Foundations of synergetics. II: Complex patterns
This textbook is the second of two volumes based on a lecture course in synergetics given at the University of Moscow. The first volume [the first author, I: Distributed active systems (1990; Zbl 0712.92001)] concerned deterministic coherent structures in active media, e.g. propagating wave fronts and pulses, spiral waves, and localized stationary dissipative patterns. The distinctive feature of a coherent pattern is the presence of long-range order in space and time. Chaotic patterns, treated in this second volume, are characterized by short-range order with respect to time and/or spatial coordinates. Chaotic behavior may be an autonomous property of an active system which does not require external random perturbations or it may be the result of selective amplification of fluctuations induced by weak external noise. Both processes are discussed extensively. In essence, volume 2 of Foundations of Synergetics is a didactic review which links the nonlinear nondeterministic ``new math'' of the last 25 years to classical studies in dynamics and noise theory. The authors present such topics as strange attractors, fractals, bifurcations, the Fokker-Planck equation, Brownian motion, population growth and extinction, and catastrophe theory in the context of chaos. Even today the relationships among these and kindred subjects are far from clear, but this commendable book represents one of the few efforts to unite them to the extent that it is possible to do so. Chapters 1-7 give a concise introduction to chaotic dynamics in small systems. Chapters 8-12 treat noise-induced complex patterns. The first chapter is introductory. A chapter-by-chapter summary of the rest of the book follows. Chapter 2. Unpredictable dynamics. The phase-space representation of Hamiltonian (non-dissipative) systems is reviewed. Ergodicity is a necessary but not sufficient condition for chaotic motion. Mixing, in which the accessible volume of phase space ejects long irregular unbroken tentacles, is the sufficient condition to ensure the fading of temporal correlations. Chapter 3. Strange attractors. Phase-space singular points in general and attractors in particular are reviewed. Strange attractors are distinguished from simple attractors in that any two trajectories diverge exponentially with time while still belonging to the same attractor. The Lorenz model of thermal convection is used to illustrate the typical properties of a strange attractor. The use of Lyapunov exponents provides a criterion for distinguishing between complex regular motion and true chaotic motion in which initially close trajectories diverge in phase space. The autocorrelation function or the spectral density of the motion of a dynamical system can also serve as a criterion for chaos. Periodic behavior yields a periodic autocorrelation function or a line spectrum; chaotic behavior leads to an autocorrelation function that fades with time or a continuous spectrum. Chapter 4. Fractals. Fractals, i.e. self-similar patterns, having nonintegral dimensions are introduced. Fractal properties are attributed to strange attractors. The Lyapunov dimension provides an upper estimate of the fractal dimension and may coincide with it in Chapter 5. Discrete maps. Discrete models or maps in which the state of a system is allowed to change only at prescribed instants of time are described by iterative equations. Conditions are formulated under which discrete dynamics can be chaotic. The logistic equation is discussed as an example. When a parameter of a map is gradually increased, the map can undergo a transition from regular to chaotic dynamics. At certain parameter bifurcation values, the period of oscillation of the system doubles until chaos appears. Feigenbaum's rules concerning these values are discussed. Chapter 6. Routes to temporal chaos. Basic scenarios of transition to temporal chaos in dynamical systems with continuous time are considered. Bifurcation theory is reviewed. Chaotic motion is produced when, owing to a change in the value of a parameter, simple attractors in phase space are replaced by a strange attractor. Scenarios in which this occurs include (1) period doubling, (2) intermittency characterized by the alternation of chaotic behavior with almost periodic oscillations, and (3) the appearance of a sequence of toroidal attractors of increasingly higher dimension as described by Ruelle and Takens. Chapter 7. Spatio-temporal chaos. Two stages of hydrodynamic turbulence are distinguished. In early turbulence, spatial coherence is maintained, but long-range temporal order is lost; in developed turbulence, spatial coherence is also lost. Early turbulence can be described by models utilizing a small number of dynamic variables. Numerical methods are presented which make it possible to determine from experimental time series of data the minimum number of independent variables which should be employed in the construction of an effective model. This number is the embedding dimension. When the dynamical equations of an active medium are known, but time series of data are not available, no general solution to the problem of calculating the embedding dimension exists. However results have been obtained in special cases. The generalized Ginzburg-Landau equation provides an illustration. The transition from early to developed turbulence has not yet been thoroughly investigated. Specific examples are presented, but the relevant mathematical models are too complicated for an extensive analysis. Fortunately, almost the same behavior is observed in much simpler systems of discrete coupled chaotic maps. Several illustrations are provided. Chapter 8. Random processes. Probabilistic automata, i.e. Markov chains and discrete Markov processes, are introduced. The mathematical theory of continuous random processes is briefly summarized. Special attention is given to Markov processes, which have zero correlation time and no memory. It is shown that the evolution of the probability distribution of a continuous Markov process is described by the Fokker-Planck equation. Chapter 9. Active systems with noise. The mathematical framework for the analysis of noise-induced behavior in distributed active systems is presented. The application of external noise causes a dynamical system to wander in phase space. Such random wandering is superimposed on the steady drift produced by deterministic dynamics. The resulting process is similar to the Brownian motion of a particle in the presence of a steady driving force. Internal noise does not correspond to the action of any real fluctuating environment. It is introduced as a mathematical convenience which makes it possible, in some instances, to describe probabilistic discrete automata as if they were dynamical systems with continuous variables subject to external chaotic influences. Optimal fluctuations are considered. These are extremely rare large deviations from the phase-space trajectory of a deterministic system. They can visit regions of phase space that are not accessible to the deterministic trajectory. Optimal fluctuations may therefore lead to drastic consequences, e.g. catastrophic transitions in system behavior. Chapter 10. Population explosions. Noise induced complex patterns are found in explosive systems that evolve from an unstable state. In the process of such an evolution, amplitudes of many fluctuation modes increase. Since the rates of growth are generally different, there will be a relative enhancement of some fluctuations at the expense of others. If the most rapidly growing fluctuations have large characteristic length scales, this process will lead to the creation, in an unstable medium, of macroscopic random patterns. Explosive population growth in fluctuating media results in the formation of complex intermittent patterns in which small regions with very high population density randomly alternate with larger regions that are sparsely populated. The explosion threshold in different types of fluctuating media is investigated. Chapter 11. Extinction and long-time relaxation. Spatial patterns are produced by the process of extinction in several ways. In the simplest situation, particles wandering through a medium are captured or destroyed by a set of randomly distributed traps or sinks. Initially, the particle density declines exponentially with time, but, in the long-time limit, the exponential decay is replaced by a much slower time dependence due to large rare voids. Eventually, the particles are concentrated in trap-free regions, the size of which is estimated. If two kinds of particles, A and B, annihilate one another, and if their initial distribution is random, the difference in their densities obeys a simple diffusion equation. Ultimately the entire volume in which the annihilation reaction proceeds is partitioned into regions of two types: those containing only A particles and those containing only B's. In the evolution of the universe, this mechanism may have been responsible for the creation of regions filled exclusively by matter and antimatter. Chapter 12. Catastrophes. The discussion focusses on soft catastrophes described by bifurcations in which an attractor becomes unstable and gives birth to new attractors initially lying in close proximity to the original attractive set. Fluctuations determine which new attractor the system will follow after the catastrophe. Fluctuations are thus selectively enhanced, leading to the creation of large-scale complex patterns. Second-order phase transitions, exemplified by the interactions of two mutually inhibiting hostile populations, are examined in detail. Near the bifurcation, such systems exhibit strong fluctuations in the local difference of their population densities. Special situations which are examined include (1) systems which sweep through the critical transition region, (2) pairs of populations in which one has a slightly higher reproduction rate, (3) populations which reproduce in a fluctuating medium, and (4) survival and extinction of two populations which compete for the same renewable resource.
scientific article; zbMATH DE number 50431