3). Theorem 5.3: Suppose that Assumption 5.1 is fulfilled. Let be a positive number strictly larger than the one-sided uniform Lipschitz constant in Assumption 5.1. Then, for any given iniwith all , tial values , , and the adaptively coupled systems (15) and (17) can be completely synchronized with an exponential convergence rate, i.e., there exists a positive constant such that, for all
and exponentially converges to zero as tends to positive infinity.
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Proof: First, construct the function of
where is any trajectory produced by the coupled systems (15) and (17) with the initial values specified in . the theorem, and the constant vector On the one hand, the derivative of yields
(18) in which the inequality follows from Assumption 5.1. On the other hand (19) (for all ) is in which each attributed to the monotonously decreasing property of all and to their particularly selected initial values . Thus, (18) and (19) together give
(20) In what follows, it is to prove the exponential estimation in the theorem by contradiction. Without loss of generality, it is as. If the estimation in the theorem is sumed that not valid, the constant
which is the first time that the quantity transversely hits the value , exists, i.e., . Here, the positive constant is pending for determination. Accordingly
(21) and there exists a constant
, such that (22)
for any it follows that:
, because of the continuity of the function with respect to . Hence, from (20) and (21),
Set . Then, it is easy to verify that is a continuous function of with and . Therefore, according to the property of continuous functions, there exists a positive constant such as . Taking from the interval that immediately yields
Then, is strictly monotonically decreasing in the vicinity of . This clearly contradicts (22). Conse, which completes the proof of the theorem. quently, Remark 5.4: The use of Theorem 5.3 does not require the eventual boundedness of the driving signal generated by system (15). This is much weaker than the condition specified in Theorem 3.4. However, this relaxation leads to a restriction on the selection of initial values for the coupling variable in Theorem 5.3. In a particular case where the systems degenerate into or), the restriction on the dinary differential equations (i.e., initial values’ selection can be removed as long as either the LaSalle invariance principle for non-autonomous ordinary differential equations [33] or the Barb˘alat Lemma [34] is utilized in the above proof directly. Remark 5.5: Analogous to Remark 3.5, the driving signal can be selected as either an unstable equilibrium or an unstable periodic orbit of system (15). In such a case, adaptive synchronization becomes adaptive control. Naturally, Theorem 5.3 can be adapted to guaranteeing the adaptive control in one-sided unform Lipschitz time-delayed systems that either satisfy or violate the local Lipschitz condition in Assumption 2.2. Example 5.6: Consider a distributively time-delayed system whose vector field is settled as
(23) , , and for all . where is the Dirac delta function with a In particular, when compact support , the distributive system (23) becomes a system with a discrete time-varying delay. This kind of distributive systems are widely used to model those real processes in neuro-dynamics and price-fluctuations [26], [27]. In particis taken as the control target (or ular, the equilibrium the driving signal). Thus
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synchronization. The results are further illustrated through several representative examples. The local and one-sided uniform Lipschitz conditions cover a wide range of dynamical systems with discrete and distributed time delays. The established theories are suitable for ensuring the success of adaptive control in systems that are only locally Lipschitz, as a particular case of unidirectionally adaptive synchronization. The results suggest several directions for generalizations: • the only local Lipschitz condition can be replaced by the one-sided local Lipschitz condition, • multiplicative forms of different time-delay terms could be adopted in the system’s vector fields, • time-dependent density functions and time-varying discrete delays could be addressed. Also, the invariance principle for non-autonomous functional differential equations could be utilized instead of the theory of autonomous systems. ACKNOWLEDGMENT Fig. 3. (a) The divergent dynamics of the uncontrolled system (23); (b) and 0 through the (c) the successful controls of the unstable equilibrium ' () adaptive coupling in system (23) with different time-delays and initial values.
which implies that , with respect to the control target , satisfies the one-sided uniform Lipschitz condition. Howdoes not satisfy the condition in Assumption 2.2 ever, directly. Therefore, according to Theorem 5.3 and Remark 5.5, , the equilibrium for any maximal time-delay of the time-delayed system (23) can be adaptively stabilized (or completely synchronized) provided that the controlled system . is designed as system (17), with Fig. 3(a) shows the divergent dynamics of the uncontrolled , , and the density function system (23) with
This function describes time-dependent and periodic aftereffects on the state variable. Fig. 3(b) depicts a successfully adapwith initial tive control of the unstable equilibrium values: and . This is consistent with the above theoretical analysis. Also, Fig. 3(c) , presents a successful control of system (23) with , . This illustrates that the restriction on the initial values in Theorem 5.3 is not necessary but only sufficient. VI. CONCLUSION In this paper, theories are established for adaptive complete synchronization in unidirectionally coupled time-delayed systems whose vector fields are either only locally Lipschitz or one-sided uniformly Lipschitz. The adaptive coupling technique is shown to have limitations on the selection of the initial values for the coupling variables for some systems that are only locally Lipschitz. For these systems, the nonlinear degree proposed in the paper can be used to assess the success of global complete
The authors would like to thank four anonymous reviewers for their helpful and detailed comments on this work, and Dr. G. Chen and the Associate Editor Dr. R. D. Braatz for their significant suggestions leading to many improvements of the work. REFERENCES [1] M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, “Huygens’ clocks,” Proc. R. Soc. Lond. A, vol. 458, pp. 563–579, 2002. [2] S. H. Strogatz, “Exploring complex networks,” Nature, vol. 410, pp. 268–276, 2001. [3] S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou, “The synchronization of chaotic systems,” Phys. Rep., vol. 366, pp. 1–101, 2002. [4] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge, U.K.: Cambridge Univ. Press, 2001. [5] P. F. Hokayem and M. W. Spong, “Bilateral teleoperation: An historical survey,” Automatica, vol. 42, pp. 2035–2057, 2006. [6] H. Fujisaka and T. Yamada, “Stability theory of synchronized motion in coupled-oscillator systems,” Prog. Theor. Phys., vol. 69, pp. 32–43, 1983. [7] C. W. Wu and L. O. Chua, “Synchronization in an array of linearly coupled dynamical systems,” IEEE Trans. Circuits Syst. I, vol. 42, no. 8, pp. 430–447, Aug. 1995. [8] G. Chen and X. Dong, From Chaos to Order: Methodologies, Persperctives and Applications. Singapore: World Scientific, 1998. [9] K. M. Cuomo, A. V. Oppenheim, and S. H. Stogatz, “Synchronization of Lorenz based chaotic circuits with applications to communications,” IEEE Trans. Circuits Syst. II, vol. 40, no. 10, pp. 626–633, Oct. 1993. [10] H. Dedieu and M. J. Ogorzalek, “Identifiability and identification of chaotic systems based on adaptive synchronization,” IEEE Trans. Circuits Syst. I, vol. 44, no. 10, pp. 948–962, Oct. 1997. [11] Z. H. Guan, G. Chen, X. Yu, and Y. Qin, “Robust decentralized stablization for a cloass of large-scale time-delay uncertain impulsive dynamical systems,” Automatica, vol. 38, pp. 2075–2084, 2002. [12] C. S. Zhou and J. Kurths, “Noise-induced phase synchronization and synchronization transitions in chaotic oscillators,” Phys. Rev. Lett., vol. 88, p. 230602, 2002. [13] W. Lin and G. Chen, “Using white noise to enhance synchronization of coupled chaotic systems,” Chaos, vol. 16, p. 013134, 2006. [14] W. Lin, “Realization of synchronization in time-delayed systems with stochastic perturbation,” J. Phys. A: Math. Theor., vol. 41, p. 235101, 2008. [15] K. Lian, P. Liu, T. Chiang, and C. Chiu, “Adaptive synchronization design for chaotic systems via a scalar driving signal,” IEEE Trans. Circuits Syst. I, vol. 49, no. 1, pp. 17–27, Jan. 2002.
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Wei Lin (M’10) was born in Shanghai, China, in 1976. He received the B.Sc. and Ph.D. degrees in applied mathematics from Fudan University, Shanghai, China, in 1998 and 2003, respectively, with specialization in dynamical systems, bifurcation and chaos theory, and chaos control and synchronization. In 2003, he joined the School of Mathematical Sciences, Fudan University, where presently, he is an Associate Professor in applied mathematics. From 2004 to 2005, he was a Postdoctoral Research Fellow at the Department of Mathematics and Statistics, York University, Toronto, ON, Canada. Currently, he is the Vice Dean of the School of Mathematical Sciences and the Deputy Dean of the Centre for Computational Systems Biology, Fudan University. Since 2008, he has held the Staff Scientist position at the CAS-MPG Partner Institute for Computational Biology, Shanghai. He is now the Deputy Secretary of the Shanghai Society of Nonlinear Sciences. He is the Guest Associate Editor of the International Journal of Bifurcation and Chaos. His current research interests include nonlinear dynamical systems, bifurcation and chaos theory, hybrid systems, stochastic differential and difference equations, chaos control and synchronization, complex networks, and computational systems biology. Dr. Lin received the Shanghai Science and Technology Rising Star from Shanghai Science and Technology Committee, China, in 2007.
Huanfei Ma (S’07) was born in Suzhou, China, in 1981. He received the B.Sc. degree in mathematics from Fudan University, Shanghai, China, in 2005 where he is currently pursuing the Ph.D. degree at the school of Computer Science. He is now the member of the Shanghai Key Laboratory of Intelligent Information Processing, and also a member of the Centre for Computational Systems Biology, Fudan University. His research interests include theory of adaptive control and synchronization in various systems, parameters identifications, neural networks, and computational systems biology.
