Chaos, Fractals, and Dynamics

P. Fischer

• Editore: CRC Press
• Anno: 2020
• ISBN:
  9781000154221
• Edizione: 1
• Argomento: Mathematics
• Categoria: Mathematics General

This book contains eighteen papers, all more-or-less linked to the theory of dynamical systems together with related studies of chaos and fractals. It shows many fractal configurations that were generated by computer calculations of underlying two-dimensional maps.

PART I

1

Chaostrophes, Intermittency, and Noise

Ralph H. Abraham

Mathematics Board University of California Santa Cruz, California

Dedicated to René Thom

In 1972, we proposed the blue sky catastrophe for periodic limit sets. Here, we describe one for chaotic limit sets. This provides a pathway to chaos quite different from the usual ones, which are all sequences of subtle bifurcations. Further, models for intermittency and noise amplification are given, based on hysteresis loops in a serially coupled chain of dynamical schemes.

PART A. SUBTLE AND CATASTROPHIC BIFURCATIONS

The classification of bifurcations into these two types was suggested in 1966 by Thom(1972), and given explicit treatment (under the names leaps and wobbles) by the author (Abraham, 1976). In this part we have two goals: to define catastrophic borders in the control space of a complex dynamical scheme, and to discuss an example of a chaostrophe, that is, a catastrophic border for the domain of a chaotic attractor, in the context of a serial chain of three oscillators.

A1. PARTITIONS AND BORDERS

We consider a vector field depending upon a parameter, also known as a metabolic field, or dynamical scheme. Let C and M be manifolds of finite dimension, X(M) a space of vectorfields on M, and _F: C + X(M) the dynamical scheme. If B(M) is the subset of X(M) consisting of structurally unstable vectorfields, then the bifurcation set of the scheme, B is the inverse image of B(M) under F.

Imagining the phase portrait of F(c) in {c}xM for each c in c creates a control-phase portrait of F in CxM. We wish to concentrate on the attractors (in the sense of probability, for example) in this portrait, along with their basins and separators (the complements of the basins, elsewhere called separatrices). Let A denote the locus of attraction, the union of all the attractors of the scheme, and denote the locus of separation, the union of all the separators of the scheme.

A relatively open subset of the locus of attraction will be called an attractrix. This is usally called a branch of the attractive surface in static catastrophe theory. A relatively open subset of the locus of separation, similarly, will be called a separatrix. This is also known as a branch of the repelling surface in static catastrophe theory.

We assume that the scheme, is generic in any reasonable sense. Specifically, it is as transversal to B(M) as possible, and over each point b in the bifurcation set, there is a single bifurcation event in the phase portrait of F(b). We see in examples that this bifurcation event normally involves a single attractrix and a single separatrix, or it involves no attractrix. Thus, B may be divided in two parts. Here, we will be interested in the attractrix bifurcations only, in which an attractrix and a separatrix are involved. Further, we will discuss only the hypersurfaces contained in this part of the bifurcation set, which we call the attractrix bifurcation hypersurfaces in the control manifold, C. And finally, these may be isolated hypersurfaces in the bifurcation set, or they may be hypersurfaces of accumulation, from one or both sides. We will refer to one of these isolated attractrix bifurcation hypersurfaces as a border, if an attractor appears or disappears during the bifurcation occuring across it. Otherwise, we call it a partition. The borders belong to the boundaries of the domains of attraction, the regions of control space in which certain attractors exist. These domains are the shadows (images in C under the projection from C x M onto the first factor) of attractrices, and the borders are shadows of boundaries of attractrices. Borders may always be oriented, by a normal vectorfield pointing toward the exterior of the region it bounds. Partitions belong to the interiors of the domains of attraction, and may radiate inward from a border. Precise definitions are given in Section A5.

A2. STANDARD EXAMPLES WITH ONE CONTROL

Specializing the preceding definitions to the case in which the control space, C, is a line or circle, yields the most important examples. Thus, F:C-→X(M) generic arc or generic loop, the bifurcation set, B, is zero-dimensional, every point is a hypersurface, and we fasten attention upon the isolated points at which an attractrix appears or disappears. These are the borders in this context.

In case the dimension of the state space, M, is two, everything is known about the bifurcations of generic arcs. The attractrices correspond to static or periodic attractors. The separatrices are generated by the insets of limit points and cycles of saddle type. The isolated points of the attractrix bifurcation set belong to a known list of possible models, while the accumulation points (also called thick bifurcations, Abraham and Shaw, 1983) ar...