Measures of Complexity and Chaos

ISBN-13: 9781475706253 / Angielski / Miękka / 2012 / 486 str.

Neal B. Abraham; Alfonso M. Albano; Anthony Passamante

Opis

This volume serves as a general introduction to the state of the art of quantitatively characterizing chaotic and turbulent behavior. It is the outgrowth of an international workshop on "Quantitative Measures of Dynamical Complexity and Chaos" held at Bryn Mawr College, June 22-24, 1989. The workshop was co-sponsored by the Naval Air Development Center in Warminster, PA and by the NATO Scientific Affairs Programme through its special program on Chaos and Complexity. Meetings on this subject have occurred regularly since the NATO workshop held in June 1983 at Haverford College only two kilometers distant from the site of this latest in the series. At that first meeting, organized by J. Gollub and H. Swinney, quantitative tests for nonlinear dynamics and chaotic behavior were debated and promoted 1). In the six years since, the methods for dimension, entropy and Lyapunov exponent calculations have been applied in many disciplines and the procedures have been refined. Since then it has been necessary to demonstrate quantitatively that a signal is chaotic rather than it being acceptable to observe that "it looks chaotic." Other related meetings have included the Pecos River Ranch meeting in September 1985 of G. Mayer Kress 2) and the reflective and forward looking gathering near Jerusalem organized by M. Shapiro and I. Procaccia in December 1986 3). This meeting was proof that interest in measuring chaotic and turbulent signals is widespread."

Spis treści

Complexity and Chaos.- Chaotic Metamorphoses.- I. Characterizing Temporal Complexity: Chaos.- A. Measuring Dimensions, Entropies and Lyapunov Exponents.- Measures of Dimensions from Astrophysical Data.- Some Remarks on Nonlinear Data Analysis of Physiological Time Series.- Hierarchies of Relations Between Partial Dimensions and Local Expansion Rates in Strange Attractors.- Experimental Study of the Multifractal Structure of the Quasiperiodic Set.- Statistical Inference Theory for Measures of Complexity in Chaos Theory and Nonlinear Science.- Practical Remarks on the Estimation of Dimension and Entropy from Experimental Data.- Chaotic Behavior of the Forced Hodgkin-Huxley Axon.- Chaotic Time Series Analysis Using Short and Noisy Dta Sets: Applications to a Clinical Epilepsy Seizure.- Measuring Complexity in Terms of Mutual Information.- Estimating Lyapunov Exponents From Approximate Return Maps.- Estimating Local Intrinsic Dimensionality Using Thresholding Techniques.- Seeking Dynamically Connected Chaotic Variables.- On Problems Encountered with Dimension Calculations.- Systematic Errors in Estimating Dimensions from Experimental Data.- Analyzing Periodic Saddles in Experimental Strange Attractors.- Time Evolution of Local Complexity Measures and Aperiodic Perturbations of Nonlinear Dynamical Systems.- Analysis of Local Space/Time Statistics and Dimensions of Attractors Using Singular Value Decompositon and Information Theoretic Criteria.- Entropy and Correlation Time in a Multimode Dye Laser.- Dimension Calculation Precision with Finite Data Sets.- Chaos in Childhood Epidemics.- Measurement of f(?) for Multifractal Attractors in Driven Diode Resonator Systems.- Is there a Strange Attractor in a Fluidized Bed?.- Statistical Error in Dimension Estimators.- B. Other Measures.- Dynamical Complelxity of Strange Sets.- Characterization of Complexity by Aperiodic Driving Forces.- Stabilization of Prolific Populations Through Migration and Long-lived Propagules.- Complex Behavior of Systems Due to Semi-stable Attractors: Attractors That Have Been Destablized but Which Still Temporarily Dominate the Dynamics of a System.- Universal Properties of the Resonance Curve of Complex Systems.- The Effects of External Noise on Complexity in Two-dimensional Driven Damped Dynamical System.- Chaos on a Catastrophe Manifold.- Topolgical Frequencies in Dynamical Systems.- Phase Transitions Induced by Deterministic Delayed Forces.- Mutual Information Functions Versus Correlation Functions in Binary Sequences.- Reduction of Complexity by Optimal Driving Forces.- Symbolic Dynamical Resolution of Power Spectra.- Relative Rotation Rates for Driven Dynamical Systems.- Stretching Folding Twisting in the Driven Damped Duffing Device.- Characterizing Chaotic Attractors Underlying Single Mode Laser Emission by Quantitative Laser Field Phase Measurements.- C. Characterizing Homoclinic Chaos.- Shil’nikov Chaos: How to Characterize Homoclinic and Heteroclinic Behavior.- Time Series Near Codimension Two Global Bifurcations.- Characterization of Shil’nikov Chaos in a CO2 Laser Containing a Saturable Absorber.- Symmetry-breaking Homoclinic Chaos.- Time Return Maps and Distributions for the Laser with Saturable Absorber.- D. Building Models from Data.- Unfolding Complexity in Nonlinear Dynamical Systems.- Inferring the Dynamic; Quantifying Physical Complexity.- Symbolic Dynamics from Chaotic Time Series.- Modelling Dynamical Systems from Real-world Data.- Extraction of Models from Complex Data.- Quantifying Chaos with Predictive Flows and Maps: Locating Unstable Periodic Orbits.- II. Characterizing Spatio- Temp Oral Complexity: Turbulence.- A. Theoretical.- Defect-induced Spatio-temporal Chaos.- Lyapunov Exponents, Dimension and Entropy in Coupled Lattice Maps.- Phase Dynamics, Phase Resettiing, Correlation Functions and Coupled Map Lattices.- Characterization of Spatiotemporal Structures in Lasers: A Progress Report.- Amplitude Equations for Hexagonal Patterns of Convection in Non-Boussinesq Fluids.- Fractal Dimensions in Coupled Map Lattices.- Weak Turbulence and the Dynamics of Topological Defects.- Pattern Cardinality as a Characterization of Dynamical Complexity.- B. Experimental.- Characterizing Spatiotemporal Chaos in Electrodeposition Experiments.- Characterizing Space-time Chaos in an Experiment of Thermal Convection...- Characterizing Dynamical Complexity in Interfacial Waves.- Characterization of Irregular Interfaces: Roughness and Self-affine Fractals.- The Field Patterns of a Hybrid Mode Laser: Detecting the “Hidden” Bistability of the Optical Phase Pattern.- Contributors.