Developments In Nonlinear Dynamics Research Articles

Application of The Classical Soliton Theory to Study the Dynamics in PassiveDispersed and Active-Relaxation Media During the Formation of a Twist in A Toroidal Plasma

In this paper we consider the possibility of using analog models to study the behavior of the evolution and dynamics of toroidal vortex flows. The current model based on coupled van der Pol generators is used as the main one. In this robot, based on nonlinear dynamics display Bernoulli presented the evolution of the Torus. The use of analog models makes it possible to trace the dynamics of vortex flows. In General, the presented results allow us to confidently assert that both in the experiment and in the theoretical consideration we are dealing with the same object - a strange attractor of the Smale – Williams type in a non-Autonomous oscillatory system. The available data suggest that it is an attractor of hyperbolic type, although, strictly speaking, this statement needs mathematical proof. The appearance of an example of a physical system with a hyperbolic chaotic attractor is of fundamental importance for the further development of nonlinear dynamics and its applications. This is, in a sense, a "breakthrough into the hyperbolic realm." Based on the inherent property of coarseness of hyperbolic attractors, we can build other examples of systems with hyperbolic chaos. The presence of such physical systems opens up opportunities for the application of a deeply studied branch of mathematics – hyperbolic theory, and also translates into practice the problem of comparative study of hyperbolic and non-hyperbolic chaos in theory and experiment. This system can be used to study toroidal vortex processes.

Application of The Classical Soliton Theory to Study the Dynamics in PassiveDispersed and Active-Relaxation Media During the Formation of a Twist in A Toroidal Plasma

In this paper we consider the possibility of using analog models to study the behavior of the evolution and dynamics of toroidal vortex flows. The current model based on coupled van der Pol generators is used as the main one. In this robot, based on nonlinear dynamics display Bernoulli presented the evolution of the Torus. The use of analog models makes it possible to trace the dynamics of vortex flows. In General, the presented results allow us to confidently assert that both in the experiment and in the theoretical consideration we are dealing with the same object - a strange attractor of the Smale – Williams type in a non-Autonomous oscillatory system. The available data suggest that it is an attractor of hyperbolic type, although, strictly speaking, this statement needs mathematical proof. The appearance of an example of a physical system with a hyperbolic chaotic attractor is of fundamental importance for the further development of nonlinear dynamics and its applications. This is, in a sense, a "breakthrough into the hyperbolic realm." Based on the inherent property of coarseness of hyperbolic attractors, we can build other examples of systems with hyperbolic chaos. The presence of such physical systems opens up opportunities for the application of a deeply studied branch of mathematics – hyperbolic theory, and also translates into practice the problem of comparative study of hyperbolic and non-hyperbolic chaos in theory and experiment. This system can be used to study toroidal vortex processes.

Special issue dedicated to the memory of Paul Waltman

This volume is dedicated to the memory of Paul Waltman. Many of the authors of articles contained here were participants at the NCTS International Conference on Nonlinear Dynamics with Applications to Biology held May 28-30, 2014 at National Tsing-Hua University, Hsinchu, Taiwan. The purpose of the conference was to survey new developments in nonlinear dynamics and its applications to biology and to honor the memory of Professor Paul Waltman for his influence on the development of Mathematical Biology and Dynamical Systems. Attendees at the conference included Paul's sons Fred and Dennis, many of Paul's former doctoral and post-doctoral students, many others who, although not students of Paul, nevertheless were recipients of Paul's valuable advice and council, and many colleagues from all over the world who were influenced by Paul's mathematics and by his personality. We thank the NCTS for its financial support of the conference and Dr. J.S.W. Wong for supporting the conference banquet. For more information please click the “Full Text” above.

MEASURING AND CONTROLLING THE CHAOTIC MOTION OF PROFITS

The study of economic systems has generated deep interest in exploring the complexity of chaotic motions in economy. Due to important developments in nonlinear dynamics, the last two decades have witnessed strong revival of interest in nonlinear endogenous business chaotic models. The inability to predict the behavior of dynamical systems in the presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we study a specific economic model from the literature. More precisely, a system of three ordinary differential equations gather the variables of profits, reinvestments and financial flow of borrowings in the structure of a firm. Firstly, using results of symbolic dynamics, we characterize the topological entropy and the parameter space ordering of kneading sequences, associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. Finally, we show that complicated behavior arising from the chaotic firm model can be controlled without changing its original properties and the dynamics can be turned into the desired attracting time periodic motion (a stable steady state or into a regular cycle). The orbit stabilization is illustrated by the application of a feedback control technique initially developed by Romeiras et al. [1992]. This work provides another illustration of how our understanding of economic models can be enhanced by the theoretical and numerical investigation of nonlinear dynamical systems modeled by ordinary differential equations.

Steady-state behaviors in nonlinear systems with an application to robust disturbance rejection

Despite the widespread intuitive appeal of the concept of steady-state response and its use in shaping the asymptotic behavior of control systems, this concept has only been rigorously defined for finite-dimensional, linear time invariant systems. In this paper, we investigate this concept for nonlinear systems, following some classical developments in nonlinear dynamics. As an application, we show how the concept in question plays a role of paramount importance in the design of control laws for asymptotic tracking and disturbance attenuation.

Downward causation in fluid convection

Recent developments in nonlinear dynamics have found wide application in many areas of science from physics to neuroscience. Nonlinear phenomena such as feedback loops, inter-level relations, wholes constraining and modifying the behavior of their parts, and memory effects are interesting candidates for emergence and downward causation. Rayleigh–Bénard convection is an example of a nonlinear system that, I suggest, yields important insights for metaphysics and philosophy of science. In this paper I propose convection as a model for downward causation in classical mechanics, far more robust and less speculative than the examples typically provided in the philosophy of mind literature. Although the physics of Rayleigh–Bénard convection is quite complicated, this model provides a much more realistic and concrete example for examining various assumptions and arguments found in emergence and philosophy of mind debates. After reviewing some key concepts of nonlinear dynamics, complex systems and the basic physics of Rayleigh–Bénard convection, I begin that examination here by (1) assessing a recently proposed definition for emergence and downward causation, (2) discussing some typical objections to downward causation and (3) comparing this model with Sperry’s examples.

Birth and Early History of Nonlinear Dynamics in Economics

Since the 1980s, nonlinear dynamic modelling is becoming a popular methodology in economics. However, it is not as new as many researchers seem to believe. Before the linear approach dominated economic theory around the 1950s, many economists were actively involved in the development of nonlinear models, this tendency being particularly strong during the period 1930-1950. The main objective of this essay is to offer a systematic and comprehensive survey of the early developments in nonlinear dynamics in economics, ranging form Frisch’s original impulse and propagation model in 1933, to Goodwin’s formalisation of the limit cycle in 1951.

The Development of Nonlinear Dynamics in Astronomy

We present the historical development of Nonlinear Dynamical Astronomy with emphasis on the “third integral” and its applications. The new era started with the use of computers, and of formal analytical developments in the spirit of Poincare. Most dynamical systems were found to contain both ordered and chaotic orbits. The transition from order to chaos is discussed. Recent developments refer to the dynamical spectra, integrals of notion in self-consistent models, systems of 3 or more degrees of freedom, chaos in relativity and cosmology, and chaos in quandum mechanics.

LIVING ON THE EDGE OF CHAOS: POPULATION DYNAMICS OF FENNOSCANDIAN VOLES

Although ecologists were among the first scientists to study the implications of chaotic dynamics, there is still no widely accepted example of chaos in a field population. If current developments in nonlinear dynamics are to be of relevance to population ecology, however, it is necessary to determine whether chaos occurs in nature. Here we bring together two lines of evidence suggesting that the dynamics of vole populations in northern Fennoscandia may be chaotic: nonlinear analysis of time-series data, and a model based on biological mechanisms. We review how a mechanistic population model can be used to produce short-term population forecasts, even when one of the species is not observed, and then use this benchmark of model accuracy to support our choice of a mechanistic population model for vole dynamics. By adjusting model parameters to optimize the forecasting accuracy on the vole time series, we can obtain parameter estimates for the model. In tests on simulated data, this new approach to parameter estimation for stochastic dynamic models yields parameter estimates such that the fitted model produces dynamics qualitatively and quantitatively similar to those observed. However, some parameters are poorly identified (i.e., many different parameter combinations yield nearly identical model dynamics). For the best-fit parameters, the model dynamics are characterized by quasi-chaotic behavior; the global Lyapunov exponent is very close to (and statistically undistinguishable from) zero, but the median local Lyapunov exponent is significantly positive. Our findings of recurrent short-term chaos suggest that vole dynamics in northern Fennoscandia are characterized by a blend of order and irregularity. The irregularity is the same type as would occur in a noisy chaotic system (i.e., exogenous noise amplification by sensitive dependence on initial conditions) but only occurs during specific portions of the oscillation.

Oscillatory brain theory: a new trend in neuroscience.

During the Decade of the Brain, science is coming to terms with its ultimate problem: understanding the mechanisms by which the immense number of neurons in the human interact to produce the higher cognitive functions. The analysis of the brain's natural frequencies opens a new window toward a combined analysis of sensory and cognitive functions at the level of single neurons and the field potentials. In the last decade, our research group has been strongly involved in the development of nonlinear dynamics and with the oscillatory processes of neural assemblies. Recently, three new volumes extensively described this new evolution in neuroscience by concluding that a new integrative neurophysiology and a new brain theory is needed in order to confront the problems recognized in this decade of the brain. This article provides a brief outline of how the findings in the last 20 years of research have led to such a development.

NEO Orbits and Nonlinear Dynamics: A Brief Overview and Interpretations

ABSTRACT: The development of nonlinear dynamics is outlined with emphasis on the concepts leading to the development of a numerical technique known as “fuzzy boundary theory” or FBT. Origins of this methodology are a direct outgrowth of concepts derived from the Hamiltonian formulation, KAM theory, and “Mather regions.” The significance of FBT as a computational methodology for NEO studies includes the weak capture and fast resonance shifting applied to NEO orbits and spacecraft rendezvous as described by a paper by Belbruno in this volume. Further implications of the FBT methodology suggest that some NEO capture cross‐sections with planetary bodies may have to be recomputed.

Correlating dimension for a gas-liquid contactor

Recent developments in non-linear dynamics and complex phenomena have resulted in mathematical tools which are potentially useful in the control and design of bubble column reactors. It is conjectured that analysis of conductivity time series, measured in a bubble column, indicated a low-dimensional chaotic attractor. The cross-sectional average correlation dimension varied with the liquid and the gas flow rate. The cross-sectional average correlation dimension characterized the global bubble column hydrodynamics and, more specifically, the average bubble size.

Differentiation of alpha coma from awake alpha by nonlinear dynamics of electroencephalography

The electroencephalogram, as a probe of scalp-recorded electrical activity arising from the human cortex, provides useful information because of its temporal and spatial organization. Recent developments in nonlinear dynamics suggest that an object can be constructed in an n-dimensional space out of a temporal sequence of data such as an EEG signal and that its organization is characterized by the dimensionality of the object (in this case, human brain activity). We have carried out an analysis of a set of alpha coma EEG patterns in comparison to the awake alpha EEG patterns of normal volunteers and patients. Alpha coma recorded from a single channel is visually indistinguishable from normal resting alpha due to its similar frequency spectrum (a broad-band spectrum with 1/f characteristics). Our results show that alpha coma dimensionality, however, differs from that of normal alpha in that it has a greater variability over different temporal segments of EEG. Single channel recordings in 7 patients with alpha coma were differentiable from those of 10 subjects with “normal” EEGs. Through dynamic analysis of the EEG, novel methods of signal extraction from EEG may become evident and applicable to clinical practice.

Patterns of attractor dimensions of sleep EEG

Low dimensional chaos is a property of many physiological oscillatory systems including the brain. Time series of sleep EEG records have been analyzed in the framework of recent developments in nonlinear dynamics. One of the characteristics of a chaotic time series is its attractor dimension. The running attractor dimension of a chaotic time series may reflect changes in states more accurately than manually scored records. In the present study the attractor dimensions of consecutive EEG segments of five sleep records were analyzed. The block of the EEG segment (window) was shifted by various lengths along the entire sleep data of each subject thus producing a running attractor dimension curve for each record. The attractor dimension values for different sleep stages were significantly different. The pattern of the running attractor dimension closely matched the scored hypnograms in these five sleep records.

A nonlinear perspective in understanding the neurodynamics of EEG

The developments in nonlinear dynamics and the theory of chaos have considerably altered our perception and analysis of many complex systems, including the brain. This paper reviews the physical and dynamical aspect of brain's electrical activity from this new perspective and indicates possible future directions. The importance of emerging trends of nonlinear dynamics and chaos to neurobiology has been discussed in the context of various states of consciousness and behaviour. In the past, EEG analysis has been confined to descriptive stochastic statistics and any understanding of the transitional process of brain activities was either nonexistent or not amenable for investigation. With the developments in nonlinear dynamics, the chaotic dynamical parameters and trajectory behaviour will find their use as feature detection techniques in EEG. Furthermore, nonlinear dynamics provides a model for EEG generation and temporal prediction which will help in determining the nature of neuronal processes governing various states of brain activity. The formalism of globally coupled dynamic systems will find applications in modelling the transitional states of EEG.

Recent developments in business cycle theory

This article examines recent developent in cycle theory, comparing the contributions of Keynesian, Marxian, and neoclassical equilibrium theorists. First the weaknesses of linear-stochastic theory, about which all partisans of nonlinear dynamics agree, are discussed. Secondly, the contributions of nonlinear dynamics to the competing streams of cycle theory will be illustrated. Thirdly, it is argued that perfect foresight theory may have long-term difficulties in accomodating developments in nonlinear dynamics, chaos theory especially.

Fragility and Deterministic Modelling in the Exact Sciences

The theoretical framework adopted in the exact sciences, for constructing and testing deterministic theories on the one hand, and modelling and analysis of observed phenomena on the other, is often implicitly assumed to be that of structural stability. In view of recent developments in nonlinear dynamics, it is argued here that in general it may not be possible to assume strict determinism and structural stability simultaneously; either strict determinism holds, in which case the fragility framework may turn out to be the appropriate framework for the study of certain phenomena in the exact sciences, or ‘structural stability’ is restored at the expense of introducing stochasticity. In this sense a certain degree of indeterminacy may be unavoidable even at the classical level.

Invariant surfaces and phase space flux in three-dimensional surface diffusion

Investigation of the classical three-dimensional motion of a helium atom on a (001) xenon surface uncovered new invariant manifolds that are in close accord with recent theoretical developments in nonlinear dynamics. The relationship between these manifolds, phase space flux, and recrossing in transition-state theory is explored. General theory and computational methods are also discussed.