Low-dimensional Deterministic Chaos Research Articles

Identification of Chaos in Financial Time Series to Forecast Nonperforming Loan

This paper discusses the importance of modeling financial time series as a chaotic dynamic rather than a stochastic system. The dynamical properties of a financial time series of an economic institution in Iran were analyzed to identify the potential occurrence of the low-dimensional deterministic chaos. This paper applies several classic nonlinear techniques for detecting the chaotic nature of the time series of loan payment portion and proposes a modified nonlinear predictor scheme for forecasting the future levels of the nonperforming loan. The auto mutual information was implemented to estimate the delay time dimension, and Cao’s approach, along with correlation dimension methodology, quantified the embedding dimension of the time series. The results reveal a low embedding dimension implying the chaotic nature exists in the financial data. The maximum Lyapunov exponent measure is also adopted to investigate the divergence or convergence of the trajectories. Since positive Lyapunov exponents are revealed, the long-term unpredictability of the time series is proved. Lastly, a modified nonlinear local approximator is developed to forecast the short-term history of the time series. Numerical simulations are provided to illustrate the adopted nonlinear techniques. The results reported in this paper could have implications for commercial bank managers who could use the nonlinear models for early detection of the possible nonperforming loans before they become uncontrollable.

Transition to chaos in wave memory dynamics in a harmonic well: Deterministic and noise-driven behavior.

A walker is the association of a sub-millimetric bouncing drop moving along with a co-evolving Faraday wave. When confined in a harmonic potential, its stable trajectories are periodic and quantised both in extension and mean angular momentum. In this article, we present the rest of the story, specifically the chaotic paths. They are chaotic and show intermittent behaviors between an unstable quantised set of attractors. First, we present the two possible situations we find experimentally. Then, we emphasise theoretically two mechanisms that lead to unstable situations. It corresponds either to noise-driven chaos or low-dimensional deterministic chaos. Finally, we characterise experimentally each of these distinct situations. This article aims at presenting a comprehensive investigation of the unstable paths in order to complete the picture of walkers in a two dimensional harmonic potential.

Nonlinear dynamics of a buoyancy-induced turbulent fire.

We conduct a numerical study on the dynamic behavior of a buoyancy-induced turbulent fire from the viewpoints of symbolic dynamics, complex networks, and statistical complexity. Here, we consider two classes of entropies: the permutation entropy and network entropy in ε-recurrence networks, both of which evaluate the degree of randomness in the underlying dynamics. These entropies enable us to capture the significant changes in the dynamic behavior of flow velocity fluctuations. The possible presence of two important dynamics, low-dimensional deterministic chaos in the near field dominated by the motion of large-scale vortices and high-dimensional chaos in the far field forming a well-developed turbulent plume, is clearly identified by the multiscale complexity-entropy causality plane.

Nonlinear Forecasting of the Generalized Kuramoto–Sivashinsky Equation

The emergence of pattern formation and chaotic dynamics is studied in the one-dimensional (1D) generalized Kuramoto–Sivashinsky (gKS) equation by means of a time-series analysis, in particular, a nonlinear forecasting method which is based on concepts from chaos theory and appropriate statistical methods. We analyze two types of temporal signals, a local one and a global one, finding in both cases that the dynamical state of the gKS solution undergoes a transition from high-dimensional chaos to periodic pulsed oscillations through low-dimensional deterministic chaos while increasing the control parameter of the system. Our results demonstrate that the proposed nonlinear forecasting methodology allows to elucidate the dynamics of the system in terms of its predictability properties.

Scaling and universality in the human voice.

Speech is a distinctive complex feature of human capabilities. In order to understand the physics underlying speech production, in this work, we empirically analyse the statistics of large human speech datasets ranging several languages. We first show that during speech, the energy is unevenly released and power-law distributed, reporting a universal robust Gutenberg-Richter-like law in speech. We further show that such 'earthquakes in speech' show temporal correlations, as the interevent statistics are again power-law distributed. As this feature takes place in the intraphoneme range, we conjecture that the process responsible for this complex phenomenon is not cognitive, but it resides in the physiological (mechanical) mechanisms of speech production. Moreover, we show that these waiting time distributions are scale invariant under a renormalization group transformation, suggesting that the process of speech generation is indeed operating close to a critical point. These results are put in contrast with current paradigms in speech processing, which point towards low dimensional deterministic chaos as the origin of nonlinear traits in speech fluctuations. As these latter fluctuations are indeed the aspects that humanize synthetic speech, these findings may have an impact in future speech synthesis technologies. Results are robust and independent of the communication language or the number of speakers, pointing towards a universal pattern and yet another hint of complexity in human speech.

A Comparison of Bayesian Methods and Artificial Neural Networks for Forecasting Chaotic Financial Time Series

Most recent empirical work implies that the presence of low-dimensional deterministic chaos increases the complexity of the financial time series behavior. In this study we propose the Generalized Multilayer Perceptron (GMLP), and the Bayesian inference via Markov Chain Monte Carlo (MCMC) method for parameter estimation and one-step-ahead prediction. By out-of-sample prediction approach, these proposed methods are compared to autoregressive integrated moving average (ARIMA) models which have been used as a benchmark. The deterministic Mackey-Glass equation with errors that follow an ARCH (p) process (MG-ARCH (p)) is applied to generate the data set used in this study. It turns out that GMLP outperforms the other two forecasting methods using RMSE, MAPE, and MAE criteria of forecasting accuracy.

Dynamic Behavior of Buoyancy-Induced Flame Oscillation Under Swirling Flow by a Use of Nonlinear Time Series Analysis in Combination with Surrogate Data Method

The authors experimentally investigated the dynamic behavior of buoyancy-induced flame oscillation under a swirling flow produced by rotating a cylindrical burner tube, focusing on the characterization of the complex dynamics in flame front by a use of nonlinear time series analysis. A Wayland method (R. Wayland et al., 1993), which quantifies the degree of parallelism of trajectories in phase space constructed from time series data of flame front fluctuations, is applied as a sophisticated nonlinear time series analysis in this work. To reveal whether or not the complex dynamics is deterministic chaos, a quantitative method for discussing the null hypothesis that the irregular components of the flame front fluctuations represent a stochastic process (i.e., a surrogate data method; T. Schreiber and A. Schmitz, 1996), is applied in this work. A sophisticated nonlinear time series analysis in combination with a surrogate data method, which has not been widely applied to the study of combustion phenomena, clearly demonstrates that the dynamic behavior undergoes a significant transition from periodic oscillation to low-dimensional deterministic chaos with increasing rotational Reynolds number.

Research on characterization of wireless LANs traffic

In this paper, we employ actual wireless data that draw from well known archives of network traffic traces and investigate the characterization of the wireless LANs traffic. Firstly, useful preliminary information regarding the general wireless traffic dynamics is obtained using one standard statistical technique named Fourier power spectrum. Then the estimation of the parameters, such as the correlation dimension, the largest Lyapunov exponent and the principal components analysis indicate the existence of low-dimensional deterministic chaos in wireless traffic time series. Our results also show that the parameters selection of the phase space reconstruction influence the value of the correlation dimension and the largest Lyapunov exponent, but can not influence on diagnosis of chaotic nature of wireless traffic.

Chaos in the pulse spacing of passive Q-switched all-solid-state lasers

We report the experimental and theoretical verification that, in a diode-pumped Nd:YAG+Cr:YAGQ-switched laser, the instabilities in the pulse spacing ("jitter") are ruled by low-dimensional deterministic chaos. From our experimental time series, we determine the embedding and fractal dimensions of the attractor, as well as the values of the Lyapunov exponents. We also present a simplified theoretical description in terms of a map of the same universality class as the logistic map, which explains the bifurcations' cascade and the period-three window of stability observed. The achieved characterization of the dynamics and its main parameters opens a door to effective ways to reduce the jitter, which is of practical interest, through mechanisms of control of chaos. Conversely, the difficulty in the prediction of the interpulse spacing makes this system attractive for high power, robust FM chaotic laser cryptography in free-space propagation.

Low dimensional chaos in the AT and GC skew profiles of DNA sequences

This paper investigates the existence of low-dimensional deterministic chaos in the AT and GC skew profiles of DNA sequences. It has taken DNA sequences from eight organisms as samples. The skew profiles are analysed using continuous wavelet transform and then nonlinear time series methods. The invariant measures of correlation dimension and the largest Lyapunov exponent are calculated. It is demonstrated that the AT and GC skew profiles of these DNA sequences all exhibit low dimensional chaotic behaviour. It suggests that chaotic properties may be ubiquitous in the DNA sequences of all organisms.

Nonlinear analysis on dynamic behavior of buoyancy-induced flame oscillation under swirling flow

From the viewpoint of nonlinear dynamics, the dynamic behavior of buoyancy-induced flame oscillation has been experimentally investigated under a swirling flow produced by rotating a cylindrical burner tube. As the rotational Reynolds number increases, the dynamic behavior undergoes a significant transition from periodic oscillation to low-dimensional deterministic chaos, through quasi-periodic oscillation. This is clearly demonstrated by nonlinear time series analysis based on chaos theory. The motion of the vortical structure around the burner tube due to the centrifugal instability associated with a rotating Taylor–Couette flow plays an important role in the onset of low-dimensional deterministic chaos.

Failure in distinguishing colored noise from chaos using the “noise titration” technique

Identifying chaos in experimental data-noisy data-remains a challenging problem for which conclusive arguments are still very difficult to provide. In order to avoid problems usually encountered with techniques based on geometrical invariants (dimensions, Lyapunov exponent, etc.), Poon and Barahona introduced a numerical titration procedure which compares one-step-ahead predictions of linear and nonlinear models [Proc. Natl. Acad. Sci. U.S.A. 98, 7107 (2001)]. We investigate the aformentioned technique in the context of colored noise or other types of nonchaotic behaviors. The main conclusion is that in several examples noise titration fails to distinguish such nonchaotic signals from low-dimensional deterministic chaos.

CA Simulations of 2D Stellar Atmosphere Pulsations

We develop a new version of a Cellular Automaton (CA) for the simulation of the dynamics of a stellar atmosphere sitting on top of an inert core, and specified by the following physical input parameters: mass, radius and luminosity of core, and mass of atmosphere. The CA incorporates various parametrised simulation schemes of the instability mechanism (essentially ionisation). The initial state in all of our numerical experiments is a radially symmetric atmosphere of exponential density run and uniform temperature (input parameters: density scale– height and temperature of atmosphere). The initial atmosphere is not in hydrostatic and thermal equilibrium. After a transient stage, the system relaxes, for certain ranges of the parameters of the instability mechanism, towards a state of nontrivial dynamical behaviour: Local heat–driven circulations are set up which may range from nearly stationary and spatially symmetric cellular patterns to temporally and spatially irregularly fluctuating velocity fields. The traditional radial symmetry of the density pattern is broken, so that the star acquires a globally non–spherical shape. The residual non–stationary component, when integrated over the star to produce the counterpart of an observational velocity curve of a variable star, shows an irregular cyclic behaviour which does not have the signature of low–dimensional deterministic chaos.

Detection of low-dimensional chaos in wind time series

In the present work we investigated the existence of low-dimensional deterministic chaos in wind time series. The time series were obtained from the New Anchialos (Greece) Air Base measurement station. In a first place we used the raw data without any noise filtering. Characteristic times were extracted using power spectrum and average mutual information function. The estimation of invariant measures, such as the correlation dimension and Lyapunov exponents indicate the possible existence of a low-dimensional attractor. After noise removal with the use of the local projective method the analysis indicates in a more clear way the existence of a low-dimensional attractor. In addition, the null hypothesis was tested for the dynamical characteristics of the wind time series by using the surrogate data test and the corresponding results provide significant evidence for the existence of low-dimensional chaotic dynamics underlying the wind time series.

BISTABLE BEHAVIOR OF A KERR LENS MODE LOCKED Ti:SAPPHIRE LASER

Kerr lens mode locked Ti :Sapphire lasers can operate in at least two pulsed modes. Several models were developed with the aim to describe the characteristics of these modes. Those based on iterative maps, can reproduce the structurally stable properties of each mode but are unable to describe the interaction between modes. In this paper, we present a numerical simulation based on a complete map equation that makes possible to accurately describe the bistability experimentally observed in the laser. With the numerical time series we determine that the bistable behavior corresponds to low dimensional deterministic chaos and calculate that the embedding dimension of the attractor is three.

On the quest for chaotic attractors in hydrological processes

In the last two decades, several researchers have claimed to have discovered low-dimensional determinism in hydrological processes, such as rainfall and runoff, using methods of chaotic analysis. However, such results have been criticized by others. In an attempt to offer additional insights into this discussion, it is shown here that, in some cases, merely the careful application of concepts of dynamical systems, without doing any calculation, provides strong indications that hydrological processes cannot be (low-dimensional) deterministic chaoti. Furthermore, it is shown that specific peculiarities of hydrological processes on fine time scales, such as asymmetric, J-shaped distribution functions, intermittency, and high autocorrelations, are synergistic factors that can lead to misleading conclusions regarding the presence of (low-dimensional) deterministic chaos. In addition, the recovery of a hypothetical attractor from a time series is put as a statistical estimation problem whose study allows, among others, quantification of the required sample size; this appears to be so huge that it prohibits any accurate estimation, even with the largest available hydrological records. All these arguments are demonstrated using appropriately synthesized theoretical examples. Finally, in light of the theoretical analyses and arguments, typical real-world hydrometeorological time series, such as relative humidity, rainfall, and runoff, are explored and none of them is found to indicate the presence of chaos.

Parameter estimation of feedback gain in a stochastic model of renal hemodynamics: differences between spontaneously hypertensive and Sprague-Dawley rats

Proximal tubular pressure shows periodic self-sustained oscillations in normotensive rats but highly irregular fluctuations in spontaneously hypertensive rats (SHR). Although we have suggested that the irregular fluctuations in SHR represent low-dimensional deterministic chaos in tubuloglomerular feedback (TGF), they could also arise from other mechanisms, such as intrinsic instabilities in preglomerular vessels or inputs from neighboring, coupled nephrons. To test this possibility, we applied a parameter estimation procedure to a model of TGF, where a stochastic process was added to represent mechanisms not included explicitly in the model. In its deterministic version, the model can have chaotic dynamics arising from TGF. The model introduces random fluctuations into a parameter that determines the gain of TGF. The model shows a rich variety of dynamics ranging from low-dimensional deterministic oscillations and chaos to high-dimensional random fluctuations. To fit the data from normotensive rats, the model must introduce only a small variation in the feedback gain, and its estimates of that gain agree well with experimental values. These results support the use of the deterministic model of nephron dynamics in normotensive rats. In contrast, the irregular tubular pressure fluctuations in SHR were best described by a model dominated by random parameter fluctuations. The results point to the failure of simple mathematical models of nephron dynamics adequately to describe processes that are important for the irregular tubular pressure fluctuations and the need to consider other factors, such as differences in vascular function or nephron-nephron interactions, in further work on this problem.

Chaos in computer performance

Modern computer microprocessors are composed of hundreds of millions of transistors that interact through intricate protocols. Their performance during program execution may be highly variable and present aperiodic oscillations. In this paper, we apply current nonlinear time series analysis techniques to the performances of modern microprocessors during the execution of prototypical programs. Our results present pieces of evidence strongly supporting that the high variability of the performance dynamics during the execution of several programs display low-dimensional deterministic chaos, with sensitivity to initial conditions comparable to textbook models. Taken together, these results show that the instantaneous performances of modern microprocessors constitute a complex (or at least complicated) system and would benefit from analysis with modern tools of nonlinear and complexity science.

On control of chaos in the Kerr lens mode locked Ti:Sapphire laser

In this paper, I present the numerical demonstration of a simple way to reduce the pulse duration of a Kerr lens mode locked Ti:Sapphire laser. The core of the method lies in the low dimensional deterministic chaos that displays the laser. By adding an acousto optic modulator and applying an adequate control algorithm, it is possible to reduce the pulse duration by a factor of two with a laser that otherwise operates in the range of 40fs.

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.