No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Abstract methods for synchronization and applications
Abstract
In this work we present some mathematical methods to obtain uniform estimates for attractors and to study the synchronization of two similar dynamical systems. We give sufficient conditions to control the coupling devices in order to accomplish the synchronization, even when each of the systems moves chaotically. Some examples that include systems of coupled lasers and coupled Lorenz equations are discussed.
... Overview of the theory can be found in the monographs [5,36,38,42,46,49] and references therein. The synchronization of coupled parabolic systems was studied in the papers [14,16,17,31,44]. The synchronization of coupled hyperbolic models was studied by Chueschov [27]. ...
... for any full trajectory y(t) = (u(t), v(t), u t (t), v t (t)) from A . This implies the uniform bound for the supremum in (44). Using the estimate (23) we obtain the corresponding bound for the dissipation integral in (44). ...
... This implies the uniform bound for the supremum in (44). Using the estimate (23) we obtain the corresponding bound for the dissipation integral in (44). Now, since (H, S (t)) is quasi-stable uniformly in > 0, Theorem Appendix A.10 (ii) implies that there exists a constant R > 0 independent of > 0 such that ...
... Bounded synchronization, in a sense different from the one in [66], for asymmetrical coupled networks was investigated in [59]. In [41,42], an approach was developed to analyze approximate synchronization of a coupled system comprising two nonidentical systems of nonlinear differential equations, where a uniform estimate of the synchronization error with respect to the parameters of the systems was derived. Chaotic approximate synchronization between nonidentical master-slave systems was addressed in [20,24,52]. ...
... To summarize, the equations considered in the studies of approximate synchronization originated from motivations in physics, networks, or engineering, largely carry linear, and/or diffusive (or akin to diffusive) coupling functions. This can be seen from the above-mentioned papers [18,24,41,52,59,66] for the ones without delays, and [19,20] for the ones with delays. On the other hand, depending on the methodologies employed, some of those investigations conclude local dynamics, whereas others established global dynamics. ...
... System in the form of (4) is still very general, and is commonly considered in the studies of synchronization problems in the literature. For example, the systems that were studied in [18,30,33,36,41,54,58,59,63,65,66] are in the form of (4), or similar to (4). Usually, network system (2) or (4) is called homogeneous if it comprises identical subsystems, i.e., F i ≡ F j , for all i, j ∈ N , and called heterogeneous otherwise. ...
... (Affraimovich et al., 1997), for example, proves synchronization for a class of coupled systems, including chaotic systems with linear coupling. Abstract results, robustness with respect to parameter variation and uniform dissipativeness were obtained in (Rodrigues, 1996). or infinite dimensional systems, ______________________________ 1 mijolaro@sel.eesc.usp.br, 2 luis@sel.eesc.usp.br, 3 ngbretas@sel.eesc.usp.br. ...
... or infinite dimensional systems, ______________________________ 1 mijolaro@sel.eesc.usp.br, 2 luis@sel.eesc.usp.br, 3 ngbretas@sel.eesc.usp.br. some results were presented in (Rodrigues, 1996), (Affraimovich et al., 1997) and (Labouriau and Rodrigues, 2003). ...
... One approach employed to prove synchronization, first proposed in (Rodrigues, 1996), consists of two parts. First, an estimate of the attractor, uniform with respect to the coupling parameter, is obtained then, in a second stage, synchronization is studied into this attractor estimate and an estimate of the coupling parameter, which is sufficient to guarantee synchronization, is obtained. ...
The asymptotic behavior of a class of nonlinear coupled dynamical systems is studied in the present paper. Conditions of the coupling parameters guarantee synchronization between these systems are provided for every situation starting in a positively invariant region of initial conditions. This results guarantees synchronization even so the systems are unstable or do not have global dissipativeness. Synchronization of two coupled nonlinear pendulums that represent two-generators in a simple electrical power system is proved in the proposed theory.
... The estimates for synchronization appear in section 4. We follow a procedure similar to the general result in [20] adapted to the weaker hypotheses used here. The result is illustrated in section 5 by a numerical simulation of two coupled Hodgkin-Huxley equations. ...
... Abstract results on synchronization, the robustness with respect to parameter variation and uniform dissipativeness were obtained in [20,2]. Some methods deal with synchronization in specific classes of problems, like for instance, synchronization of chaotic systems [4], master-slave synchronization in chaotic systems [18,14,25] and robustness with respect to parameter variation [11,12]. ...
... Before establishing synchronization we find a region in R (n+1) that attracts the flow of (HHC). We modify the technique of uniform dissipativity developed of [2,20] in order to obtain results for asymmetric coupling k 1 = k 2 . ...
We study a class of differential equations modelling the electrical activity in biological systems. This class includes the Hodgkin-Huxley equations for the nerve impulse, as well as models for other excitable tissue, like muscle fibers, pacemakers and pancreatic cells. We show that when two of these equations are coupled their solutions always synchronize. Synchronization takes place regardless of the initial condition if the coupling is strong enough, and even for two equations with different parameter values, coupled asymetrically. We find a bounded region in phase space that attracts the flow globally and thus contains all points with recurrent behaviour. The size of the region can be calculated from the parameters in the equations. Thus we show that the system is uniformly dissipative. We obtain explicit bounds for this region in terms of the parameters as a tool for establishing synchronization. These estimates are also obtained for the uncoupled equations.
... Mathematical methods for the study of synchronization in different contexts have been developed by several authors. Abstract results and the robustness with respect to parameter variation and uniform dissipativeness were obtained in Rodrigues [1] and Afraimovich-Rodrigues [2]. Some methods deal with synchronization in specific classes of problems. ...
... So this paper we apply Rodrigues's analytical method to the simplified Morris-Lecar model and follow a procedure similar to the general result in [1,14], but make more strict proof such as the inequality (11) and (40) compared to the inequality in Rodrigues's paper and use weaker hypotheses to analyze the simplified Morris-Lecar model. We also simplify the analysis process of synchronization in the case of symmetric parameters and asymmetric parameters. ...
... Proof. We follow a procedure similar to that of [14,1,2], and start by obtaining estimates for our case, similar to those in the hypotheses of these papers. From now on assume n and z are in the interval [0, 1] and thatj(u, v)j 6 R. Since all variables are in a compact set, we may obtain uniform Lipschitz constants for the functions involved. ...
The Morris–Lecar equations explicitly modeling the flow of potassium and calcium ions are a two-dimensional description of neuronal spike dynamics. Some research shows that two coupled Morris–Lecar equations can be synchronized, and synchronization takes place regardless of the initial condition if the coupling is strong enough, and even for two equations with different parameter values, coupled asymmetrically. This paper finds a bounded region in phase space that attracts the flow globally and thus contains all points with recurrent behavior. The size of the region can be calculated from the parameter values in the equations and is proportional to external current. And we obtain explicit bounds for this region in terms of the parameter values as a tool for establishing synchronization.
... There are now several monographs [30, 36, 38] in this field, which contain extensive lists of references. In the case of infinite dimensional systems the synchronization problem has been studied in [10, 33] for the case of coupled (deterministic) parabolic systems. The synchronization of stochastic stationary solutions (i.e. ...
... The main reason is that the backward time estimate in (4) cannot be obtained in the case when the master equation is parabolic. In the purely parabolic case the approach to synchronization relies on the construction of appropriate Lyapunov type functions [10, 33]. Our approach is alternative in some sense and covers another, in comparison with [10, 33], kind of problems. ...
... In the purely parabolic case the approach to synchronization relies on the construction of appropriate Lyapunov type functions [10, 33]. Our approach is alternative in some sense and covers another, in comparison with [10, 33], kind of problems. ...
We deal with abstract systems of two coupled nonlinear stochastic (infinite dimensional) equations subjected to additive white noise type process. This kind of systems may describe various interaction phe-nomena in a continuum random medium. Under suitable conditions we prove the existence of an exponentially attracting random invari-ant manifold for the coupled system and show that this system can be reduced to a single equation with modified nonlinearity. This result means that under some conditions we observe (nonlinear) synchro-nization phenomena in the coupled system. Our applications include stochastic systems consisting of (i) parabolic and hyperbolic equations, (ii) two hyperbolic equations, and (iii) Klein-Gordon and Schrödinger equations. We also show that the random manifold constructed con-verges to its deterministic counterpart when the intensity of noise tends to zero.
... Most of the research efforts have been devoted to the study of chaos control and chaos synchronization problems in low-dimensional nonlinear dynamical systems [2][3][4][5][6][7][8][9][10]. Synchronizing high dimensional systems in which state variables depend not only on time but also on the spatial position remains a challenge. ...
... where ∇ is the gradient vector, is the is the unit outer normal to Ω, and is an auxiliary variable for integration. Then, using the assumption given in (6), the condition given in (5), and the fact that − is a negative definite matrix, we obtain ...
Synchronization and control in high dimensional spatial-temporal systems have received increasing interest in recent years. In this paper, the problem of complete synchronization for reaction-diffusion systems is investigated. Linear and nonlinear synchronization control schemes have been proposed to exhibit synchronization between coupled reaction-diffusion systems. Synchronization behaviors of coupled Lengyel-Epstein systems are obtained to demonstrate the effectiveness and feasibility of the proposed control techniques.
... These sources deal mainly with finite-dimensional systems. For infinite dimensional systems the synchronization problem has been studied in [4,7,8,25,38] for coupled parabolic systems, while synchronization in Berger plates was considered in [31,32,33]. General (deterministic) infinite-dimensional second order in time models were studied in [14] (for the deterministic second order ODE see also [1] and [25]), while master-slave synchronization of coupled parabolic-hyperbolic PDE systems was considered in [12,13] (see also [20] for the stochastic case). ...
... in which problem (38) can be rewritten in the form w tt + γw t − ∆w + 2κw + λ sin w cos z = 0, (40a) z tt + γz t − ∆z + λ cos w sin z = f (x) +Ẇ , ...
The asymptotic synchronization at the level of global random attractors is investigated for a class of coupled stochastic second order in time evolution equations. The main focus is on sine-Gordon type models perturbed by additive white noise. The model describes distributed Josephson junctions. The analysis makes extensive use of the method of quasi-stability. © 2016, Southwest Missouri State University. All rights reserved.
... There are now quite a few monographs [4,28,30,34,38,40] in this field, which contain extensive lists of references. In the case of infinite dimensional systems the synchronization problem has been studied in [5,7,8,24,36] for coupled parabolic systems. Synchronization in Berger plates (they are a particular case of our abstract models) was considered in [31,32,33]. ...
... Now we apply the results above to synchronization. We switch on the interaction operators K of the standard (see, e.g., [8,24,31,36] and the references therein) symmetric form. Moreover we suppose that the damping operator D 0 has a diagonal structure. ...
We study asymptotic synchronization at the level of global attractors in a
class of coupled second order in time models which arises in dissipative wave
and elastic structure dynamics. Under some conditions we prove that this
synchronization arises in the infinite coupling intensity limit and show that
for identical subsystems this phenomenon appears for finite intensities. Our
argument involves a method based on "compensated" compactness and
quasi-stability estimates. As an application we consider the nonlinear
Kirchhoff, Karman and Berger plate models with different types of boundary
conditions. Our results can be also applied to the nonlinear wave equations in
an arbitrary dimension. We consider synchronization in sine-Gordon type models
which describes distributed Josephson junctions.
... Therefore every solution of (3.17) enters in A R in finite time and stays there in the future. In order to study the synchronization, either Theorem 2.3 of [20] or Theorem 3.1 of [1] can be used. With this aim, it is convenient to rewrite (3.17) system in the following form, ...
... Following Theorem 2.3 of [20] or Theorem 3.1 of [1], an exponential decay for the evolutions operator of &_&2k +_ 0 ...
The objective of this work is to obtain uniform estimates, with respect to parameters, of the attractor and of the basin of attraction of a dynamical system and to apply these results to analyze the roughness of the synchronization of two subsystems. These estimates are obtained through a uniform version of the invariance principle of La Salle which is stated and proved in this work.
... Synchronization phenomenon, which was discovered in physics, biology, and social science areas [1][2][3], has been paid more attention due to its extensive applications in secure communications, modeling brain activity, and optimization of nonlinear system performance [4,5]. Synchronization of deterministic coupled dissipative systems has been investigated [6][7][8][9]. ...
In this paper, a random coupled Ginzburg–Landau equation driven by colored noise on unbounded domains is considered, in which the nonlinear term satisfies a local Lipschitz condition. It is shown that the random attractor of such a coupled Ginzburg–Landau equation is a singleton set, and the components of solutions are very close when the coupling parameter becomes large enough.
... Zhen Li zhenleemath@163.com 1 A mathematical model for such systems is often dealt with in an asymptotic sense. For ordinary differential equations (ODEs), provided that asymptotically stable equilibria and appropriate attractors are considered, synchronization of coupled dissipative systems has been investigated mathematically in the case of autonomous systems by [5,17] and nonautonomous systems by [1,11,12]. Recently, the influence of noise on synchronization of coupled dissipative stochastic differential equations (SDEs) has been studied, within the framework of random attractors and stochastic stationary solutions in random dynamic systems. ...
In this paper, we mainly construct a connection between synchronized systems and multi-scale equations, and then use the averaging principle as an intermediate step to obtain synchronization. This strategy solves the synchronization problem of dissipative stochastic differential equations, regardless of the structure of the noise. Moreover, the averaging principle of stationary solutions is also investigated, which is different from the averaging principle of solutions with the fixed initial values.
... Afraimovich and Rodrigues [2] investigated the synchronization of coupled dissipative model in the case of autonomous model. Asymptotically stable equilibria and general attractors have been investigated mathematically by Carvalho, et al. [3] and Rodrigues [4]. Caraballo, et al. [5] proved synchronization of systems under additive noise and gave another matter of random attractors. ...
We discuss the discretization influence on multiplicative noise dissipative system. Moreover, using a drift-implicit Euler formula with discretization, we find the synchronization of multiplicative noise dissipative model.
... Rodrigues and his coauthors [4,5] investigated mathematically the autonomous systems, including asymptotically stable equilibria and general attractors. They not only showed that the coupled trajectories converged to each other as time increases but also obtained the global attractor of the coupled system. ...
This work is about the synchronization of nonlinear coupled dynamical systems driven by -stable noise. Firstly, we provide a novel technique to construct the relationship between synchronized system and slow-fast system. Secondly, we show that the slow component of original systems converges to the mild solution of the averaging equation under sense. Finally, using the results of averaging principle for stochastic dynamical system with two-time scales, we show that the synchronization effect is persisted provided equilibria are replaced by stationary random solutions.
... In addition, based on the consideration of the equilibrium points with parameters, a mathematical model for such systems is often dealt with in an asymptotic sense. For example, the synchronization of coupled dissipative systems has been investigated mathematically in the case of autonomous systems by [5,21] and nonautonomous systems by [1,12,13], provided that asymptotically stable equilibria and appropriate attractors are considered in ordinary differential equations (ODEs). ...
... Chaotic behavior in nonlinear dynamical systems is a very attractive phenomenon which has been extensively investigated and studied in the last decades. Recent developments in methods and techniques for controlling and synchronizing chaotic systems have seen remarkable growth in the chaos literature [1][2][3][4][5][6][7][8] . Recently, investigating and studying the chaotic dynamics of systems including fractional differential equations, where the fractional derivative is considered on a continuous time scale, has become an interesting research area. ...
This paper is concerned with a fractional Caputo-difference form of Ikeda map. The dynamics of the proposed map is investigated numerically through phase plots and bifurcation diagrams considered from different perspectives. In addition, a stabilization controller is proposed and the asymptotic convergence of the states is established using the stability theory of linear fractional-order discrete systems. Furthermore, a new synchronization scheme is introduced whereby a new 2D fractional-order chaotic map is considered as the master system and the fractional-order Ikeda map is considered as the response system. Experimental investigations and numerical simulations are also provided to confirm the main findings of the study.
... Here H * d is the Hausdorff semi-distance on R d . The following result [2,9,13] ensures the existence of a random attractor. ...
The synchronization of stochastic differential equations with both additive noise and linear multiplicative noise is investigated in pathwise sense, which generalize the results of [5] and [6]. In our situation, we can deal with the synchronization of the systems with mixed type noise, where one system has the additive noise, another has the linear multiplicative noise.
... In particular, synchronization provides an explanation for the emergence of spontaneous order in the dynamical behavior of coupled systems, which in isolation may exhibit chaotic dynamics. The synchronization of coupled dissipative systems has been investigated mathematically in the case of autonomous systems by [1,3,13], including asymptotically stable equilibria and general attractors, such as chaotic attractors. Analogous results also hold for nonautonomous systems [8], but require a new concept of a nonautonomous attractor. ...
The synchronization of stochastic differential equations (SDEs) with additive noise is investigated in pathwise sense, moreover convergence rate of synchronization is obtained. The optimality of the convergence rate is illustrated through examples.
... Let u(t), v(t) ∈ R d be two functions defined in [t 0 , ∞) (t 0 ∈ R), u(t), v(t) are said to be synchronized if lim t→∞ ∥u(t)−v(t)∥ = 0. Synchronization of deterministic coupled systems has been investigated both for autonomous systems and non-autonomous systems (see e.g. [3,19,10,25]). For coupled systems of Itö stochastic differential equations with various Gaussian noises (in terms of Brownian c ⃝ 2015 Diogenes Co., Sofia pp. ...
This paper is devoted to the study of synchronous phenomena of the solutions to system driven by fractional environmental noises on finite lattice. Under certain dissipative and integrability conditions, we obtain the synchronization between two solutions, and among different components of solutions when the coupling coefficient tends to infinity. This indicates that no matter how large the intensity and what kinds of the noises perturb the system, the synchronization persists.
... Synchronization of two coupled continuous nonlinear systems has been studied by many authors, including Rodrigues [12], Affraimovich & Rodrigues [2], Carvalho, Dlotko & Rodrigues [3], Rodrigues, Alberto & Bretas [13, 14], to name a few. Synchronization has also been used by Labouriau & Rodrigues to study the coupled system of Hodgkin-Huxley equations [7]. ...
We study synchronization of a coupled discrete system consisting of a Master System and a Slave System. The Master System usually exhibits chaotic or complicated behavior and transmits a signal with a chaotic compo-nent to the Slave System. The Slave System then recovers the original signal and removes the chaotic component. To ensure secured communication, the Master and the Slave systems must synchronize independent of the variation of the systems parameters and initial conditions. Here we develop a general approach and obtain some general results for synchronization of such coupled systems naturally arising from discretization of well-know continuous systems, and we illustrate general results with two specific examples: the discretized Lorenz system and a discretized nonlinear oscillator. We also present some simulations using MatLab to illustrate our discussions.
... The general results in our paper can be regarded as the generalization of the two above papers and analogue of the uniform invariance principle for continuous dynamical systems developed previously in [Rodrigues et al., 2000 and [Gameiro & Rodrigues, 2001]. We also obtain some previous related results in [Rodrigues, 1996] for autonomous systems, in [Affraimovich & Rodrigues, 1998] for nonautonomous systems and in [Carvalho et al., 1998] for infinite dimensional parabolic systems. ...
In this series of papers, we study issues related to the synchronization of two coupled chaotic discrete systems arising from secured communication. The first part deals with uniform dissipativeness with respect to parameter variation via the Liapunov direct method. We obtain uniform estimates of the global attractor for a general discrete nonautonomous system, that yields a uniform invariance principle in the autonomous case. The Liapunov function is allowed to have positive derivative along solutions of the system inside a bounded set, and this reduces substantially the difficulty of constructing a Liapunov function for a given system. In particular, we develop an approach that incorporates the classical Lagrange multiplier into the Liapunov function method to naturally extend those Liapunov functions from continuous dynamical system to their discretizations, so that the corresponding uniform dispativeness results are valid when the step size of the discretization is small. Applications to the discretized Lorenz system and the discretization of a time-periodic chaotic system are given to illustrate the general results. We also show how to obtain uniform estimation of attractors for parametrized linear stable systems with nonlinear perturbation.
... However from 1990, with the work of Pecora and Carroll [2], the interest for the phenomenon has increased enormously and different techniques, based on some well establish theories (e.g. Lyapunov functions [3,4], exponential dichotomies theory [5], are some examples) have been applied. On the other hand, for spatially extended systems the synchronization is a very complicated problem and several techniques, in order to achieve this, have been implemented. ...
In this article a technique to achieve synchronization in spatially extended systems is introduced. The basic idea behind this method is to map a system of partial differential equations (PDEs) into a high-dimensional space where the representation of this PDE is an ordinary differential equation. By using semi-group theory, we are able to find conditions that ensure the synchronization of two systems of non-identical reaction–diffusion equations with a master–slave coupling.
... Mathematical methods for studying this type of synchronization have been developed, in particular, in [7][8][9][10]. In [11,12], the application of the uniform invariance principle to the synchronization problem was demonstrated. Rigorous results on coupled lattices of nonlinear oscillators are given in [13]. ...
In this article we derive conditions for complete synchronization of two symmetrically coupled identical systems of ordinary differential equations and differential-delay equations. Using Lyapunov function approach we give an estimate of the region of attraction of the synchronized solution. We also established that complete synchronization is robust with respect to small perturbations of the identical systems.
... We are especially interested in the case when the coupling in the system leads to the coincidence of the limiting dynamics of the plates. The synchronization of coupled dissipative equations has been investigated mathematically by Rodrigues [3], Afraimovich and Rodrigues [4], Kloeden [5], Caraballo and Kloeden [6]. Substantial results were obtained for the case of finite dimensional systems. ...
The dynamical system arising in the study of nonlinear oscillations of a number of coupled Berger plates is considered. The dependence of the long-time behavior of the trajectories of the system on the properties of the coupling operator is studied. It is shown that the global attractor of the dynamical system is continuous with respect to the coupling parameter γ expressing the intensity of plate interaction. When γ→∞ it converges upper semicontinuously to the attractor of the system generated by the projection of the vector field of the coupled system on the kernel of the coupling operator. For the particular case of 3-diagonal coupling operator the synchronization phenomenon at the level of attractors is stated for large values of γ as well as the absence of synchronization for γ small. The case of cluster synchronization is also considered.
... In the recent book of Strogatz [4], a number of its diversity of occurrence and an extensive list of references can be found. Let ( ), ( ) ∈ R be two functions defined in [ 0 , ∞) ( 0 ∈ R) and ( ), ( ) are said to be synchronized if lim → ∞ ‖ ( ) − ( )‖ = 0. Synchronization of deterministic coupled systems has been investigated both for autonomous systems and nonautonomous systems (see, e.g., [7][8][9][10]). For coupled systems of Itö stochastic differential equations with various Gaussian noises (in the terms of Brownian motion), the synchronization of solutions has been considered in the papers Caraballo and Kloeden [11], Caraballo et al. [12], Caraballo et al. [13] and Chueshov and Schmalfuß [14]. In [15], Shen et al. showed the synchronization of solutions for more general systems with multiplicative noise. ...
The synchronization of the solutions to coupled stochastic systems of N-Marcus stochastic ordinary differential equations which are driven by α-stable Lévy noises is investigated . We obtain the synchronization between two solutions and among different components of solutions under certain dissipative conditions. The synchronous phenomena persist no matter how large the intensity of the environment noises. These results generalize the work of two Marcus canonical equations in X. M. Liu et al.' s (2010).
... Synchronization of coupled dissipative systems has been investigated mathematically for continuous time systems generated by ordinary differential or parabolic differential equations by Rodrigues and his coauthors [1] [2] [5] and by Kloeden [4]. In particular, in the autonomous case, Rodrigues et al. showed that the coupled trajectories converge to each other as time increases for sufficiently large coupling coefficient but also that the global attractor of the coupled system converges upper semi continuously as the coupling parameter increases to the diagonal of the product of the global attractor of a system generated by the average of the vector fields of the original uncoupled systems. ...
The synchronization of two discrete time dynamical systems is considered, where the systems are described in terms of first order difference equations, each of which satisfies a global dissipativity condition and hence has a global attractor. It is shown that the coupled trajectories converge to each other as time increases for sufficiently large coupling coefficient and that the global attractor of the coupled system converges upper semi continuously as the coupling parameter increases to the diagonal of the product of the global attractor of a discrete time dynamical system for which the defining function is the average of those of the original uncoupled systems.
... In particular, synchronization provides an explanation for the emergence of spontaneous order in the dynamical behavior of coupled systems, which in isolation may exhibit chaotic dynamics. The synchronization of coupled dissipative systems has been investigated mathematically in the case of autonomous systems by Afraimovich and Rodrigues [1], Carvalho et al. [6] and Rodrigues [13], both for asymptotically stable equilibria and general attractors, such as chaotic attractors. Analogous results also hold for nonautonomous systems [8], but require a new concept of a nonautonomous attractor. ...
It is shown that the synchronization of noisy dissipative systems is preserved when a drift-implicit Euler scheme is used for the discretization. In particular, in this case the order of discretization and synchronization can be exchanged.
... In this article we focus attention on the synchronization of coupled dissipative systems. This has been investigated mathematically by Afraimovich & Rodrigues [1], Carvalho et al. [6] and Rodrigues [19] for asymptotically stable equilibria and for general attractors, such as chaotic attractors. Analogous results also hold for nonautonomous systems [10] and noisy systems [4,14], but require new concepts of nonautonomous attractors and random attractors. ...
Recent results on the dissipative synchronization of nonanutonomous and random dynamical systems are discussed as well as new mathematical ideas and tools from the theories of nonautonomous and random dynamical systems that are needed for their formulation and investigation (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
... A readable descriptive account of its diversity of occurrence can be found in the recent book of Strogatz (2003), which contains an extensive list of references. The synchronization of coupled dissipative systems has been investigated mathematically in the case of autonomous systems by Rodrigues (1996), Afraimovich & Rodrigues (1998) and Carvalho et al. (1998, both for asymptotically stable equilibria and general attractors, such as chaotic attractors. Analogous results also hold for non-autonomous systems (Kloeden 2003), but require a new concept of a non-autonomous attractor. ...
It is shown that the synchronization of dissipative systems persists when they are disturbed by additive noise, no matter how large the intensity of the noise, provided asymptotically stable stationary-stochastic solutions are used instead of asymptotically stable equilibria.
This paper focuses on achieving pathwise synchronization in stochastic differential equations with linear multiplicative rough noises, which are fractional Brownian rough paths with Hurst parameter H ∈ ( 1 3 , 1 2 ). Using rough paths theory, a useful transformation is introduced to convert the equations into random differential equations. Stability and dynamical behavior of the solutions to the equations are discussed, and pathwise synchronization of the solutions to the coupled system is proven. Also we have verified the synchronization results in Hölder space. And at the end, two alternative forms of noises are considered, and synchronization results are presented. Moreover, numerical simulations are provided to illustrate the results.
This paper is devoted to the synchronization of stochastic differential equations driven by the linear multiplicative fractional Brownian motion with Hurst parameter H∈(12,1). We use equivalent transformations to prove that the differential equation has a unique stationary solution, which generates a random dynamical system. Moreover, the system has the pathwise singleton set random attractor. We then establish the synchronization of the coupled differential equations and provide numerical simulation results. At the end, we discuss two specific noise forms and present the corresponding synchronization results.
In this chapter we mainly discuss phenomena in the synchronization of deterministic dissipative systems at the level of global attractors. This type of synchronization means that all dynamical (phase) components of a coupled system are attracted by limiting structures of the same form.
The purpose of this paper is to study the behavior of the solutions of two synchronized chaotic systems when the solutions switch from the first to the second system and vice-versa. The initial condition is chosen in the first system and the solutions travels for time t ∈ [0, h], where h > 0. The value of the solution at time h is then chosen as the initial condition for the solution of the second system and this solution travels for time t ∈ [h, 2h]. The value of the solution at time 2h is then chosen as the initial value for the solution of the first system and so on. The first system is composed of two subsystems, Master and Slave that are synchronized. We present applications using the Lorenz, Chua and Chen systems. Some simulations using Matlab are presented.
In this work, we analyze the synchronization role in a 1D array of coupled continuous chaotic elements. The ordinary differential equations, which represent each chaotic element, have a stable linear part perturbed by a bounded nonlinear function. A theorem is established to give sufficient condition to obtain synchronization of the array. Then, the above results are applied to a 2D network of coupled Wilson-Cowan neural oscillators showing how to solve scene segmentation problems by rapid chaotic synchronization and desynchronization. Computer simulations confirm the mathematical analysis.
We consider the synchronization of the solutions to coupled stochastic
systems of N-stochastic ordinary differential equations (SODEs) driven by
Non-Gaussian L\'evy noises (. We discuss the synchronization
between two solutions and among different components of solutions under certain
dissipative and integrability conditions. Our results generalize the present
work obtained in Liu et al (2010) and Shen et al (2010).
The dynamical system generated by a system describing nonlinear oscillations of two coupled Berger plates with nonlinear interior damping and clamped boundary is considered. The dependence of the long-time behavior of the system trajectories on the coupling parameter γ is studied in the case of (i) same equations for both plates of the system and damping possibly degenerate at zero; and (ii) different equations and damping non-degenerate at any point. Ultimate synchronization at the level of attractors is proved for both cases, which means that the global attractor of the system approaches the diagonal of the phase space of the system as γ→ ∞. In case (ii) the structure of the upper limit of the attractor is studied. It coincides with the diagonal of the product of two samples of the attractor to the dynamical system generated by a single plate equation. If both the equations describing the plate dynamics are the same and the damping functions are non-degenerate at any point we prove the synchronization phenomenon for finite large γ. System synchronization rate is exponential in this case.
The synchronization of Stratonovich stochastic differential equations (SDE) with a one-sided dissipative Lipschitz drift and linear multiplicative noise is investigated by transforming the SDE to random ordinary differential equations (RODE) and synchronizing their dynamics. In terms of the original SDE, this gives synchronization only when the driving noises are the same. Otherwise, the synchronization is modulo exponential factors involving Ornstein–Uhlenbeck processes corresponding to the driving noises. Moreover, this occurs no matter how large the intensity coefficients of the noise.
The problem of nonlinear oscillations of two Berger plates occupying bounded domains Ω in different parallel planes and coupled by internal subdomains Ω1⊂Ω is considered. A dynamical system generated by the problem in the space H=[H02(Ω)]2×[L2(Ω)]2 is studied. The long-time behavior of the trajectories of the system and its dependence on the value of the coupling parameter γ is described in terms of the system global attractor. In particular, we prove a synchronization phenomenon at the level of attractor for the system. Namely, we consider a (limiting) dynamical system generated by a suitable second order in time evolution equation in the space H̃ consisting of the elements from H with coordinates equal for the values of the spatial variable x from the closed set Ω1¯: H̃={y=(y1,y2,y3,y4)∈H:y1(x)=y2(x),y3(x)=y4(x),x∈Ω1¯}, and prove that the attractor of the system describing oscillations of two partially coupled Berger plates approaches the attractor of the limiting system as γ tends to the infinity.
The dynamical system arising in the study of nonlinear oscillations of two coupling plates is considered. The conditions under which the dynamical system possesses a global attractor are obtained. The continuous dependence of the attractor of the system on the coupling parameter is established. The phenomenon of synchronization of plate oscillations is studied. In particular, we describe values of the coupling parameter for which the system either synchronizes or does not synchronize at the level of attractors.
In this work, using chaotic systems, we study the role of synchronization on codification and decodification of messages. We first present a general result that is useful to prove uniform dessipativeness for nonautonomous systems of ordinary differential equations. Then some theorems are established to give sufficient conditions to obtain synchronization of coupled systems. The above results are applied to some specfic coupled systems, namely, coupled Lorenz systems, coupled Duffing's equations, coupled Chua's systems, etc., showing how to code and decode message using chaotic systems. One of our main results is to obtain the robustness of the synchronization with respect to parameter variation.
We consider an abstract system of two coupled nonlinear (infinite dimensional) equations. This kind of systems may describe various interaction phenomena in a continuum medium. Under some conditions we prove the existence of an exponentially attracting invariant manifold for the coupled system and show that this system can be reduced to a single equation with modified nonlinearity. This result means that under some conditions we observe (nonlinear) synchronization phenomena in the coupled system. As applications we consider coupled systems consisting of (i) parabolic and hyperbolic equations, (ii) two hyperbolic equations, and (iii) Klein–Gordon and Schrödinger equations.
In most applications, the synchronization of systems evolving under a chaotic regime requires the construction of identical systems or subsystems. In practical applications, systems should be created so that they match as closely as possible. Moreover, in real devices parameters can fluctuate resulting in loss of synchronization. In this paper, we consider a master–slave system of ordinary differential equations which are not identical. Considering bounded solutions of the master equation, we use those as an input in the slave equation. By using exponential dichotomies techniques we establish conditions that ensure synchronization.
We consider the synchronization of solutions to coupled systems of the conjugate random ordinary differential equations (RODEs) for the N -Stratronovich stochastic ordinary dif-ferential equations (SODEs) with linear multiplicative noise (N ∈ N). We consider the synchronization between two solutions and among different components of solutions under one-sided dissipative Lipschitz conditions. We first show that the random dynam-ical system generated by the solution of the coupled RODEs has a singleton sets random attractor which implies the synchronization of any two solutions. Moreover, the single-ton sets random attractor determines a stationary stochastic solution of the equivalently coupled SODEs. Then we show that any solution of the RODEs converge to a solution of the averaged RODE within any finite time interval as the coupled coefficient tends to infinity. Our results generalize the work of two Stratronovich SODEs in [9].
In this paper, a new approach combining synchronization techniques and trajectory sensitivity analysis is proposed to estimate the parameters of electrical synchronous generator models. The main contribution of this combination is the improvement on the numerical robustness of the estimation algorithm. The synchronization technique consists of coupling the real system with the mathematical model. The coupling is of unilateral type, that is, some outputs of the real system are considered inputs of the mathematical model. With a convenient choice of coupling, parameters that would have difficulties to be estimated by the traditional approach (due in general to very low sensitivity) can be more easily estimated. The proposed methodology is appropriate for on-line applications and does not require special operating conditions. Indeed, it can naturally deal with generator model nonlinearities. The methodology was successfully applied to estimate the parameters of a two-axis synchronous generator model. Synchronization improved the numerical robustness of the estimation algorithm allowing the simultaneous estimation of the d and q axis parameters even for bad initial guesses of parameters
The object of this work is to obtain uniform estimates, with
respect to parameters, of the attractor and of the basin of attraction
of a dynamical system and to apply these results to analyze the
roughness of the synchronization of two subsystems. These estimates are
obtained through an uniform version of the invariance principle of La
Salle which is stated and proved in this work
In this paper we prove that diffusively coupled abstract semilinear parabolic systems synchronize. We apply the abstract results obtained to a class of ordinary differential equations and to reaction diffusion problems. The technique consists of proving that the attractors for the coupled differential equations are upper semicontinuous with respect to the attractor of a limiting problem, explicitly exhibited, in the diagonal.
The authors describe the conditions necessary for synchronizing a
subsystem of one chaotic system with a separate chaotic system by
sending a signal from the chaotic system to the subsystem. The general
scheme for creating synchronizing systems is to take a nonlinear system,
duplicate some subsystem of this system, and drive the duplicate and the
original subsystem with signals from the unduplicated part. This is a
generalization of driving or forcing a system. The process can be
visualized with ordinary differential equations. The authors have build
a simple circuit based on chaotic circuits described by R. W. Newcomb et
al. (1983, 1986), and they use this circuit to demonstrate this chaotic
synchronization
We study self-synchronization of digital phase-locked loops (DPLL's) and the chaotic synchronization of DPLL's in a communication system which consists of three or more coupled DPLL's. Triangular wave signals, convenient for experiments, are employed. Numerical and experimental studies of two loops are in good agreement, giving bifurcation diagrams that show quasiperiodic, locked, and chaotic behavior. The approach to chaos does not show the full bifurcation sequence of sinusoidal signals. For studying synchronization to a chaotic signal, the chaotic carrier is generated in a subsystem of two or more self-synchronized DPLL's where one of the loops is stable and the other is unstable. The receiver consists of a stable loop. We verified numerically and experimentally that the receiver may synchronize with the transmitter if the stable loop in the transmitter and receiver are nearly identical and the synchronization degrades with noise and parameter variation. We studied the phase space where synchronization occurs, and quantify the deviation from synchronization using the concept of mutual information.
This paper is devoted to the problem of synchronization of dynamical systems in chaotic oscillations regimes. The authors attempt to use the ideas of synchronization and its mechanisms on a certain class of chaotic oscillations. These are chaotic oscillations for which one can pick out basic frequencies in their power spectra.
The physical and computer experiments were carried out for the system of two coupled auto-oscillators. The experimental installation permitted one to realize both unidirectional coupling (external synchronization) and symmetrical coupling (mutual synchronization). An auto-oscillator with an inertial nonlinearity was chosen as a partial subsystem. It possesses a chaotic attractor of spiral type in its phase space. It is known that such chaotic oscillations have a distinguished peak in the power spectrum at the frequency f 0 (basic frequency). In the experiments, one could make the basic frequencies of partial oscillators equal or different. The bifurcation diagrams on the plane of control parameters "detuning" and "coupling" were constructed and analyzed.
The results of investigations permit one to conclude that classical ideas of synchronization can be applied to chaotic systems of the mentioned type. Two mechanisms of chaos synchronization were established: 1) basic frequency locking and 2) basic frequency suppression. The bifurcational background of these mechanisms was created using numerical analysis on a computer. This allowed one to analyze the evolution of different oscillation characteristics under the influence of synchronization.
A new type of synchronized chaos conditioned by the threshold synchronization of relaxation oscillators with chaotic behaviour is studied experimentally. It is shown that for a certain parameter ratio the pulses generated by the chaotic oscillator may be synchronized by periodic and chaotic pulse sequences generated by the drive oscillator. It is found that the threshold synchronization regime enables one to detect modulation of chaotic signals.
Two methods of synchronization of chaotic oscillations in electronic circuits are considered. An example of application of synchronized chaos for secure communication is demonstrated by means of relaxation oscillators.
The general stability theory of the synchronized motions of the coupled-oscillator systems is developed with the use of the extended Lyapunov matrix approach. We give the explicit formula for a stability parameter of the synchronized state Psiunif. When the coupling strength is weakened, the coupled system may exhibit several types of non-synchronized motion. In particular, if Psiunif is chaotic, we always get a transition from chaotic Psiunif to a certain non-uniform state and finally the non-uniform chaos. Details associated with such transition are investigated for the coupled Lorenz model. As an application of the theory, we propose a new experimental method to directly measure the positive Lyapunov exponent of intrinsic chaos in reaction systems.
The objective of this work is to discuss the existence, bifurcation, and regularity, with respect to time and parameters, of bounded solutions of infinite dimensional equations. The authors present an application of their results to the study of homoclinic solutions of a nonlinear forced beam equation. They use the approach of alternative method and semigroup theory.
It is shown that the unstable steady state of a multimode laser system may be stabilized by the occasional proportional feedback technique for dynamical control of chaotic systems. The range of pump excitations over which stabilization can be maintained is extended by more than an order of magnitude through application of a procedure for tracking the unstable steady state as the pump excitation is slowly varied.
A simple model for synchronous firing of biological oscillators based on C. S. Peskin’s model of the cardiac pacemaker [Mathematical aspects of heart physiology (1975; Zbl 0301.92001), pp. 268-278] is studied. The model consists of a population of identical integrate-and-fire oscillators. The coupling between oscillators is pulsatile: when a given oscillator fires, it pulls the others up by a fixed amount, or brings them to the firing threshold, whichever is less. The main result is that for almost all initial conditions, the population evolves to a state in which all the oscillators are firing synchronously. The relationship between the model and real communities of biological oscillators is discussed; examples include populations of synchronously flashing fireflies, crickets that chirp in unison, electrically synchronous pacemaker cells, and groups of women whose menstrual cycles become mutually synchronized.
Incluye bibliografía e índice
We examine the mutual coherence and phase dynamics of two solid-state lasers, generated adjacent to each other in a single Nd:YAG rod. The coupling of the lasers is varied by changing the separation of the pump beams. A model is formulated to interpret the experimental results, and theoretical predictions are obtained that are in excellent agreement with the measurements.
We demonstrate that two identical chaotic systems can be made to synchronize by applying small, judiciously chosen, temporal-parameter perturbations to one of them. This idea is illustrated with a numerical example. Other issues related to synchronization are also discussed.
A multimode, autonomously chaotic solid-state laser system has been controlled by the technique of occassional proportional feedback, related to the control scheme of Ott, Grebogi, and Yorke. We show that complex periodic wave forms can be stabilized in the laser output intensity. A detailed model of the system is not necessary. Our results indicate that this control technique may be widely applicable to autonomous, higher-dimensional chaotic systems, including globally coupled arrays of nonlinear oscillators.
Models for the excitation of collective modes in a system of nonlinear classical oscillators, initially out of phase, are discussed. The oscillators may be coupled in a dissipative or conservative manner. The analysis is based on the results of recent studies dealing with the problem of the free excitation of a coherent pulse, analogous to "superradiance" in two-level quantum systems. Several physical examples from the realms of electrodynamics and acoustics are discussed. The processes discussed here may be thought of as chaos-order transitions, provided that "chaos" is understood not as a stochastic nature of an individual oscillator but as the absence of a coherent component in their collective field.
Experimental synchronization of chaotic lasers, (to appear). 1201 N. F. Rul'kov and A. R. Volkovskii, Synchronized chaos i n electronic circuits
- R Roy
- K S Thornburg
- Jr
R. Roy and K. S. Thornburg Jr., Experimental synchronization of chaotic lasers, (to appear). 1201 N. F. Rul'kov and A. R. Volkovskii, Synchronized chaos i n electronic circuits, Proceedings of SPIE conference on Exploiting chaos and nonlinearities, San Diego, July (1993).
Mutual synchronization of chaotic oscillations i n the Lorenz systems
- N N Verichev
N. N. Verichev, Mutual synchronization of chaotic oscillations i n the Lorenz systems, Methods of the Qual. Theory of Diff. Eq. (In Russian). Gorky, 47-57 (1986).