Ralph Abraham

COMPLEX DYNAMICAL SYSTEMS

Complex dynamical systems theory and system dynamics diverged at some point in the recent past, and should reunite. This is a concise introduction to the basic concepts of complex dynamical systems, in the context of the new mathematical theories of chaos and bifurcation.

A DOUBLE LOGISTIC MAP

Several endomorphisms of a plane have been constructed by coupling two logistic maps. Here we study the dynamics occurring in one of them, a twisted version due to J. Dorband, which (like the other models) is rich in global bifurcations. By use of critical curves, absorbing and invariant areas are determined, inside which global bifurcations of the attracting sets (fixed points, closed invariant curves, cycles or chaotic attractors) take place. The basins of attraction of the absorbing areas are determined together with their bifurcations.

The Genesis of Complexity

The theories of complexity comprise a system of great breadth. But what is included under this umbrella? Here we attempt a portrait of complexity theory, seen through the lens of complexity theory itself. That is, we portray the subject as an evolving complex dynamical system, or social network, with bifurcations, emergent properties, and so on. This is a capsule history covering the twentieth century. Extensive background data may be seen at www.visual-chaos.org/complexity.

ORDER AND CHAOS IN THE TORAL LOGISTIC LATTICE

Cellular dynamical systems, alias lattice dynamical systems, emerged as a new mathematical structure and modeling strategy in the 1980s. Based, like cellular automata, on finite difference methods for partial differential equations, they provide challenging patterns of spatiotemporal organization, in which chaos and order cooperate in novel ways. Here we present initial findings of our exploration of a two-dimensional logistic lattice with the Massively Parallel Processor (MPP) at NASAs Goddard Space Flight Center, a machine capable of 200 megaflops per second. A video tape illustrating these findings is available.

Complex dynamics and the social sciences

Abstract Complex dynamical systems theory is an evolution of nonlinear dynamics, developed for modeling and simulation of biological systems. Here, we speculate on the potential of this strategy for the emerging theory of social systems, and the implications for the future of our own planetary society.

Mathematics and evolution: A manifesto

Abstract This paper deals with various possibilities for the role of mathematical modeling and computer simulation in attempting to deal with the crises of evolution. Brief introductions to some concepts of holarchic dynamics are included.

Webometry: Measuring the complexity of the world wide web

The explosive growth of the WWW may be viewed as the neurogenesis phase in the embryogenesis of a new planetary civilization. To empower this emergent phenomenon with self‐reflection, we propose strategies for the visualization of the complexity of the WWW, seen as a neural net. The pointwise fractal dimension of a massive matrix is the basis of our strategy.

NONLINEAR RESONANCE IN BASIN PORTRAITS OF TWO COUPLED SWINGS UNDER PERIODIC FORCING

A model of a simple electric power supply network involving two generators connected by a transmission network to a bus is studied by numerical simulation. In this model, the bus is supposed to maintain a voltage of fixed amplitude, but with a small periodic fluctuation in the phase angle. In such a case, traditional analysis using direct methods is not applicable. The frequency of the periodic fluctuation is varied over a range of values near a nonlinear resonance of the two-generator network. When the bus fluctuation frequency is away from resonance, the system has several attractors; one is a small-amplitude periodic oscillation corresponding to synchronized, quasi-normal operation (slightly swinging), while others are large amplitude periodic oscillations which, if realized, would correspond to one or both generators operating in a desynchronized steady state. When the bus fluctuation frequency approaches resonance, a new periodic attractor with large amplitude oscillations appears. Although it does correspond to a synchronized steady state, this attractor has a disastrously large amplitude of oscillation, and represents an unacceptable condition for the network. Basin portraits show that this resonant attractor erodes large, complicated regions of the basin of the safe operating condition. Under conditions of small periodic fluctuation in bus voltage, this basin erosion would not be detected by traditional analysis using direct methods. Further understanding of such complicated basin structures will be essential to correctly predict the stability of electric power supply systems.

ATTRACTOR AND BASIN PORTRAITS OF A DOUBLE SWING POWER SYSTEM

In a normal power system, many generators are operating in synchrony. That is, they all have the same speed or frequency, the system frequency. In case some accident occurs, a situation might arise in which one or more generators are running at a different speed, much faster than the system frequency. They are said to be running away or stepping out, or in a state of accelerated stepping out. We have been engaged in a series of studies of this situation, and have found global attractor-basin portraits. In the course of this program, we have observed the phenomenon of decelerated stepping out, in which one or more generators deviate from the system frequency toward lower speeds. These kinds of behavior cannot be explained with the well-known model involving one generator operating on to an infinite bus. Rather, we require a model in which robust subsystems — for example, generator/motor combination, which we call swing pairs — are connected by interconnecting transmission lines. In this more general context, the deviant behaviors we are considering may be regarded as forms of desynchronization of subsystems. We therefore begin this paper with the derivation of a new mathematical model, in which there is no infinite bus nor fixed system frequency. In the simple case of two subsystems (each a swing pair) weakly coupled by an interconnecting transmission line, we develop a system of seven differential equations which include the variation of frequency in a fundamental way. We then go on to study the behavior of this model, using our usual methods of computer simulation to draw the attractor-basin portraits. We have succeeded in finding both accelerated and decelerated stepping out in this new model. In addition, we discovered an unexpected subharmonic swinging of the whole system.

DYNASIM. EXPLORATORY RESEARCH IN BIFURCATIONS USING INTERACTIVE COMPUTER GRAPHICS

The crucial early experiments regarding bifurcation theory and applications as the experimental branch of differentiable dynamics may be described in three overlapping periods. The period qfdirecr observation may be much older than we think, but let us say it begins with the musician, Chladni, contemporary of Beethoven, who observed bifurcations of thin plate vibrations. Much can still be learned from his work, painstakingly reproduced by Waller .’ Analogous phenomena discovered in fluids by Faradap are still actively ~ t u d i e d . ~ ~ These experiments, so valuable because the medium is real, suffer from inflexibility-especially in choosing initial conditions. The next wave of bifurcation experiments, which I shall call the analog period, begins with the triode oscillator. The pioneering works of van der Pol, with improvements by Hayashi, produced a flexible analog computer, and institutionalized the subharmonic bifurcations. These devices oHPr exceptional speed of convergence. but even with the recent development of modular electronics, only a limited class of dynamical systems are tractable. The development of the early computing machines ushered in the digital period. Well-known numerical methods were implemented from the start, and graphical (CRT) output began to appear in the literature by 1962. The pioneer papers of lor en^.^ and Stein and Ulam,6 are still studied. By 1967, the Association for Computing Machinery recognized this new field with a symposium entitled “Interactive Systems for Experimental Applied Mathematics.”’ Special systems for experimental math that have evolved since the Culler-Fried device of 1961 are fully described by Smith.8 The current state of the art is now readily available i n the form of a very large general purpose computer with BASIC or APL language and a color video graphics terminal. The currently available terminals of this type are listed in TABLE 1. An equivalent, less expensive system would replace the large computer with a minicomputer. and a fast array processor. Such systems now exist a t several institutions. These devices are extremely flexible, accommodating a very wide class of dynamical systems, but suffer from the cost/resolution quandary: high resolution implies either a vast machine (high capital costs) or long run times (high operating costs.) Our experiences over the past three years with forced oscillation machines in all three categories (direct observation of fluids, analog systems, and digital com-