Motomasa Komuro

The double scroll

A detailed analysis is given of the geometric structure of a chaotic attractor observed from an extremely simple autonomous electrical circuit. It is third order, reciprocal, and has only one nonlinear element: a 3-segment piecewise-linear resistor. Extensive laboratory measurements from this circuit and a detailed geometrical analysis and computer simulation reveal the following rather intricate anatomy of the associated strange attractor. In addition to a microscopically infinite sheet-like composition the attractor has a macroscopic double-scroll structure, i.e., two sheetlike objects are curled up together into spiral forms with infinitely many rotations. (See frontispiece.) The chaotic nature of this circuit is further confirmed by calculating its associated Lyapunov exponents and Lyapunov dimension. The double-scroll attractor has one positive, one zero and one negative Lyapunov exponent. The Lyapunov dimension turns out to be a fractal between 2 and 3 which agrees with the observed structures. The power spectra of the three associated state variables obtained by both measurement and computer simulation show a continuous broad spectrum typical of chaotic systems.

GLOBAL BIFURCATION ANALYSIS OF THE DOUBLE SCROLL CIRCUIT

An in-depth analysis is made of the global 2-parameter bifurcation structures of the double scroll circuit in terms of their homoclinic, heteroclinic, and periodic orbits. Many fine details are uncovered via a 3-dimensional unfolding of the 2-parameter bifurcation structures. Major findings are: (i) The parameter sets which give rise to the homoclinic and heteroclinic orbits (homoclinic and heteroclinic bifurcation sets) studied in this paper are found to be all connected to each other via only one family of periodic orbits. (ii) Moreover, the structure of the windows of this family essentially determines the global structure of the periodic windows of the double scroll circuit. These bifurcation analyses are accomplished by deriving first the relevant bifurcation equations in exact analytic form and then solving these nonlinear equations by iterations. No numerical integration formula for differential equations are used.

MULTIPLE HOMOCLINIC BIFURCATIONS FROM ORBIT-FLIP I: SUCCESSIVE HOMOCLINIC DOUBLINGS

The purpose of this and forthcoming papers is to obtain a better understanding of complicated bifurcations for multiple homoclinic orbits. We shall take one particular type of codimension two homoclinic orbits called orbit-flip and study bifurcations to multiple homoclinic orbits appearing in a tubular neighborhood of the original orbit-flip. The main interest of the present paper lies in the occurrence of successive homoclinic doubling bifurcations under an appropriate condition, which is a part of the entire bifurcation for multiple homoclinic orbits. Since this is a totally global bifurcation, we need the aid of numerical experiments for which we must choose a concrete set of ordinary differential equations that exhibits the desired bifurcation. In this paper we employ a family of continuous piecewise-linear vector fields for such a model equation. In order to explain the cascade of homoclinic doubling bifurcations theoretically, we also derive a two-parameter family of unimodal maps as a singular limit of the Poincare maps along homoclinic orbits. We locate bifurcation curves for this family of unimodal maps in the two-dimensional parameter space, which basically agree with those for the piecewise-linear vector fields. In particular, we show, using a standard technique from the theory of unimodal maps, that there exists an infinite sequence of doubling bifurcations which corresponds to the sequence of homoclinic doubling bifurcations for the piecewise-linear vector fields described above. Since our unimodal map has a singularity at a boundary point of its domain of definition, the doubling bifurcation is slightly different from that for standard quadratic unimodal maps, for instance the Feigenbaum constant associated with the accumulation of the doubling bifurcations is different from the standard value 4.6692.…

Bifurcation equations of continuous piecewise-linear vector fields

This paper provides global equations for certain bifurcation sets of continuous piecewise-linear vector fields. Homoclinic and heteroclinic bifurcations for singular points, and saddle-node, period-doubling and Hopf bifurcations for periodic orbits are studied. The equations are numerically solved to describe the structure of bifurcation sets.

Homoclinic linkage: a new bifurcation mechanism

The observation of homoclinic linkage in the double-scroll circuit is reported. Different homoclinicities are linked together via the same periodic orbit. A discussion of the Shilnikov condition is presented.<<ETX>>

Bifurcations of Continuous Piecewise-Linear Vector Fields

This chapter provides several fundamental theorems for continuous piecewise-linear vector fields. In Section 2.2, we will give a definition of continuous piecewise-linear mappings. And we will give a fundamental representation theorem (Standard Form) for an arbitrary continuous piecewise-linear mapping.

Bifurcations Observed from Electronic Circuits

The purpose of this chapter is to give concrete experimental as well as numerical results of bifurcation phenomena in electronic circuits together with reasonable theoretical justifications. All circuits described are so simple that high school students can build them. All of them except for one, in addition, behave within audible frequencies. We strongly recommend that the reader build at least one of them, and take a look at and listen to the bifurcations. It is a lot of fun.

Fundamental Concepts in Bifurcations

The objective of this chapter is to give some fundamental notions and results in the theory of dynamical systems as well as their bifurcations, which are important in the qualitative study of dynamical systems and are used in the other two chapters, so that readers can become familiar with those important underlying ideas without referring to other textbooks or articles. Since the other chapters mainly deal with vector fields, that is, continuous dynamical systems, the description of this chapter also places more emphasis on continuous dynamical systems than on discrete dynamical systems, although one section is devoted to these discrete systems.