Preservation of hyperbolic equilibrium points and synchronization in dynamical systems (original) (raw)

Tutorial and Review on the Synchronization of Chaotic Dynamical Systems

sestindia.org

1990s researchers have realized that chaotic systems can be synchronized. The phenomenon chaotic synchronization is immediately interesting because of its high potential for applications. Our purpose in this paper is to collect results from various areas in a review article format with a tutorial emphasis. Chaotic dynamical systems having features, sensitive dependence on initial conditions and topological mixing along with periodic orbit must be dense. In order to optimize the result of synchronization is also a motivating factor for the study of this phenomenon, there is possibility of chaos control. In this review paper we considered the synchronization of chaotic dynamical systems, both in continuous and discrete time along with preliminary notions on nonlinear dynamics.

Various Synchronization Schemes for Chaotic Dynamical Systems (A Classical Survey)

International Journal

The phenomenon of chaotic system synchronization is very interesting because of its high potential of applications. In this paper results from various synchronization schemes in the form of a survey articles with a tutorial emphasis has been presented. Chaotic dynamical systems having features such as sensitive dependence on initial conditions and topological mixing along with periodic dense orbits. In order to optimize the results of synchronization is also a motivating factor for the study of this phenomenon.

Robust synchronization of chaotic systems

Physical Review E, 2000

The question of robustness of synchronization with respect to small arbitrary perturbations of the underlying dynamical systems is addressed. We present examples of chaos synchronization demonstrating that normal hyperbolicity is a necessary and sufficient condition for the synchronization manifold to be smooth and persistent under small perturbations. The same examples, however, show that in real applications normal hyperbolicity is not sufficient to give quantitative bounds for deformations of the synchronization manifold, i.e., even in the case of normal hyperbolicity two almost identical systems may cause large synchronization errors.

Chaos synchronization in a hyperbolic dynamical system with long-range interactions

2008

We show that the threshold of complete synchronization in a lattice of coupled non-smooth chaotic maps is determined by linear stability along the directions transversal to the synchronization subspace. As a result, the numerically determined synchronization threshold agree with the analytical results previously obtained [C. Anteneodo et al., Phys. Rev. E 68, 045202(R) (2003)] for this class of systems. We present both careful numerical experiments and a rigorous mathematical explanation confirming this fact, allowing for a generalization involving hyperbolic coupled map lattices.

Fundamentals of synchronization in chaotic systems, concepts, and applications

1997

The field of chaotic synchronization has grown considerably since its advent in 1990. Several subdisciplines and ''cottage industries'' have emerged that have taken on bona fide lives of their own. Our purpose in this paper is to collect results from these various areas in a review article format with a tutorial emphasis. Fundamentals of chaotic synchronization are reviewed first with emphases on the geometry of synchronization and stability criteria. Several widely used coupling configurations are examined and, when available, experimental demonstrations of their success ͑generally with chaotic circuit systems͒ are described. Particular focus is given to the recent notion of synchronous substitution-a method to synchronize chaotic systems using a larger class of scalar chaotic coupling signals than previously thought possible. Connections between this technique and well-known control theory results are also outlined. Extensions of the technique are presented that allow so-called hyperchaotic systems ͑systems with more than one positive Lyapunov exponent͒ to be synchronized. Several proposals for ''secure'' communication schemes have been advanced; major ones are reviewed and their strengths and weaknesses are touched upon. Arrays of coupled chaotic systems have received a great deal of attention lately and have spawned a host of interesting and, in some cases, counterintuitive phenomena including bursting above synchronization thresholds, destabilizing transitions as coupling increases ͑short-wavelength bifurcations͒, and riddled basins. In addition, a general mathematical framework for analyzing the stability of arrays with arbitrary coupling configurations is outlined. Finally, the topic of generalized synchronization is discussed, along with data analysis techniques that can be used to decide whether two systems satisfy the mathematical requirements of generalized synchronization. © 1997 American Institute of Physics. ͓S1054-1500͑97͒02904-2͔

Chaos synchronization of nonlinear dynamical systems via a novel analytical approach

Alexandria Engineering Journal, 2018

This paper deals with the synchronization between two non-identical 4-D hyperchaotic systems. The nonlinear control technique is used for synchronization. The stability analysis of the error dynamics system is done by (i) Lyapunov's second method and (ii) Cardano's method. Four different expressions of the controller are presented in the paper and a comparison between the two methods are given. We notice that the Cardano's method is better than the Lyapunov approach. Finally, theoretical and numerical simulations are given to illustrate and verify the results.

Control of continuous-time chaotic (hyperchaotic) systems: F-M synchronisation

International Journal of Automation and Control, 2019

In this paper, a new type of chaos synchronisation between different dimensional chaotic systems is proposed. The novel scheme is called F-M synchronisation, since it combines the inverse generalised synchronisation with the matrix projective synchronisation. In particular, the proposed approach enables F-M synchronisation to be achieved between n-dimensional master system and m-dimensional slave system in different dimensions. The technique, which exploits nonlinear controllers, stability property of integer-order linear continuous-time dynamical systems and Lyapunov stability theory, proves to be effective in achieving the F-M synchronisation. Finally, simulation results are reported, with the aim to illustrate the capabilities of the novel scheme proposed herein.

Generalized synchronization of chaos in directionally coupled chaotic systems

Physical Review E - PHYS REV E, 1995

Synchronization of chaotic systems is frequently taken to mean actual equality of the variables of the coupled systems as they evolve in time. We explore a generalization of this condition, which equates dynamical variables from one subsystem with a function of the variables of another subsystem. This means that synchronization implies a collapse of the overall evolution onto a subspace of the system attractor in full space. We explore this idea in systems where a response system y(t) is driven with the output of a driving system x(t), but there is no feedback to the driver. We lose generality but gain tractability with this restriction. To investigate the existence of the synchronization condition y(t)=P(x(t)) we introduce the idea of mutual false nearest neighbors to determine when closeness in response space implies closeness in driving space. The synchronization condition also implies that the response dynamics is determined by the drive alone, and we provide tests for this as well. Examples are drawn from computer simulations on various known cases of synchronization and on data from nonlinear electrical circuits. Determining the presence of generalized synchronization will be quite important when one has only scalar observations from the drive and from the response systems since the use of time delay (or other) embedding methods will produce "imperfect" coordinates in which strict equality of the synchronized variables is unlikely to transpire. PACS number(s): 05.45. +b, 84.30. Ng, 07.05.t variables of the two systems are proportional to each other. There are two categories of systems where this 'Electronic address: rulkov Ihamilton.

Optimized synchronization of chaotic and hyperchaotic systems

Physical Review E, 2010

A method of synchronization is presented which, unlike existing methods, can, for generic dynamical systems, force all conditional Lyapunov exponents to go to −ϱ. It also has improved noise immunity compared to existing methods, and unlike most of them it can synchronize hyperchaotic systems with almost any single coupling variable from the drive system. Results are presented for the Rossler hyperchaos system and the Lorenz system.