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
Limits to the experimental detection of nonlinear synchrony
… REVIEW-SERIES E-, 2002
Chaos synchronization is often characterized by the existence of a continuous function between the states of the components. However, in coupled systems without inherent symmetries, the synchronization set might be extremely complicated. For coupled invertible systems, the synchronization set can be nondi®erentiable; in the more severe case of coupled noninvertible systems, the synchronization set will in general be a multivalued relation. We will discuss how existing methods for detecting synchronization will be hampered by these features.
Complete periodic synchronization in coupled systems
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2008
Recently, complete chaotic synchronization in coupled systems has been well studied. In this paper, we study complete synchronization in coupled periodic oscillators with diffusive and gradient couplings. Eight typical types of critical curve for the transverse Lyapunov exponent of standard mode, which give rise to different synchronization-desynchronization patterns, are classified. All possible desynchronous behaviors including steady state, periodic state, quasiperiodic state, lowdimensional chaotic state, and two types of high-dimensional chaotic state are identified, and two classical synchronization-desynchronizaiton bifurcations-the shortest wavelength bifurcation and Hopf bifurcation from synchronous periodic state-are classified.
IFAC-PapersOnLine, 2015
In this paper, synchronous behaviour is induced in a group of identical dynamical oscillators by applying a synchronizing signal, which is generated by an exogenous system. Specifically, a second order dynamical system driven by a pulse train is designed. The output of this system is further applied to a group of identical oscillatory systems, forcing them to synchronize. The novelty in this approach is that the synchronizing signal does not require any information from the dynamics of the systems to synchronize, i.e. the systems are uncoupled and they synchronize not because of an interaction between them but because of the influence of the synchronizing exogenous signal. In the analysis, the proposed synchronizing signal is applied to a set of chaotic systems, to a set of biological systems, and to a set of mechanical oscillators.
Dynamics of generalized bidirectional synchronization
International Journal of Dynamics and Control, 2015
In this paper, two different methods for two(bi)directional generalized synchronization of chaotic systems are proposed. The coupling based scheme provides a strategy to design suitable coupling functions to achieve synchronization. The controller based scheme relies on designing a suitable control input for achieving generalized synchronization and works irrespective of the nature of coupling between the systems. The two proposed schemes are applied to identical as well as non identical Sprott N and Sprott Q chaotic systems to illustrate their effectiveness. Numerical simulations are performed for empirical evidence of the theoretical work.
Synchronization of Coupled Extended Dynamical Systems: A Short Review
International Journal of Bifurcation and Chaos, 2003
We briefly review the synchronization properties of cross-coupled spatially extended dynamical systems, with particular emphasis on elementary cellular automata and Kauffman networks subject to stochastic coupling. We also discuss the main results for the joint evolution of deterministically cross-coupled Ginzburg–Landau equations and neural networks. Both numerical and analytical approaches are addressed, and the main differences with the synchronization of zero-dimensional systems are highlighted. New results are presented characterizing the critical behavior at the synchronization transition of coupled Kauffman networks.
Synchronization by reactive coupling and nonlinear frequency pulling
Physical Review E, 2006
We present a detailed analysis of a model for the synchronization of nonlinear oscillators due to reactive coupling and nonlinear frequency pulling. We study the model for the mean field case of allto-all coupling, deriving results for the initial onset of synchronization as the coupling or nonlinearity increase, and conditions for the existence of the completely synchronized state when all the oscillators evolve with the same frequency. Explicit results are derived for Lorentzian, triangular, and top-hat distributions of oscillator frequencies. Numerical simulations are used to construct complete phase diagrams for these distributions. PACS numbers: 85.85.+j, 05.45.-a, 05.45.Xt, 62.25.+g
Nonlinear control and synchronization of a class of nonlinear coupled dynamical systems
Journal of Control Theory and Applications, 2013
In this paper, a control problem for a class of nonlinear coupled dynamical systems is proposed and a continuous nonlinear feedback control law is designed using direct Lyapunov method to solve the proposed control problem. Moreover, synchronization problem for a special case of this class nonlinear coupled dynamical systems is concerned. Numerical examples show the effectiveness and advantage of the designed continuous nonlinear control law and derived synchronization result.