Recent Advances in Nonlinear Dynamics and Synchronization (original) (raw)
Synchronization: A Universal Concept in Nonlinear Science
American Journal of Physics, 2002
What is NOT synchronization? 1 4 .3 Synchronization : an overview of different cases 1 8 .3 .1 Terminological remarks 2 2 .4 Main bibliography 23 Part I: Synchronization without formulae Chapter 2 Basic notions : the self-sustained oscillator and its phase 2 7 2 .1 Self-sustained oscillators : mathematical models of natural systems 2 8 2 .1 .1 Self-sustained oscillations are typical in nature 28 2 .1 .2 Geometrical image of periodic self-sustained oscillations: limit cycle 2 9 2 .2 Phase : definition and properties 3 1 2 .2 .1
Dynamic Analysis and Synchronization for a Generalized Class of Nonlinear Systems
3rd IFAC Conference on Analysis and Control of Chaotic Systems (2012), 2012
This paper presents a generalized differential equation structure which gives rise to new nonlinear, chaotic systems and also encompasses many well known systems, i.e. Lorenz, Chen, Lü, Rössler, Sprott and others. Throughout the paper we shall analyze several properties from a few of the systems derived from this general structure. Some of the systems described in the article, to the extent of the authors knowledge, have not been published. The analytical and numerical results derived from these analysis have shown evidence of chaotic behavior. We will also address the possibility of quasi-simultaneous synchronization for some members of this class of chaotic systems.
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
Synchronization phenomena in complex dynamical systems
. The nonlinear dynamics and the synchronization characteristics of two different coupled oscillatory systems are studied both from a theoretical modelling and from a parametric analysis point of view, with the support of an effective computational approach. First the synchronous lateral excitation phenomenon occurring in bridge-pedestrians systems is addressed, then the complex dynamical interactions occurring in physical systems moldable as chains of nonlinearly coupled chaotic pendulums subjected to harmonic parametric excitations are examined.
Robust synchronization of a class of nonlinear systems
Proceedings of the American Control Conference
In this paper we present a new algorithm to synchronize two nonlinear systems. The systems may differ in structure, parameter values, and have structural uncertainties; therefore, the algorithm is robust in this sense. The conditions on the systems for applying this algorithm are the following: They must have the same order, and be in integrator chain form. Furthermore, we must know the bounds on the parametric uncertainties and on additional terms due to non modeled dynamics. The algorithm is based on the master/slave synchronization scheme. The coupling signal is designed through the sliding mode control technique. The results are illustrated with an experiment where a Duffing circuit and a simple pendulum are synchronized; both systems exhibit a chaotic attractor when they are not coupled.
Synchronization problems via a nonlinear feedback approach
In this paper we consider three different synchronization problems consisting in designing a nonlinear feedback unidirectional coupling term for two (possibly chaotic) dynamical systems in order to drive the trajectories of one of them, the slave system, to a reference trajectory or to a prescribed neighborhood of the reference trajectory of the second dynamical system: the master system. If the slave system is chaotic then synchronization can be viewed as the control of chaos; namely the coupling term allows to suppress the chaotic motion by driving the chaotic system to a prescribed reference trajectory. Assuming that the entire vector field representing the velocity of the state can be modified, three different methods to define the nonlinear feedback synchronizing controller are proposed: one for each of the treated problems. These methods are based on results from the small parameter perturbation theory of autonomous systems having a limit cycle, from nonsmooth analysis and from the singular perturbation theory respectively. Simulations to illustrate the effectiveness of the obtained results are also presented.
Physical Review Letters, 1996
Necessary and sufficient conditions for the occurrence of generalized synchronization of unidirectionally coupled dynamical systems are given in terms of asymptotic stability. The relation between generalized synchronization, predictability, and equivalence of dynamical systems is discussed. All theoretical results are illustrated by analytical and numerical examples. In particular, the existence of generalized synchronization in the case of parameter mismatch between coupled systems leads to a new interpretation of recent experimental results. Furthermore, the possible application of generalized synchronization for attractor reconstruction in nonlinear time series analysis is discussed.
2009
The phenomena of de-synchronization, synchronization, and forced oscillation has been investigation using describing function theory for a two input and two output nonlinear system containing saturation-type nonlinearities and subjected to high-frequency deterministic signal for the purpose of limit cycle quenching. The analytical results have been compared with the results of digital simulation Matlab-Simulink for a typical example varying the nonlinear element. ( ) ( ) du u ' X J Bu J u WSEAS TRANSACTIONS on SYSTEMS and CONTROL