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Synchronization of Discrete Time Dynamical Systems
Abstract
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.
... Different synchronization methods and techniques have been used to study synchronization between two similar integer order systems (Ojo and Ogunjo 2012), two dissimilar integer order systems of same dimension (Motallebzadeh et al. 2012;Femat and Solís-Perales 2002), two similar or dissimilar systems with different dimensions Ogunjo 2013), three or more integer order system (compound, combination-combination synchronization Ojo et al. 2014b, a;Mahmoud et al. 2016, discrete systems (Liu 2008Kloeden 2004;Ma et al. 2007), fractional order system of similar dimension (Lu 2005), fractional order synchronization of different dimension (Bhalekar 2014;Khan and Bhat 2017), circuit implementation of synchronization (Adelakun et al. 2017), impulsive anti-synchronization of fractional order (Meng et al. 2018), discrete fractional order systems (Liu 2016), multiswitching (Ogunjo et al. 2018), and synchronization between integer order and fractional order systems (Chen et al. 2012). ...
The paper investigates a new hybrid synchronization called modified hybrid synchronization (MHS) via the active control technique. Using the active control technique, stable controllers which enable the realization of the coexistence of complete synchronization and anti-synchronization in four identical fractional order chaotic systems were derived. Numerical simulations were presented to confirm the effectiveness of the analytical technique.
... The phenomenon is called synchronization. Clearly, analogous results are fulfilled for more general autonomous attractors [2,3] and for nonautonomous model [9] with suitably defined nonautonomous attractors as well as discrete time systems [10]. Caraballo, et al. [5] illustrated that the synchronization effect persists under additive noise, provided equilibria are replaced by stationary random solutions. ...
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.
... Different synchronization methods and techniques have been used to study synchronization between two similar integer order systems [26], two dissimilar integer order systems of same dimension [20,8], two similar or dissimilar systems with different dimensions [27,28,22], three or more integer order system (compound, combination-combination synchronization) [25,24,19], discrete systems [14,12,18], fractional order system of similar dimension [16], fractional order synchronization of different dimension [2,11], circuit implementation of synchronization [1] and synchronization between integer order and fractional order systems [3]. ...
The paper investigates a new hybrid synchronization called modified hybrid synchronization (MHS) via the active control technique. Using the active control technique, stable controllers which enable the realization of the coexistence of complete synchronization, anti-synchronization and project synchronization in four identical fractional order chaotic systems were derived. Numerical simulations were presented to confirm the effectiveness of the analytical technique.
... One of the topics in discrete dynamical systems theory, which deals with the evolution in discrete time steps of certain quantities over time (several examples are provided in [15,16,19]), consists of the study of certain dynamic invariants associated with expressions of the form ...
A fractal structure is a countable family of coverings which displays accurate information about the irregularities that a set presents when being explored with enough level of detail. It is worth noting that fractal structures become especially appropriate to provide new definitions of fractal dimension, which constitutes a valuable measure to test for chaos in dynamical systems. In this paper, we explore several approaches to calculate the fractal dimension of a subset with respect to a fractal structure. These models generalize the classical box dimension in the context of Euclidean subspaces from a discrete viewpoint. To illustrate the flexibility of the new models, we calculate the fractal dimension of a family of self-affine sets associated with certain discrete dynamical systems.
... This phenomenon is known as dissipative synchronization. Analogous results hold for more general autonomous attractors [1,6] as well as for nonautonomous systems [8] with appropriately defined nonautonomous attractors and discrete time systems [9]. Moreover, it was shown in [3] that the synchronization effect is preserved under additive noise provided equilibria are replaced by stationary random solutions. ...
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.
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)
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 synchronization of two nonautonomous dynamical systems is considered, where the systems are described in terms of a skew-product formalism, i. e., in which an inputed autonomous driving system governs the evolution of the vector field of a differential equation with the passage of time. It is shown that the coupled trajectories converge to each other as time increases for sufficiently large coupling coefficient and also that the component sets of the pullback attractor of the coupled system converges upper semi continuously as the coupling parameter increases to the diagonal of the product of the corresponding component sets of the pullback attractor of a system generated by the average of the vector fields of the original uncoupled systems.
The second edition has greatly benefited from a sizable number of comments and suggestions I received from users of the book. I hope that I have corrected all the er rors and misprints in the book. Important revisions were made in Chapters I and 4. In Chapter I, we added two appendices (global stability and periodic solutions). In Chapter 4, we added a section on applications to mathematical biology. Influenced by a friendly and some not so friendly comments about Chapter 8 (previously Chapter 7: Asymptotic Behavior of Difference Equations), I rewrote the chapter with additional material on Birkhoff's theory. Also, due to popular demand, a new chapter (Chapter 9) under the title "Applications to Continued Fractions and Orthogonal Polynomials" has been added. This chapter gives a rather thorough presentation of continued fractions and orthogonal polynomials and their intimate connection to second-order difference equations. Chapter 8 (Oscillation Theory) has now become Chapter 7. Accordingly, the new revised suggestions for using the text are as follows. The diagram on p. viii shows the interdependence of the chapters The book may be used with considerable flexibility. For a one-semester course, one may choose one of the following options: (i) If you want a course that emphasizes stability and control, then you may select Chapters I, 2, 3, and parts of 4, 5, and 6. This is perhaps appropriate for a class populated by mathematics, physics, and engineering majors.
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.
This book unites the study of dynamical systems and numerical solution of differential equations. The first three chapters contain the elements of the theory of dynamical systems and the numerical solution of initial-value problems. In the remaining chapters, numerical methods are formulted as dynamical systems and the convergence and stability properties of the methods are examined. Topics studied include the stability of numerical methods for contractive, dissipative, gradient and Hamiltonian systems together with the convergence properties of equilibria, periodic solutions and strage attractors under numerical approximation. This book will be an invaluable tool for graduate students and researchers in the fields of numerical analysis and dynamical systems.