Encounter with Chaos: Self-Organized Hierarchical Complexity in Semiconductor Experiments

ISBN-13: 9783540558453 / Angielski / Miękka / 1992 / 289 str.

Joachim Peinke;Jürgen Parisi;Otto E. Rossler

Opis

Our life is a highly nonlinear process. It starts with birth and ends with death; in between there are a lot of ups and downs. Quite often, we believe that stable and steady situations, probably easy to capture by linearization, are paradisiacal, but already after a short period of everyday routine we usually become bored and seek change, that is, nonlinearities. If we reflect for a while, we notice that our life and our perceptions are mainly determined by nonlinear phenomena, for example, events occurring suddenly and unexpectedly. One may be surprised by how long scientists tried to explain our world by models based on a linear ansatz. Due to the lack of typical nonlinear patterns, although everybody experienced nonlinearities, nobody could classify them and, thus, - study them further. The discoveries of the last few decades have finally provided access to the world of nonlinear phenomena and have initiated a unique inter- disciplinary field of research: nonlinear science. In contrast to the general tendency of science to become more branched out and specialized as the result of any progress, nonlinear science has brought together many different disciplines. This has been motivated not only by the immense importance of nonlinearities for science, but also by the wonderful simplicity ohhe concepts. Models like the logistic map can be easily understood by high school students and have brought revolutionary new insights into our scientific under- standing.

Spis treści

1 Introductory Remarks.- Problems.- 2 Semiconductor Physics.- 2.1 Fundamentals of Nonlinear Dynamics.- 2.1.1 Historical Remarks.- 2.1.2 Plasma Ansatz.- 2.1.3 Negative Differential Conductivity.- 2.1.4 Transport Mechanisms.- 2.2 Recent Experimental Progress.- 2.3 Model Experimental System.- 2.3.1 Material Characterization.- 2.3.2 Experimental Set-up.- 2.4 Experimental Results.- 2.4.1 Static Current-Voltage Characteristics.- 2.4.2 Temporal Instabilities.- 2.4.3 Spatial Structures.- 2.4.4 Spatio-Temporal Behavior.- Problems.- 3 Nonlinear Dynamics.- 3.1 Basic Ideas and Definitions.- 3.2 Fixed Points.- 3.2.1 Fundamental Bifurcations.- 3.2.2 Catastrophe Theory.- 3.2.3 Experiments.- 3.3 Periodic Oscillations.- 3.3.1 The Periodic State.- 3.3.2 Bifurcations to Periodic States.- 3.3.3 Experiments.- 3.4 Quasiperiodic Oscillations.- 3.4.1 The Quasiperiodic State.- 3.4.2 Bifurcations to Quasiperiodic States.- 3.4.3 Influence of Nonlinearities.- 3.4.4 Experiments.- 3.5 Chaotic Oscillations and Hierarchy of Dynamical States.- 3.5.1 The Chaotic State.- 3.5.2 Characterization Methods.- 3.5.3 Bifurcations to Chaotic States.- 3.6 Spatio-Temporal Dynamics.- Problems.- 4 Mathematical Background.- 4.1 Basic Concepts in the Theory of Dynamical Systems.- 4.1.1 Dissipative Dynamical Systems and Attractors.- 4.1.2 Invariant Probability Measures.- 4.1.3 Invariant Manifolds.- 4.1.4 Chaos.- 4.2 Scaling Behavior of Attractors of Dissipative Dynamical Systems.- 4.2.1 Scale Invariance.- 4.2.2 Symbolic Dynamics.- 4.2.3 Analogy with Statistical Mechanics.- 4.2.4 Partition Function for Chaotic Attractors of Dissipative Dynamical Systems.- 4.2.5 Discussion of the Partition Function.- 4.3 Generalized Dimensions, Lyapunov Exponents, Entropies.- 4.3.1 Definition of the Scaling Functions for Sampling Processes.- 4.3.2 Relation Between the Scaling Functions and a Thermodynamical Formalism.- 4.3.3 Relation Between the Scaling of the Support and the Scaling of the Measure (Generic Case).- 4.3.4 Discussion of Nonanalyticities.- 4.3.5 Evidence of Phase-Transition-Like Behavior in Experimental Observations.- 4.4 Evaluation of Experimental Systems.- 4.4.1 Embedding of a Time Series.- 4.4.2 Lyapunov Exponents from the Dynamical Equations and from Time Series.- 4.4.3 Results and Stability of Results.- 4.4.4 Comparison with Other Methods.- 4.5 High-Dimensional Systems.- 4.5.1 Singular-Value Decomposition.- 4.5.2 The Modified Approach.- 4.5.3 Results on Simulated Data.- 4.6 Lyapunov Exponents, Rotation Numbers and the Degree of Mappings.- 4.6.1 Characterization of Solutions of Dynamical Systems via the Degree of Mapping.- 4.6.2 Riemannian Motions on Manifolds of Constant Negative Curvature.- 4.7 Conclusions.- Problems.- References.