Definition Of Chaos Research Articles

Regular and chaotic quantum motions

In the framework of Bohm's interpretation of quantum mechanics, particles are subject to both classical and quantum forces and actually follow physical trajectories. A natural definition of chaos in quantum mechanics is therefore possible and it is measured by a quantum Lyapunov exponent. In order to test the effectiveness of this new definition of quantum chaoticity, we have comparatively studied a classical model and a truncated N-level quantum model of a hydrogen atom acted upon by an external electromagnetic field. We find direct evidence for the existence of quantum chaotic motions, for a chaotic transition and for a quantum attenuation of classical chaos in a broad range of external field amplitudes.

CLARIFYING CHAOS: EXAMPLES AND COUNTEREXAMPLES

Over the past fifteen years there have been various attempts to define chaos. In an effort to find a universally acceptable definition we began constructing new examples of chaotic systems in the hope that the salient features of chaos could be captured. Our efforts to date have failed and the examples we have constructed seem to suggest that no such definition exists. However, these examples have proved to be valuable in spite of our inability to hone a universal definition of chaos from them. Consequently, we present this list of examples and their significance. Some interesting conclusions that we can draw from them are: It is possible to construct simple closed form solutions of chaotic one-dimensional maps; sensitive dependence on initial conditions, the most widely used definition of chaos, has many counterexamples; there are invertible chaotic dynamical systems defined by simple differential equations that do not have horseshoes; three important properties that are thought to characterize chaos, continuous power spectral density, exponentially sensitive dependence on initial conditions, and exponential loss of information (Chaitin’s concept of algorithmic complexity), are independent. Chaos seems to be tied to our notion of rates of divergence of orbits or degradation of information such as is found in systems with positive Lyapunov exponents. The reliance on rates seems to open the door to a pandora’s box of rates, both higher and lower than exponential. The intuitive notion of pseudo-randomness, a practical feature of chaos, is present in examples that do not have positive Lyapunov exponents. And in general, nonlinear polynomial rates of degradation of information are also quite “unpredictable”. We conclude that it appears that for any given definition of chaos, there may always be some “clearly” chaotic systems which do not fall under that definition, thus making chaos a cousin to Gödel’s undecidability.

The Role of Transitivity in Devaney's Definition of Chaos

The Role of Transitivity in Devaney's Definition of Chaos

The Role of Transitivity in Devaney's Definition of Chaos

The Role of Transitivity in Devaney's Definition of Chaos

Chaos in Ecology: Is Mother Nature a Strange Attractor?

We review the role of chaos and the study of chaos in ecology. We use sensitive dependence on initial conditions as the best heuristic definition of chaos. This definition forms the common theme for our review of approaches for demonstrating the importance of chaos in ecology. We emphasize that this

Defining Chaos

This paper considers definitions of classical dynamical chaos that focus primarily on notions of predictability and computability, sometimes called algorithmic complexity definitions of chaos. I argue that accounts of this type are seriously flawed. They focus on a likely consequence of chaos, namely, randomness in behavior which gets characterized in terms of the unpredictability or uncomputability of final given initial states. In doing so, however, they can overlook the definitive feature of dynamical chaos—the fact that the underlying motion generating the behavior exhibits extreme trajectory instability. I formulate a simple criterion of adequacy for any definition of chaos and show how such accounts fail to satisfy it.

𝜔-chaos and topological entropy

We present a new concept of chaos, ω \omega -chaos, and prove some properties of ω \omega -chaos. Then we prove that ω \omega -chaos is equivalent to positive entropy on the interval. We also prove that ω \omega -chaos is equivalent to the definition of chaos given by Devaney on the interval.

Chaos

AbstractThis is an introductory article on Chaos giving the definition of Chaos, source and tools of Chaos, routes to chaos, measurement of chaos, chaos through resonance. The Duffing equation has been mentioned as a problem of double resonance leading to chaos and finally some problems relating to chaos in the Solar system have been described.

On Devaney's Definition of Chaos

(1992). On Devaney's Definition of Chaos. The American Mathematical Monthly: Vol. 99, No. 4, pp. 332-334.

On Devaney's Definition of Chaos

On Devaney's Definition of Chaos

On Maps with Dense Orbits and the Definition of Chaos

On Maps with Dense Orbits and the Definition of Chaos

Chaos, Topology, and Social Organization

This paper attempts to apply chaos theory to social organization. It begins with a mathematical definition of chaos. Specifically, the geometric concept of attractor is explored; phase space and Poincaré maps are discussed and applied to the concept of attractor; and nonlinearity is conceptually defined. We then apply the mathematics of chaos to social systems by showing how a nonlinear equation might be used to describe organization and how data derived from a simple univariate equation can be converted into a multivariate Poincaré map. The conclusion section identifies three approaches to analyzing chaos in social organization: metaphorical analysis, mathematical modeling, and data collection. Finally, possible uses of chaos are explored, as are the shortcomings of such analysis.

Book review:Chaotic behavior of deterministic dissipative systems

This book contains a survey of theoretical and experimental aspects of the chaotic behavior of deterministic systems. It is an introductory text aimed at the level of graduate students and researchers from a variety of fields (mathematics, physics, biology, and engineering). The monograph provides a comprehensive bibliography on the subject up to the date of publication. In my opinion, one of the most interesting features of the book is its detailed description of experiments in which chaos constitutes a significant backdrop. The book consists of seven chapters and two technical appendices. The introductory Chapter 1 discusses the significance of chaos as a model of many seemingly random processes in nature and defines the class of dissipative systems described in greater detail later in the book. Chapter 2 discusses basic notions of dynamical systems and their asymptotic behavior, including the definitions of chaos and strange attractors. Chapter 3 deals with the transition from order to chaos, illustrated by a number of examples. Chapter 4 discusses numerical methods for studies of parametric dependences, bifurcations, and chaos. Chapters 5 and 6 survey experimental observations of chaotic behavior, including a number of experiments and data from mechanical and electromechanical systems, semiconductors, lasers, chemical and biological systems, and hydrodynamics. Some of the experiments and data are based on the authors' work. Finally, Chapter 7 deals with spatiotemporal behavior in distributed systems, presenting a brief review on results in the areas of cellular automata, coupled map lattices, and partial differential equations:

Macro Imaging of Rock Cores: Preparation, Approach, and Usefulness

Petrographic image analysis (PIA) has extracted useful data from rock thin sections taken from rock cores. These data include porosity, permeability, and recovery potential. The usual procedure is to analyze images of rock thin sections viewed through a microscope using frame-grabbing hardware and software, which are then processed using a wide array of statistical/deterministic algorithms. The results are functions relating: the imaged parameters to rock petro-physics; the definition of diagenetic classification schemes, or even the foundation for definition of chaos functions describing 3-D pore networks. Normally, only the pore network serves as input into the approach previously described. On the macro scale, however, a different collection of images is used as input. The procedure involves imaging sections of polished rock core which represent areas as small as 2 x 2 in. or as large as 5 x 5 in. The variability observed normally on this scale includes large pores, fractures, bedding structures; in carbonates it includes mineralogy changes, fossil content, and various types of textures. The ability to differentiate any of these characteristics depends generally on the quality of hardware (resolution) and the ability to partition the rock image by defining colors or patterns. To increase discrimination power the rockmore » slab should be stained and the pores and fractures enhanced. Examples of processing carbonate cores will be presented. Problems and the overall usefulness of the macro approach to core analysis for reservoir analysis will also be highlighted.« less

Chaos theory for the biomedical engineer

A brief introduction to chaos theory is provided. Definitions of chaos and attributes of chaos and fractals are discussed. Several general examples are examined, and fractals are introduced with a brief look at the Mandelbrot and Julia sets. Biomedical examples of chaotic behaviour and fractals are presented.

Does an Unstable Keynesian Unemployment Equilibrium in a Non-Walrasian Dynamic Macroeconomic Model Imply Chaos?

Simonovits (1982) introduced a non-Walrasian dynamic macromodel which is based on the disequilibrium model with inventory dynamics due to Honkapohja and Ito (1980). This model describes transactions on two markets (one market for labor and one market for goods) and contains five parameters. Obviously, there is an equilibrium which is the so-called Keynesian unemployment equilibrium point. For suitable choices of the parameters, this equilibrium point is locally stable and, according to Simonovits (1982), the Keynesian unemployment equilibrium point is also globally stable, i.e., the trajectories of all initial values converge to the equilibrium point. For many parameter values the equilibrium point is unstable; this is in fact the case when the product of the two eigenvalues at the equilibrium point is greater than 1. Simonovits' (1982) conjecture says: If the Keynesian unemployment equilibrium point is unstable, then there is First, we recall some definitions of chaos. We restrict our attention to systems whose trajectories are bounded. A first definition is the following. A system is chaotic if there exist infinitely many periodic points with different period and if there is an uncountable set of aperiodic points. This kind of chaos is nowadays called Li-Yorke chaos; see Li and Yorke (1975) and Diamond (1976). Before turning to a second definition, we need the notion of dependence on initial values. A system may be said to have sensitive dependence on initial values if we can find, with probability p (where 0 < p < 1), a point x such that for every open neighborhood U of x there is a point y in U such that the trajectories of x and y will not be close

Chaos: principles and implications in biology.

In this paper we review some of the basic principles of the theory of dynamical systems. We introduce the reader to the definition of chaos and strange attractors and we discuss their implications in biology.

A precise definition of chaos

A precise definition of chaos