Showing papers in "International Journal of Bifurcation and Chaos in 2001"
TL;DR: Sufficient conditions for master–slave synchronization of Lur'e systems are presented for a known time-delay in the master and slave systems, and a delay-dependent synchronization criterion is given based upon a new Lyapunov–Krasovskii function.
Abstract: In this paper time-delay effects on the master–slave synchronization scheme are investigated. Sufficient conditions for master–slave synchronization of Lur'e systems are presented for a known time-delay in the master and slave systems. A delay-dependent synchronization criterion is given based upon a new Lyapunov–Krasovskii function. The derived criterion is a sufficient condition for global asymptotic stability of the error system, expressed by means of a matrix inequality. The feedback matrix follows from solving a nonlinear optimization problem. The method is illustrated for the synchronization of Chua's circuits, 5-scroll attractors and hyperchaotic attractors.
221 citations
TL;DR: This work first proves asymptotic compactness and then establishes the existence of global attractors and the upper semicontinuity of the global attractor is also obtained when the lattice differential equations are approached by finite-dimensional systems.
Abstract: We study the asymptotic behavior of solutions for lattice dynamical systems. We first prove asymptotic compactness and then establish the existence of global attractors. The upper semicontinuity of the global attractor is also obtained when the lattice differential equations are approached by finite-dimensional systems.
215 citations
TL;DR: A theory of stochastic chaos is developed, in which aperiodic outputs with 1/f2 spectra are formed by the interaction of globally connected nodes that are individually governed by point attractors under perturbation by continuous white noise.
Abstract: A fundamental tenet of the theory of deterministic chaos holds that infinitesimal variation in the initial conditions of a network that is operating in the basin of a low-dimensional chaotic attractor causes the various trajectories to diverge from each other quickly. This "sensitivity to initial conditions" might seem to hold promise for signal detection, owing to an implied capacity for distinguishing small differences in patterns. However, this sensitivity is incompatible with pattern classification, because it amplifies irrelevant differences in incomplete patterns belonging to the same class, and it renders the network easily corrupted by noise. Here a theory of stochastic chaos is developed, in which aperiodic outputs with 1/f2 spectra are formed by the interaction of globally connected nodes that are individually governed by point attractors under perturbation by continuous white noise. The interaction leads to a high-dimensional global chaotic attractor that governs the entire array of nodes. An example is our spatially distributed KIII network that is derived from studies of the olfactory system, and that is stabilized by additive noise modeled on biological noise sources. Systematic parameterization of the interaction strengths corresponding to synaptic gains among nodes representing excitatory and inhibitory neuron populations enables the formation of a robust high-dimensional global chaotic attractor. Reinforcement learning from examples of patterns to be classified using habituation and association creates lower dimensional local basins, which form a global attractor landscape with one basin for each class. Thereafter, presentation of incomplete examples of a test pattern leads to confinement of the KIII network in the basin corresponding to that pattern, which constitutes many-to-one generalization. The capture after learning is expressed by a stereotypical spatial pattern of amplitude modulation of a chaotic carrier wave. Sensitivity to initial conditions is no longer an issue. Scaling of the additive noise as a parameter optimizes the classification of data sets in a manner that is comparable to stochastic resonance. The local basins constitute dynamical memories that solve difficult problems in classifying data sets that are not linearly separable. New local basins can be added quickly from very few examples without loss of existing basins. The attractor landscape enables the KIII set to provide an interface between noisy, unconstrained environments and conventional pattern classifiers. Examples given here of its robust performance include fault detection in small machine parts and the classification of spatiotemporal EEG patterns from rabbits trained to discriminate visual stimuli.
202 citations
TL;DR: It is shown that, despite what has been conjectured in the literature, a symmetric and unforced relay feedback system can exhibit asymmetric periodic solutions.
Abstract: This paper is concerned with the bifurcation analysis of linear dynamical systems with relay feedback. The emphasis is on the bifurcations of the system periodic solutions and their symmetry. It is shown that, despite what has been conjectured in the literature, a symmetric and unforced relay feedback system can exhibit asymmetric periodic solutions. Moreover, the occurrence of periodic solutions characterized by one or more sections lying within the system discontinuity set is outlined. The mechanisms underlying their formation are carefully studied and shown to be due to an interesting, novel class of local bifurcations.
172 citations
TL;DR: Experimental results show that chaotic and hyperchaotic systems can be synchronized by impulses sampled from one or two state variables.
Abstract: Experimental results show that chaotic and hyperchaotic systems can be synchronized by impulses sampled from one or two state variables. In this paper, we study the conditions under which chaotic a...
165 citations
TL;DR: The variation of the Lyapunov exponent, and the qualitative and quantitative aspects of its local fluctuation properties, have emerged as an important means of studying fractal attractors, and this analysis finds useful application here.
Abstract: Aperiodic dynamics which is nonchaotic is realized on Strange Nonchaotic Attractors (SNAs). Such attractors are generic in quasiperiodically driven nonlinear systems, and like strange attractors, are geometrically fractal. The largest Lyapunov exponent is zero or negative: trajectories do not show exponential sensitivity to initial conditions. In recent years, SNAs have been seen in a number of diverse experimental situations ranging from quasiperiodically driven mechanical or electronic systems to plasma discharges. An important connection is the equivalence between a quasiperiodically driven system and the Schrodinger equation for a particle in a related quasiperiodic potential, showing a correspondence between the localized states of the quantum problem with SNAs in the related dynamical system. In this review we discuss the main conceptual issues in the study of SNAs, including the different bifurcations or routes for the creation of such attractors, the methods of characterization, and the nature of dynamical transitions in quasiperiodically forced systems. The variation of the Lyapunov exponent, and the qualitative and quantitative aspects of its local fluctuation properties, have emerged as an important means of studying fractal attractors, and this analysis finds useful application here. The ubiquity of such attractors, in conjunction with their several unusual properties, suggests novel applications.
161 citations
TL;DR: A reapproach to chaotic systems synchronization is presented from the perspective of passivity-based state observer design in the context of Generalized Hamiltonian systems including dissipation and destabilizing vector fields.
Abstract: A reapproach to chaotic systems synchronization is presented from the perspective of passivity-based state observer design in the context of Generalized Hamiltonian systems including dissipation and destabilizing vector fields. The synchronization and lack of synchronization of several well-studied chaotic systems is reexplained in these terms.
138 citations
TL;DR: A new method of frequency analysis for Hamiltonian Systems of 3 degrees of freedom and more is presented, based on the concept of instantaneous frequency extracted numerically from the continuous wavelet transform of the trajectories.
Abstract: We present a new method of frequency analysis for Hamiltonian Systems of 3 degrees of freedom and more. The method is based on the concept of instantaneous frequency extracted numerically from the continuous wavelet transform of the trajectories. Knowing the time-evolution of the frequencies of a given trajectory, we can define a frequency map, resonances, and diffusion in frequency space as an indication of chaos. The time-frequency analysis method is applied to the Baggott Hamiltonian to characterize the global dynamics and the structure of the phase space in terms of resonance channels. This 3-degree-of-freedom system results from the classical version of the quantum Hamiltonian for the water molecule given by Baggott [1988]. Since another first integral of the motion exists, the so-called Polyad number, the system can be reduced to 2 degrees of freedom. The dynamics is therefore simplified and we give a complete characterization of the phase space, and at the same time we could validate the results of the time-frequency analysis.
127 citations
TL;DR: A unified approach for synthesizing nonlinear circuits is presented, that is, electronic circuits for simulating nonlinear dynamics are synthesized from ordinary differential equations.
Abstract: In this paper, we present a unified approach for synthesizing nonlinear circuits. That is, we synthesize electronic circuits for simulating nonlinear dynamics. One advantage of our approach is that we can directly synthesize nonlinear circuits from ordinary differential equations. A large variety of chaotic nonlinear systems (Chua's circuit, hyperchaotic system, Lorenz system, Rossler system, etc.) are realized by using several analog circuit elements.
114 citations
TL;DR: An invariant manifold based chaos synchronization approach is proposed by using only a partial state of chaotic systems to synchronize the coupled chaotic systems by taking into account the inherent dynamic properties of the chaotic systems.
Abstract: An invariant manifold based chaos synchronization approach is proposed in this letter A novel idea of using only a partial state of chaotic systems to synchronize the coupled chaotic systems is presented by taking into account the inherent dynamic properties of the chaotic systems The effectiveness of the approach and idea is tested on the Lorenz system and the fourth-order Rossler system
88 citations
TL;DR: It seems that the bifurcation method of phase planes is good for the study of peakons in nonlinear integrable equations.
Abstract: Camassa and Holm [1993] recently derived a new dispersive shallow water equation known as the Camassa–Holm equation. They showed that it also has solitary wave solutions which have a discontinuous first derivative at the wave peak and thus are called "peakons". In this paper, from the mathematical point of view, we study the peakons and their bifurcation of the following generalized Camassa–Holm equation \[ u_t+2ku_x-u_{xxt}+au^mu_x = 2u_xu_{xx} + uu_{xxx} \] with a>0, k∈ℝ, m∈ℕ and the integral constants taken as zero. Using the bifurcation method of the phase plane, we first give the phase portrait bifurcation, then give the integral expressions of peakons through the bifurcation curves and the phase portraits, and finally obtain the peakon bifurcation parameter value and the number of peakons. For m=1, 2, 3, we give the explicit expressions for the peakons. It seems that the bifurcation method of phase planes is good for the study of peakons in nonlinear integrable equations.
TL;DR: In this paper, the authors proposed a method for collecting data at large scales and extrapolating to the limit of zero scale, which is a vastly reduced required number of data points for a given accuracy in the measured dimension.
Abstract: For many chaotic systems, accurate calculation of the correlation dimension from measured data is difficult because of very slow convergence as the scale size is reduced. This problem is often caused by the highly nonuniform measure on the attractor. This paper proposes a method for collecting data at large scales and extrapolating to the limit of zero scale. The result is a vastly reduced required number of data points for a given accuracy in the measured dimension. The method is illustrated in detail for one-dimensional maps and then applied to more complicated maps and flows. Values are given for the correlation dimension of many standard chaotic systems.
TL;DR: This paper shows that the Lorenz system can be transformed into a kind of nonlinear system in the so-called general strict-feedback form, and adaptive backstepping design is used to control the Lorentz system with three key parameters unknown.
Abstract: In this paper, we consider the problem of controlling chaos in the well-known Lorenz system. Firstly we show that the Lorenz system can be transformed into a kind of nonlinear system in the so-called general strict-feedback form. Then, adaptive backstepping design is used to control the Lorenz system with three key parameters unknown. By exploiting the property of the system, the resulting controller is singularity free, and the closed-loop system is stable globally. Simulation results are conducted to show the effectiveness of the approach.
TL;DR: Numerical methods for linear stability analysis of periodic solutions in codes for bifurcation analysis of small systems of ordinary differential equations (ODEs) and how the periodic Schur decomposition can be fitted into existing codes are studied.
Abstract: This paper studies numerical methods for linear stability analysis of periodic solutions in codes for bifurcation analysis of small systems of ordinary differential equations (ODEs). Popular techniques in use today (including the AUTO97 method) produce very inaccurate Floquet multipliers if the system has very large or small multipliers. These codes compute the monodromy matrix explicitly or as a matrix pencil of two matrices. The monodromy matrix arises naturally as a product of many matrices in many numerical methods, but this is not exploited. In this case, all Floquet multipliers can be computed with very high precision by using the periodic Schur decomposition and corresponding algorithm [Bojanczyk et al., 1992]. The time discretisation of the periodic orbit becomes the limiting factor for the accuracy. We present just enough of the numerical methods to show how the Floquet multipliers are currently computed and how the periodic Schur decomposition can be fitted into existing codes but omit all details. However, we show extensive test results for a few artificial matrices and for two four-dimensional systems with some very large and very small Floquet multipliers to illustrate the problems experienced by current techniques and the better results obtained using the periodic Schur decomposition. We use a modified version of AUTO97 [Doedel et al., 1997] in our experiments.
TL;DR: By means of several different Lyapunov functionals, some sufficient conditions related to the global asymptotic stability for cellular neural networks with perturbations of time-varying delays are derived.
Abstract: In this paper, the global asymptotic stability of cellular neural networks with time delay is discussed using some novel Lyapunov functionals. Novel sufficient conditions for this type of stability are derived. They are less restrictive and more practical than those currently used. As a result, the design of cellular neural networks with time delay is refined. Our work can also be generalized to cellular neural networks with time-varying delay, a topic on which little research work has been done. By means of several different Lyapunov functionals, some sufficient conditions related to the global asymptotic stability for cellular neural networks with perturbations of time-varying delays are derived.
TL;DR: The proposed approach offers a systematic design procedure for adaptive synchronization of a large class of continuous-time chaotic systems in the chaos research literature and results show the effectiveness of the approach.
Abstract: In this letter, adaptive synchronization of two uncertain chaotic systems is presented using adaptive backstepping with tuning functions. The master system is any smooth, bounded, linear-in-the-parameters nonlinear chaotic system, while the slave system is a nonlinear chaotic system in the strict-feedback form. Both master and slave systems are with key parameters unknown. Global stability and asymptotic synchronization between the outputs of master and slave systems can be achieved. The proposed approach offers a systematic design procedure for adaptive synchronization of a large class of continuous-time chaotic systems in the chaos research literature. Simulation results are presented to show the effectiveness of the approach.
TL;DR: This work suggests the importance of information exchange and state synchronization within ensembles, towards issues such as evolution, collective behavior, optimality and intelligence, in an interdisciplinary context.
Abstract: We show how coupling of local optimization processes can lead to better solutions than multistart local optimization consisting of independent runs. This is achieved by minimizing the average energy cost of the ensemble, subject to synchronization constraints between the state vectors of the individual local minimizers. From an augmented Lagrangian which incorporates the synchronization constraints both as soft and hard constraints, a network is derived wherein the local minimizers interact and exchange information through the synchronization constraints. From the viewpoint of neural networks, the array can be considered as a Lagrange programming network for continuous optimization and as a cellular neural network (CNN). The penalty weights associated with the soft state synchronization constraints follow from the solution to a linear program. This shows that the energy cost of the ensemble should maximally decrease. In this way successful local minimizers can implicitly impose their state to the others through a mechanism of master–slave dynamics resulting in a cooperative search mechanism. Improved information spreading within the ensemble is obtained by applying the concept of small-world networks. We illustrate the new optimization method on two different problems: supervized learning of multilayer perceptrons and optimization of Lennard–Jones clusters. The initial distribution of the local minimizers plays an important role. For the training of multilayer perceptrons this is related to the choice of the prior on the interconnection weights in Bayesian learning methods. Depending on the choice of this initial distribution, coupled local minimizers (CLM) can avoid overfitting and produce good generalization, i.e. reach a state of intelligence. In potential energy surface optimization of Lennard–Jones clusters, this choice is equally important. In this case it can be related to considering a confining potential. This work suggests, in an interdisciplinary context, the importance of information exchange and state synchronization within ensembles, towards issues such as evolution, collective behavior, optimality and intelligence.
TL;DR: This paper studies the qualitative behavior of a predator–prey system with nonmonotonic functional response that undergoes a series of bifurcations including the saddle-node bIfurcation, the supercritical Hopf bifircation, and the homoclinic bifURcation.
Abstract: In this paper we study the qualitative behavior of a predator–prey system with nonmonotonic functional response. The system undergoes a series of bifurcations including the saddle-node bifurcation, the supercritical Hopf bifurcation, and the homoclinic bifurcation. For different parameter values the system could have a limit cycle or a homoclinic loop, or exhibit the so-called "paradox of enrichment" phenomenon. In the generic case, the model has the bifurcation of cusp-type codimension two (i.e. the Bogdanov–Takens bifurcation) but no bifurcations of codimension three.
TL;DR: The possibility of using interval arithmetic for rigorous investigations of periodic orbits in discrete-time dynamical systems with special emphasis on chaotic systems is investigated and it is shown that methods based on interval arithmetic when implemented properly are capable of finding all period-n cycles for considerable large n.
Abstract: In this paper, we investigate the possibility of using interval arithmetic for rigorous investigations of periodic orbits in discrete-time dynamical systems with special emphasis on chaotic systems. We show that methods based on interval arithmetic when implemented properly are capable of finding all period-n cycles for considerable large n. We compare several interval methods for finding periodic orbits. We consider the interval Newton method and methods based on the Krawczyk operator and the Hansen–Sengupta operator. We also test the global versions of these three methods. We propose algorithms for computation of the invariant part and nonwandering part of a given set and for computation of the basin of attraction of stable periodic orbits, which allow reducing greatly the search space for periodic orbits. As examples we consider two-dimensional chaotic discrete-time dynamical systems, defined by the Henon map and the Ikeda map, with the "standard" parameter values for which the chaotic behavior is observed. For both maps using the algorithms presented in this paper, we find very good approximation of the invariant part and the nonwandering part of the region enclosing the chaotic attractor observed numerically. For the Henon map we find all cycles with period n ≤ 30 belonging to the trapping region. For the Ikeda map we find the basin of attraction of the stable fixed point and all periodic orbits with period n ≤ 15. For both systems using the number of short cycles, we estimate its topological entropy.
TL;DR: A mean-field model for a large array of coupled solid-state lasers with randomly distributed natural frequencies is analyzed, revealing a variety of unsteady collective states in which all the lasers' intensities vary periodically, quasiperiodically, or chaotically.
Abstract: We analyze a mean-field model for a large array of coupled solid-state lasers with randomly distributed natural frequencies. Using techniques developed previously for coupled nonlinear oscillators, we derive exact formulas for the stability boundaries of the phase locked, incoherent, and off states, as functions of the coupling and pump strength and the spread of natural frequencies. For parameters in the intermediate regime between total incoherence and perfect phase locking, numerical simulations reveal a variety of unsteady collective states in which all the lasers' intensities vary periodically, quasiperiodically, or chaotically.
TL;DR: It is shown that a denumerable set of unstable cycles can arise together with stable cycles at border-collision bifurcations at the dynamical model of a constant voltage converter, which represents a three-dimensional piecewise-smooth system of nonautonomous differential equations.
Abstract: Many problems of engineering and applied science result in the consideration of piecewise-smooth dynamical systems. Examples are relay and pulse-width control systems, impact oscillators, power converters, and various electronic circuits with piecewise-smooth characteristics. The subject of investigation in the present paper is the dynamical model of a constant voltage converter which represents a three-dimensional piecewise-smooth system of nonautonomous differential equations. A specific type of phenomena that arise in the dynamics of piecewise-smooth systems are the so-called border-collision bifurcations. The paper contains a detailed analysis of this type of bifurcational transition in the dynamics of the voltage converter, in particular, the merging and subsequent disappearance of cycles of different types, change of solution type, and period-doubling, -tripling, -quadrupling and -quintupling. We show that a denumerable set of unstable cycles can arise together with stable cycles at border-collision bifurcations. The characteristic peculiarities of border-collision bifurcational transitions in piecewise-smooth systems are described and we provide a comparison with some recent results.
TL;DR: The results on simulated data and real data (EEG and exchange rates) suggest that the test depends on the method and its parameters, the algorithm generating the surrogate data and the observational data of the examined process.
Abstract: In the analysis of real world data, the surrogate data test is often performed in order to investigate nonlinearity in the data. The null hypothesis of the test is that the original time series is generated from a linear stochastic process possibly undergoing a nonlinear static transform. We argue against reported rejection of the null hypothesis and claims of evidence of nonlinearity based on a single nonlinear statistic. In particular, two schemes for the generation of surrogate data are examined, the amplitude adjusted Fourier transform (AAFT) and the iterated AAFT (IAFFT) and many nonlinear discriminating statistics are used for testing, i.e. the fit with the Volterra series of polynomials and the fit with local average mappings, the mutual information, the correlation dimension, the false nearest neighbors, the largest Lyapunov exponent and simple nonlinear averages (the three point autocorrelation and the time reversal asymmetry). The results on simulated data and real data (EEG and exchange rates) suggest that the test depends on the method and its parameters, the algorithm generating the surrogate data and the observational data of the examined process.
TL;DR: The method can be viewed as a dynamic reconstruction mechanism, in contrast to existing static inversion methods from the theory of dynamical systems, for the synchronization of nonlinear discrete-time dynamics.
Abstract: A method is described for the synchronization of nonlinear discrete-time dynamics. The methodology consists of constructing observer–receiver dynamics that exploit at each time instant the drive signal and buffered past values of the drive signal. In this way, the method can be viewed as a dynamic reconstruction mechanism, in contrast to existing static inversion methods from the theory of dynamical systems. The method is illustrated on a few simulation examples consisting of coupled chaotic logistic equations. Also, a discrete-time message reconstruction scheme is simulated using the extended observer mechanism.
TL;DR: The possibility in relating successive low velocity impacts, especially with respect to possible low dimensional mappings for such a system, is discussed, and the types of chaotic motion that occur within the parameter range are considered.
Abstract: We consider the dynamics of a two-degree of freedom impact oscillator subject to a motion limiting constraint. These systems exhibit a range of periodic and nonperiodic impact motions. For a particular set of parameters, we consider the bifurcations which occur between differing regimes of impacting motion and in particular those which occur due to a grazing bifurcation. Unexpected resonant behavior is also observed, due to the complexity of the dynamics. We consider both periodic and chaotic chatter motions and the regions of sticking which exist. Finally we consider the types of chaotic motion that occur within the parameter range. We discuss the possibility in relating successive low velocity impacts, especially with respect to possible low dimensional mappings for such a system.
TL;DR: A method is presented for detecting weak coupling between (chaotic) dynamical systems below the threshold of (generalized) synchronization using reconstruction of mixed states consisting of delayed samples taken from simultaneously measured time series of both systems.
Abstract: A method is presented for detecting weak coupling between (chaotic) dynamical systems below the threshold of (generalized) synchronization. This approach is based on reconstruction of mixed states consisting of delayed samples taken from simultaneously measured time series of both systems.
TL;DR: Experimental investigations of synchronization and anti-synchronization of optically coupled semiconductor lasers with external cavities are presented, showing chaotic low-frequency intensity fluctuations that are (anti-) synchronized due to optical coupling by light injection.
Abstract: Experimental investigations of synchronization and anti-synchronization of optically coupled semiconductor lasers with external cavities are presented. Both lasers show chaotic low-frequency intensity fluctuations that are (anti-) synchronized due to optical coupling by light injection. For uni-directional coupling response lasers with and without external cavity are considered. In the case of bi-directional coupling synchronized fluctuations are observed even without external mirrors.
TL;DR: This letter addresses the problem of robust adaptive control for synchronization of continuous-time coupled chaotic systems, which may be subjected to disturbances and is presented to show the effectiveness of the proposed chaos synchronization methods.
Abstract: This letter addresses the problem of robust adaptive control for synchronization of continuous-time coupled chaotic systems, which may be subjected to disturbances. A general model is studied via two different approaches, using either state feedback or measured output feedback controls. Adaptive controllers are designed, in which a sliding mode structure is employed to increase the robustness of the closed-loop systems. When only output variables are measurable for synchronization, the adaptive controllers are designed by incorporating with a filter and using the so-called σ-modification technique. Several numerical examples are presented to show the effectiveness of the proposed chaos synchronization methods.
TL;DR: A novel approach for studying complete stability of piecewise-linear (PWL) CNN's based on a fundamental limit theorem for the length of the CNN trajectories, which shows that complete stability holds under hypotheses weaker than those considered in Chua & Yang, 1988.
Abstract: In recent years, the standard Cellular Neural Networks (CNN's) introduced by Chua and Yang [1988] have been one of the most investigated paradigms for neural information processing. In a wide range of applications, the CNN's are required to be completely stable, i.e. each trajectory should converge toward some stationary state. However, a rigorous proof of complete stability, even in the simplest original setting of piecewise-linear (PWL) neuron activations and symmetric interconnections [Chua & Yang, 1988], is still lacking. This paper aims primarily at filling this gap, in order to give a sound analytical foundation to the CNN paradigm. To this end, a novel approach for studying complete stability is proposed. This is based on a fundamental limit theorem for the length of the CNN trajectories. The method differs substantially from the classic approach using LaSalle invariance principle, and permits to overcome difficulties encountered when using LaSalle approach to analyze complete stability of PWL CNN's. The main result obtained, is that a symmetric PWL CNN is completely stable for any choice of the network parameters, i.e. it possesses the Absolute Stability property of global pattern formation. This result is really general and shows that complete stability holds under hypotheses weaker than those considered in [Chua & Yang, 1988]. The result does not require, for example, that the CNN has binary stable equilibrium points only. It is valid even in degenerate situations where the CNN has infinite nonisolated equilibrium points. These features significantly extend the potential application fields of the standard CNN's.
TL;DR: The results of modeling indicate that the statistical method can be applied to modeling the deterministic properties of spatiotemporal dynamics in terms of recorded data.
Abstract: Often in the analysis of spatially extended dynamic systems, we do not know an analytical model of the system dynamics, but we can provide spatiotemporal records of the characteristic state variable. The question then arises of how to extract a model of the system dynamics from the corresponding data. As a quite general solution of this problem, we propose a nonparametric statistical method of local modeling. The performance of the proposed method is demonstrated by predicting typical examples of spatiotemporal chaotic data. The results of modeling indicate that the statistical method can be applied to modeling the deterministic properties of spatiotemporal dynamics in terms of recorded data.
TL;DR: This paper discusses the bifurcation structure inside the Arnol'd tongue with zero rotation number and includes a study of nonsmooth bIfurcations that occur for large nonlinearity in the region with strange nonchaotic attractors.
Abstract: It is well known that the dynamics of the Arnol'd circle map is phase locked in regions of the parameter space called Arnol'd tongues. If the map is invertible, the only possible dynamics is either quasiperiodic motion, or phase-locked behavior with a unique attracting periodic orbit. Under the influence of quasiperiodic forcing the dynamics of the map changes dramatically. Inside the Arnol'd tongues open regions of multistability exist, and the parameter dependency of the dynamics becomes rather complex. This paper discusses the bifurcation structure inside the Arnol'd tongue with zero rotation number and includes a study of nonsmooth bifurcations that occur for large nonlinearity in the region with strange nonchaotic attractors.