Low-dimensional Chaotic Systems Research Articles

Forbidden patterns and shift systems

The scope of this paper is two-fold. First, to present to the researchers in combinatorics an interesting implementation of permutations avoiding generalized patterns in the framework of discrete-time dynamical systems. Indeed, the orbits generated by piecewise monotone maps on one-dimensional intervals have forbidden order patterns, i.e., order patterns that do not occur in any orbit. The allowed patterns are then those patterns avoiding the so-called forbidden root patterns and their shifted patterns. The second scope is to study forbidden patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. Due to its simple structure, shift systems are accessible to a more detailed analysis and, at the same time, exhibit all important properties of low-dimensional chaotic dynamical systems (e.g., sensitivity to initial conditions, strong mixing and a dense set of periodic points), allowing to export the results to other dynamical systems via order-isomorphisms.

Stochastic Modeling of Experimental Chaotic Time Series

Methods developed recently to obtain stochastic models of low-dimensional chaotic systems are tested in electronic circuit experiments. We demonstrate that reliable drift and diffusion coefficients can be obtained even when no excessive time scale separation occurs. Crisis induced intermittent motion can be described in terms of a stochastic model showing tunneling which is dominated by state space dependent diffusion. Analytical solutions of the corresponding Fokker-Planck equation are in excellent agreement with experimental data.

The Nonlinear Behavior of the Black Hole System GRS 1915+105

Using nonlinear time series analysis, along with surrogate data analysis, it is shown that the various types of long-term variability exhibited by the black hole system GRS 1915+105 can be explained in terms of a deterministic nonlinear system with some inherent stochastic noise. Evidence is provided for a nonlinear limit cycle origin of one of the low-frequency QPOs detected in the source, while some other types of variability could be due to an underlying low-dimensional chaotic system. These results imply that the partial differential equations that govern the magnetohydrodynamic flow of the inner accretion disk can be approximated by a small number (≈3-5) of nonlinear but ordinary differential equations. While this analysis does not reveal the exact nature of these approximate equations, they may be obtained in the future, after results of magnetohydrodynamic simulation of realistic accretion disks become available.

Determination of Chaotic Dynamical Correlations in Terms of Unstable Periodic Orbits

For low-dimensional chaotic systems, we find that time correlation functions can be accurately approximated by a single unstable periodic orbit. The method of this determination consists of the following two steps. First, the time correlation functions are expanded in terms of static correlation functions by making use of the expansion method recently proposed by one of the authors [H. Fujisaka, Prog. Theor. Phys. 114 (2005), 1]. Second, the static quantities are approximated in terms of an unstable periodic orbit embedded in the strange attractor. Thus the dynamical correlation functions in chaotic systems can be determined in terms of an unstable periodic orbit. Furthermore, applying this method to low-dimensional chaotic models, we prove the usefulness of the present approach.

Effect of noise on the bifurcation behavior of nonlinear dynamical systems

Abstract We argue that dynamical noise can dramatically change the dynamics of low dimensional chaotic systems. Moreover, we show that chaos tests are highly sensitive to dynamical noise and this becomes worse when the intensity of the noise increases.

Dynamical Model Reconstruction and Accurate Prediction of Power-Pool Time Series

The emergence of the power pool as a popular institution for trading of power in different countries has led to increased interest in the prediction of power demand and price. In this paper, the authors investigate whether the time series of power-pool demand and price can be modeled as the output of a low-dimensional chaotic dynamical system by using delay embedding and estimation of the embedding dimension, attractor-dimension or correlation-dimension calculation, Lyapunov-spectrum and Lyapunov-dimension calculation, stationarity and nonlinearity tests, as well as prediction analysis. Different dimension estimates are consistent and show close similarity, thus increasing the credibility of the fractal-dimension estimates. The Lyapunov spectrum consistently shows one positive Lyapunov exponent and one zero exponent with the rest being negative, pointing to the existence of chaos. The authors then propose a least squares genetic programming (LS-GP) to reconstruct the nonlinear dynamics from the power-pool time series. Compared to some standard predictors including the radial basis function (RBF) neural network and the local state-space predictor, the proposed method does not only achieve good prediction of the power-pool time series but also accurately predicts the peaks in the power price and demand based on the data sets used in the present study.

Effect of noise on estimation of Lyapunov exponents from a time series

Abstract We argue that dynamical noise can dramatically change the dynamics of low-dimensional chaotic systems. Moreover, we show that chaos tests are highly sensitive to dynamical noise and this becomes worse when the intensity of the noise increases.

Analysis of cedar pollen time series: no evidence of low-dimensional chaotic behavior

Much of the current interest in pollen time series analysis is motivated by the possibility that pollen series arise from low-dimensional chaotic systems. If this is the case, short-range prediction using nonlinear modeling is justified and would produce high-quality forecasts that could be useful in providing pollen alerts to allergy sufferers. To date, contradictory reports about the characterization of the dynamics of pollen series can be found in the literature. Pollen series have been alternatively described as featuring and not featuring deterministic chaotic behavior. We showed that the choice of test for detection of deterministic chaos in pollen series is difficult because pollen series exhibit [see text] power spectra. This is a characteristic that is also produced by colored noise series, which mimic deterministic chaos in most tests. We proposed to apply the Ikeguchi-Aihara test to properly detect the presence of deterministic chaos in pollen series. We examined the dynamics of cedar (Cryptomeria japonica) hourly pollen series by means of the Ikeguchi-Aihara test and concluded that these pollen series cannot be described as low-dimensional deterministic chaos. Therefore, the application of low-dimensional chaotic deterministic models to the prediction of short-range pollen concentration will not result in high-accuracy pollen forecasts even though these models may provide useful forecasts for certain applications. We believe that our conclusion can be generalized to pollen series from other wind-pollinated plant species, as wind speed, the forcing parameter of the pollen emission and transport, is best described as a nondeterministic series that originates in the high dimensionality of the atmosphere.

Modelling and control of chaotic processes through their Bifurcation Diagrams generated with the help of Recurrent Neural Network models: Part 1—simulation studies

Many real-world processes tend to be chaotic and also do not lead to satisfactory analytical modelling. It has been shown here that for such chaotic processes represented through short chaotic noisy time-series, a multi-input and multi-output recurrent neural networks model can be built which is capable of capturing the process trends and predicting the future values from any given starting condition. It is further shown that this capability can be achieved by the Recurrent Neural Network model when it is trained to very low value of mean squared error. Such a model can then be used for constructing the Bifurcation Diagram of the process leading to determination of desirable operating conditions. Further, this multi-input and multi-output model makes the process accessible for control using open-loop/closed-loop approaches or bifurcation control etc. All these studies have been carried out using a low dimensional discrete chaotic system of Hénon Map as a representative of some real-world processes.

Investigation of short-range cedar pollen forecasting

Pollen forecasting is of increasing interest as a way to help the general public avoid contact with allergy-inducing pollen. It was recently reported that the dynamics underlying pollen concentration series is very similar to that of low-dimensional deterministic chaos, thus opening up new avenues of development in local pollen forecasting. Our analysis of hourly cedar pollen series for two seasons showed evidence of a small degree of determinism underlying the pollen time-series dynamics. However, we could not confirm that our pollen series was generated by a low-dimensional chaotic system. The nearest-neighbor method using local constant prediction applied to hourly pollen forecasting with a 1-h lead time was effective for small to medium pollen variations, but failed to reproduce large and intermittent pollen bursts. The performance of the nearest-neighbor model was significantly improved by applying a nonlinear filter to the source dataset. Standard time-series techniques such as neural networks did not improve upon these results. The difficulty in fully characterizing and accurately forecasting the pollen series was thought to originate in the nonstationarity of the series and in the large and intermittent pollen bursts that were found to have no apparent time structure. Thus the dynamics of hourly pollen series is probably not strongly tied to a low-dimensional chaotic system.

Localized error bursts in estimating the state of spatiotemporal chaos

We consider the problem of estimating the current state of an evolving spatiotemporally chaotic system from noisy observations of the system state and a model of the system dynamics. Using a simple scheme for state estimation, we show the possible occurrence of temporally and spatially intermittent large bursts in the estimation error. We discuss the similarity of these bursts with those occurring at the bubbling transition in the synchronization of low dimensional chaotic dynamical systems. We characterize the spatial and temporal behavior of the bursts and investigate how the behavior changes as we vary the number and location of the observations.

Controlled test for predictive power of Lyapunov exponents: their inability to predict epileptic seizures.

Lyapunov exponents are a set of fundamental dynamical invariants characterizing a system's sensitive dependence on initial conditions. For more than a decade, it has been claimed that the exponents computed from electroencephalogram (EEG) or electrocorticogram (ECoG) signals can be used for prediction of epileptic seizures minutes or even tens of minutes in advance. The purpose of this paper is to examine the predictive power of Lyapunov exponents. Three approaches are employed. (1) We present qualitative arguments suggesting that the Lyapunov exponents generally are not useful for seizure prediction. (2) We construct a two-dimensional, nonstationary chaotic map with a parameter slowly varying in a range containing a crisis, and test whether this critical event can be predicted by monitoring the evolution of finite-time Lyapunov exponents. This can thus be regarded as a "control test" for the claimed predictive power of the exponents for seizure. We find that two major obstacles arise in this application: statistical fluctuations of the Lyapunov exponents due to finite time computation and noise from the time series. We show that increasing the amount of data in a moving window will not improve the exponents' detective power for characteristic system changes, and that the presence of small noise can ruin completely the predictive power of the exponents. (3) We report negative results obtained from ECoG signals recorded from patients with epilepsy. All these indicate firmly that, the use of Lyapunov exponents for seizure prediction is practically impossible as the brain dynamical system generating the ECoG signals is more complicated than low-dimensional chaotic systems, and is noisy.

The Chaotic Behavior of the Black Hole System GRS 1915+105

A modified non-linear time series analysis technique, which computes the correlation dimension $D_2$, is used to analyze the X-ray light curves of the black hole system GRS 1915+105 in all twelve temporal classes. For four of these temporal classes $D_2 $ saturates to $\approx 4-5$ which indicates that the underlying dynamical mechanism is a low dimensional chaotic system. Of the other eight classes, three show stochastic behavior while five show deviation from randomness. The light curves for four classes which depict chaotic behavior have the smallest ratio of the expected Poisson noise to the variability ($ 0.2$). This suggests that the temporal behavior of the black hole system is governed by a low dimensional chaotic system, whose nature is detectable only when the Poisson fluctuations are much smaller than the variability.

Analysis of chaotic saddles in high-dimensional dynamical systems: the Kuramoto-Sivashinsky equation.

This paper presents a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems. Our methodology is applied to the Kuramoto-Sivashinsky equation, a reaction-diffusion partial differential equation. The paper describes a novel technique that uses the stable manifold of a chaotic saddle to characterize the homoclinic tangency responsible for an interior crisis, a chaotic transition that results in the enlargement of a chaotic attractor. The numerical techniques explained here are important to improve the understanding of the connection between low-dimensional chaotic systems and spatiotemporal systems which exhibit temporal chaos and spatial coherence.

Secure communication method for fast-varying information signal based on chaotic modulation

A new secure communication method based on chaotic modulation is proposed. The parameter of the transmitting system procreated chaotic carrier is modulated by errors from an adaptive controller and information signal. At the same time, the information is directly multiplied by the chaotic carrier in transmission to drive the receiving system. In the receiver, another adaptive controller is used to maintain chaotic synchronization of the transmitting and receiving systems and to recover information signal. Since the synchronization error is independent of information signal,the chaotic modulation for large-amplitude and fast-varying information signals can be achieved. The results of theoretic analysis and numerical simulation show that the proposed method enhances the degree of security of low-dimensional chaotic systems.

On the relationship between Hamiltonian chaos and classical gravity

It is known that Hamiltonian equations of motion for low-dimensional chaotic systems are typically formulated using fractional derivatives. The evolution of such systems is governed by the fractional diffusion equation, which describes self-similar and non-Gaussian processes with strong intermittencies. We confirm, in this context, that the dynamics of a Brownian particle driven by space-time dependent fluctuations evolves towards Hamiltonian chaos and fractional diffusion. The corresponding motion of the particle has a time-dependent and nowhere vanishing acceleration. Invoking the equivalence principle of general relativity leads to the conclusion that fractional diffusion is locally equivalent to a transient gravitational field. It is shown that gravity becomes renormalizable as Newton’s constant converges towards a dimensionless quantity.

Similarity of homoclinic tangles in open planar flows

An invariance of the structure of the homoclinic tangle with respect to a simultaneous rescaling of the perturbation amplitude and the coordinates in the neighborhood of the saddle was recently studied in low dimensional chaotic Hamiltonian systems. A similar property exists in open systems with transient chaotic behaviour. The scaling constant depends on the ratio of the perturbation frequency and the eigenvalue of the linearized system at the saddle. This result can be used to analyze the structure of the mixing layers in time-dependent two-dimensional open flows. The invariance of the homoclinic tangle is demonstrated numerically for a weakly perturbed flow of ideal fluid around a cylinder.

Controlling flow turbulence

This paper investigates the viability and effectiveness of using a technique developed for low-dimensional chaotic systems to control flow turbulence governed by the Navier–Stokes equations. By using a global pinning coupling strategy, we show that turbulence can be controlled to desirable time-varying target states, including a spatially extended periodic state and a turbulent one. Exponential convergence to the target state is found and the exponential rate scales linearly to the coupling strength. The linear scaling law breaks down when localized pinning control is applied. A wavelet multiscale technique is utilized for the characterization of both the effectiveness of the present control strategy and the inverse energy transfer in two-dimensional turbulence.

A secure communication method for a high-power information signal based on chaotic masking

In this paper, we present a secure communication method for a high-power information signal based on chaotic masking. In the transmitter, an adaptive controller is adopted to pick up the change of the information signal and to inject the controller's error into the transmitting system. At the same time, the information is directly added to the chaotic signal in transmission to drive the receiving system. In the receiver, another adaptive controller is used to maintain chaotic synchronization of the transmitting and receiving systems and to recover the information signal. Since the synchronization error is independent from the information signal, the power of the information signal can be equivalent to that of the chaotic signal and the frequency of the information signal can be set within the range of the principal frequencies of the chaotic signal. The results of theoretical analysis and numerical simulation show that the presented method not only enhances the degree of security of low-dimensional chaotic systems but also significantly improves the signal-to-noise ratio at the receiving end.

Statistics of plasma fluctuations in runaway discharges in TCABR tokamak

In this work we analyze the spikes in the Ha emission, and density turbulence in runaway discharges in the TCABR tokamak. For the plasma edge density fluctuations, we find a symmetric-recurrent-type turbulence, with the same statistics found in measures of low-dimensional recurrent chaotic systems. For the plasma core fluctuations, we note that the line density oscillations are synchronized to the Ha oscillations. In addition, we suggest that both the onset of the Ha spikes and the line density oscillations are triggered by a process similar to the bifurcation responsible for the firing of neurons.