Model Of Chaotic System Research Articles

When good theories make bad predictions

Chaos-related obstructions to predictability have been used to challenge accounts of theory validation based on the agreement between theoretical predictions and experimental data (Rueger & Sharp, 1996. The British Journal for the Philosophy of Science, 47, 93–112; Koperski, 1998. Philosophy of Science, 40, 194–212). These challenges are incomplete in two respects: (a) they do not show that chaotic regimes are unpredictable in principle (i.e., with unbounded resources) and, as a result, that there is something conceptually wrong with idealized expectations of correct predictions from acceptable theories, and (b) they do not explore whether chaos-induced predictive failures of deterministic models can be remedied by stochastic modeling. In this paper we appeal to an asymptotic analysis of state space trajectories and their numerical approximations to show that chaotic regimes are deterministically unpredictable even with unbounded resources. Additionally, we explain why stochastic models of chaotic systems, while predictively successful in some cases, are in general predictively as limited as deterministic ones. We conclude by suggesting that the way in which scientists deal with such principled obstructions to predictability calls for a more comprehensive approach to theory validation, on which experimental testing is augmented by a multifaceted mathematical analysis of theoretical models, capable of identifying chaos-related predictive failures as due to principled limitations which the world itself imposes on any less-than-omniscient epistemic access to some natural systems.

Accuracy and efficiency of reduced stochastic models for chaotic Hamiltonian systems with time-scale separation

The study of the long time behavior of systems with time-scale separation is considerably facilitated if the fast degrees of freedom can be eliminated without affecting the slow dynamics. We investigate a technique in which the fluctuations due to a fast chaotic subsystem are replaced by a suitable stochastic process so that a Fokker-Planck equation for the slow variables results. The accuracy and efficiency of this technique is verified by the detailed numerical investigation of several coupled systems. The asymptotic behavior as well as transients turn out to be well modeled by the reduced dynamics. We concentrate on low-dimensional problems and cover different types of coupling schemes as well as different chaotic subsystems. As a physical application we discuss the classical dynamics of a hydrogen atom in a strong magnetic field.

Profile of David W. McLaughlin

Today, McLaughlin is a professor of Mathematics and Neural Science at New York University (NYU), as well as the university's provost. Elected to the National Academy of Sciences in 2002, McLaughlin has spent his career developing mathematical models of complex and chaotic systems, with applications from laser beams to the behavior of the visual cortex. His Inaugural Article, published in a recent issue of PNAS (1), describes a model of neurons in a single layer of the primary visual cortex, which is of sufficient lateral extent to describe certain preattentive optical illusions.

Nonlinear speech analysis using models for chaotic systems

In this paper, we use concepts and methods from chaotic systems to model and analyze nonlinear dynamics in speech signals. The modeling is done not on the scalar speech signal, but on its reconstructed multidimensional attractor by embedding the scalar signal into a phase space. We have analyzed and compared a variety of nonlinear models for approximating the dynamics of complex systems using a small record of their observed output. These models include approximations based on global or local polynomials as well as approximations inspired from machine learning such as radial basis function networks, fuzzy-logic systems and support vector machines. Our focus has been on facilitating the application of the methods of chaotic signal analysis even when only a short time series is available, like phonemes in speech utterances. This introduced an increased degree of difficulty that was dealt with by resorting to sophisticated function approximation models that are appropriate for short data sets. Using these models enabled us to compute for short time series of speech sounds useful features like Lyapunov exponents that are used to assist in the characterization of chaotic systems. Several experimental insights are reported on the possible applications of such nonlinear models and features.

Statistics of unstable periodic orbits of a chaotic dynamical system with a large number of degrees of freedom

For a simple model of chaotic dynamical systems with a large number of degrees of freedom, we find that there is an ensemble of unstable periodic orbits (UPOs) with the special property that the expectation values of macroscopic quantities can be calculated using only one UPO sampled from the ensemble. Evidence to support this conclusion is obtained by generating the ensemble by Monte Carlo calculation for a statistical mechanical model described by a space-time Hamiltonian that is expressed in terms of Floquet exponents of UPOs. This result allows us to interpret the recent interesting discovery that statistical properties of turbulence can be obtained from only one UPO [G. Kawahara and S. Kida, J. Fluid Mech. 449, 291 (2001); S. Kato and M. Yamada, Phys. Rev. E 68, 025302(R) (2003)].

Evolution of mathematical models of chaotic systems based on multiobjective genetic programming

This work is concerned with the identification of models for nonlinear dynamical systems using multiobjective evolutionary algorithms. Systems modelling involves the processes of structure selection, parameter estimation, model performance and model validation and involves a complex solution space. Evolutionary Algorithms (EAs) are search and optimisation tools founded on the principles of natural evolution and genetics, which are suitable for a wide range of application areas. Due to the versatility of these tools and motivated by the versatility of genetic programming (GP), this evolutionary paradigm is proposed for this modelling problem. GP is then combined with a multiobjective function definition scheme. Multiobjective genetic programming (MOGP) is applied to multiple, conflicting objectives and yields a set of candidate parsimonious and valid models, which reproduce the original system behaviour. The MOGP approach is then demonstrated as being applicable for system modelling with chaotic dynamics. The circuit introduced by Chua, being one of the most popular benchmarks for studying nonlinear oscillations, and the Duffing–Holmes oscillator are the systems to test the evolutionary-based modelling approach introduced in this paper.

Theoretical model and characteristics investigations of dual-channel optical chaotic communication system

A theoretical model characterizing a recent experiment of dual-channel optical chaotic communication has been presented, and is proved to be reasonable by comparing the numerical simulation with the experimental results. After deducing the transmission function of semiconductor laser by small-signal analysis, how toreasonably select the system parameters in order to realize the effective transmission of signal has been specified. Moreover, the cross talk between two channels has been analyzed quantitatively. For a 250MHz modulation message, the nume rical simulation shows that it can be hidden efficiently during the transmission in the system and decoded easily.

A MODEL-BASED SCHEME FOR ANTICONTROL OF SOME DISCRETE-TIME CHAOTIC SYSTEMS

We consider a model-based approach for the anticontrol of some discrete-time systems. We first assume the existence of a chaotic model in an appropriate form. Then by using an appropriate control input we try to match the controlled system with the chaotic system model. We also give a procedure to generate the model chaotic systems in arbitrary dimensions. We show that with this approach, controllable systems can always be chaotified. Moreover, if the system to be controlled is stable, control input can be chosen arbitrarily small.

Adaptive control of discrete-time chaotic systems: a fuzzy control approach

This paper discusses adaptive control of a class of discrete-time chaotic systems from a fuzzy control approach. Using the T–S model of discrete-time chaotic systems, an adaptive control algorithm is developed based on some conventional adaptive control techniques. The resulting adaptively controlled chaotic system is shown to be globally stable, and its robustness is discussed. A simulation example of the chaotic Henon map control is finally presented, to illustrate an application and the performance of the proposed control algorithm.

Adaptive control for chaotic systems

Control problem of chaotic system is investigated via adaptive method. A fairly simple adaptive controller is constructed, which can control chaotic systems to unstable fixed points. The precise mathematical models of chaotic systems need not be known and only the fixed points and the dimensions of chaotic systems are required to be known. Simulations on controlling different chaotic systems are investigated and the results show the validity and feasibility of the proposed controller.

Explicit analytical solution for scaling quantum graphs.

We show that scaling quantum graphs with arbitrary topology are explicitly analytically solvable. This is surprising since quantum graphs are excellent models of quantum chaos and quantum chaotic systems are not usually explicitly analytically solvable.

On extracting physical information from mathematical models of chaotic and complex systems

The paper describes a multi-structural approach to the problem indicated in the title. In analogy with quantum mechanics, at the core of the approach are two notions: states and (generalized) observables. This motivates us to distinguish between mathematical and physical dynamical systems. The first class deals with mathematical models and states (solutions) only. The second one adds physical realizations and observables, i.e. all methods of extracting useful information. Observables include: renormalization, optimal gauging, covering–coloring, discretization, tensorial measures of chaos, etc. The corresponding algorithms are correlated to Kolmogorov complexity and are intrinsic parts of observables, and thus of physical systems. The approach is illustrated by examples.

Skew-Orthogonal Polynomials and Universality of Energy-Level Correlations

Random-matrix ensembles serve as models for quantum chaotic systems. We develop the theory of skew-orthogonal polynomials to study matrix ensembles with non-Gaussian weight functions. From the asymptotic properties of these and the orthogonal polynomials, we show that the local energy level correlations in the ensembles become universal properties independent of the global level density. This provides a rigorous justification for the universality of the Gaussian ensemble results observed in quantum chaotic systems.

Secure communications of chaotic systems with robust performance via fuzzy observer-based design

This paper presents a systematic design methodology for fuzzy observer-based secure communications of chaotic systems with guaranteed robust performance. The Takagi-Sugeno fuzzy models are given to exactly represent chaotic systems. Then, the general fuzzy model of many well-known chaotic systems is constructed with only one premise variable in fuzzy rules and the same premise variable in the system output. Based on this general model, the fuzzy observer of chaotic system is given and leads the stability condition of a linear-matrix inequality problem. When taking the fuzzy observer-based design to applications on secure communications, the robust performance is presented by simultaneously considering the effects of parameter mismatch and external disturbances. Then, the error of the recovered message is stated in an H/sup /spl infin// criterion. In addition, if the communication system is free of external disturbances, the asymptotic recovering of the message is obtained in the same framework. The main results also hold for applications on chaotic synchronization. Numerical simulations illustrate that this proposed scheme yields robust performance.

UNMASKING CHAOTIC MASK BY A WAVELET MULTISCALE DECOMPOSITION ALGORITHM

A method is proposed for extracting a useful signal hidden in a deterministic chaotic signal using a wavelet multiscale decomposition algorithm that can separate two signals with different wavelet scales. The mask removal is done without resorting to chaos synchronization. It is a noise reduction method without knowing the dynamical model of chaotic system, and restricting the modulation ratio of the chaos signal and useful signal. So this method is effective for unmasking chaotic mask through the quantitative analyses and the numerical simulations of three examples.

Entropy Production and Information Gain¶in Axiom-A Systems

Anosov systems are mathematical models for chaotic systems in statistical mechanics and fluid dynamics. Most of these systems enjoy the property of positive entropy production. We introduce the concept of specific information gain (or specific relative entropy) h(μ+,μ−) in Anosov systems and prove that it is identical to the entropy production rate ep(μ+) defined by Ruelle and Gallavotti in Anosov systems. From this point of view, the entropy production rate ep(μ+ characterizes the degree of macroscopic irreversibility of the system.

Estimation of noise levels for models of chaotic dynamical systems

We investigate how far it is possible to identify and separate dynamical noise from measurement noise in observed nonlinear time series. Using Bayesian methods, we derive estimates for the two noise levels, and find that, given a good model of the dynamics, these can give accurate results even if the dynamical noise level is orders of magnitude smaller than the measurement noise level, whereas a simple calculation of root mean square error badly understates the dynamical noise. We argue that this allows better estimates of the underlying dynamical time series, and so better predictions of its future and of its fundamental dynamical properties.

Fuzzy modeling, prediction, and control of uncertain chaotic systems based on time series

A fuzzy logic-based approach is taken in this paper for modeling, prediction, and predictive control of unknown or uncertain chaotic systems. Only output data of the underlying system are required. A fuzzy predictive framework using a general structure of a linear combination of Gaussian basis functions is developed, where the basis functions are expressed as probability density functions and are empirically determined from the time-series data. A real-time one-pass learning algorithm is developed for identification of the chaotic system. Based on this framework, a fuzzy predictive controller is designed, which is especially suitable for sparse data in a real-time environment. Several simulation examples are then given for demonstration.

Constructing chaotic discrete sequences for digital communications based on correlation analysis

We introduce a model of chaotic systems that can provide a predictable probability distribution to generate discrete sequences for digital communications. Some well-known chaotic maps belong to its special cases. We generalize the spirit of a paper by Kohda and Tsuneda (1997) to explicitly express the covariance functions of the chaotic discrete sequences generated by this model. Based on the model, a scheme is proposed to construct the chaotic orthonormal discrete sequences, which are characterized by the asymptotic normalized (partial) /spl delta/-like autocovariance and near-zero cross covariance. Compared with the conventional algebraic sequences, such chaotic discrete sequences have quite a large number of choices. All empirical results well correspond to theoretic prediction.

Fuzzy modeling and adaptive control of uncertain chaotic systems

A simple and systematic approach is developed for modeling and adaptive control of an unknown (or uncertain) chaotic system, using only input–output data obtained from the underlying dynamical system. Gaussian fuzzy membership functions are used in conjunction with the least-squares principle for the modeling and control. Based on the fuzzy modeling, an adaptive controller is devised, which works through self-adjusting the means and variances of the Gaussian membership functions for adaptation. The design procedure is illustrated by using the chaotic Duffing oscillator as an example, on which simulation results demonstrate the effectiveness of the proposed methodology.