Directions in Chaos — Volume 1

The following sections are included:

• FOREWORD

• CONTENTS

Though not yet widely recognized, chaos theory now illuminates four highly significant problems in the foundations of classical dynamics. Specifically, the notions of chaos may lead to verification of the following conjectures:

1. The existence of a generalized uncertainty principle which asserts that no physical variable can be measured precisely.

2. The use of chaos to solve problems which heretofore either have not been or cannot be solved deterministically.

3. The expansion of the notion of integrability to include all systems which are ergodic (only) or mixing (only).

4. The use of judiciously chosen integrable, nearly integrable, or algorithmically integrable many-body systems to control chaos.

Classical mechanics in the past twenty years has undergone a rebirth. New ideas and techniques have been developed to understand mechanisms by which systems can undergo a transition from quasiintegral to chaotic behavior when parameters of the systems are varied. Quasiintegral behavior is destroyed in nonlinear systems due to the growth and overlap of nonlinear resonances. As a result, invariant surfaces in the phase space are destroyed and the states of the systems are free to wander in a random manner throughout the region of influence of the overlapping resonances. In this article, we analyse the phenomenon of resonance overlap using renormalization group methods, and we derive and discuss mappings which contain the essential features of the passage from regular to chaotic behavior in local regions of the phase space. To give the article a unifying theme, we use these methods to analyse the transition from regular to chaotic behavior in the conservative Duffing oscillator.

The period-doubling bifurcation route to chaos of the map x → x′ = f(λ, x) is analysed by a generalized renormalization group (RG) which scales the λ and x variables simultaneously. A fixed point of the RG is established both functionally and numerically, and a generalized RG equation is derived for the limiting function. The RG analysis is further extended to other sequences along the bifurcation chain, and subasymptotic eigenvalues are also presented.

An important application of the theory of nonlinear dynamics is the analysis of erratic experimental data. We give a review of some recent developments in the methods and problems related to the estimation of the fractal dimension of a time-series. We discuss a series of conditions which should be fulfilled for serious dimension calculations and apply these methods to data from multiperiodic signals and from (pseudo stochastic) random walks in one dimension. We show how for limited data sets finite size effects can modify the observed values of the dimension and estimate the minimal number of data points required for a given sampling frequency in order to determine the attractor dimension.

We review the recent progress directed at the prediction and observation of chaotic behavior in lasers, including lasers with an injected signal, bi-directional ring lasers, and Q-switched lasers. We discuss both theoretical and experimental advances and analyze the areas of agreement and disagreement between them. We also review some of the most common techniques that are used to discriminate between chaos and ordinary noise, in particular the measurements of dimensionality and correlation entropy.

We review progresses made in the applications of nonlinear dynamical techniques to the studies of IR multiphoton excitation of molecules in contact with a bath. An overall picture emerges by approaching this problem from the classical, semiclassical and quantum mechanical points of view, although differences exist among the results of these three approaches. Possible experimental tests of the bistable and chaotic behavior and remaining questions are discussed.

To understand transitions to chaos in higher dimensions, we have investigated two common routes: period doubling and quasiperiodicity. Period doubling in four-dimensional volume-preserving maps and quasiperiodicity with three incommensurate frequencies are studied.

The following sections are included:

• COMPLEXITY AND STATISTICAL MECHANICS OF DETERMINISTIC BEHAVIOR

• LATTICE DYNAMICAL SYSTEMS: PROTOTYPES FOR SPATIO-TEMPORAL COMPLEXITY
  • Restriction to Spatial Lattices
  • Architectural Relationship to Other Spatially-Extended Systems
  • Choice of Local Dynamics
  • Coupling Classes
  • Boundary Conditions
  • Initial Conditions

• PERIOD-DOUBLING LATTICES
  • Dependence of Domain Behavior on Domain Size
  • Kink Self-similarity

• SPATIAL MODE INSTABILITY: THE TRANSITION FROM TORUS TO CHAOS

• SOLITON TURBULENCE

• INTERMITTENCY LATTICES
  • Spatial Pomeau-Manneville Intermittency
  • Intermittency via Pattern Competition

• OPEN FLOW LATTICES
  • Spatial Period-Doubling
  • Spatial Amplification of Fluctuations
  • Domain Walls and Kink Dynamics

• TRANSIENT SPATIAL CHAOS

• SUMMARY

• ACKNOWLEDGMENTS

• REFERENCES