Chaos in abstract kinetics: two prototypes.

Abstract

“Spiral-type chaos” and “screw-type chaos” constitute two simple types of nonperiodic oscillatory flow in 3-variable continuous systems. The former type is exhibited, for example, by auniversal system in the switching mode, when the regimens of flow on the two stable branches of the slow manifold in state space are made to differ in an appropriate manner. Screw-type chaos occurs in ahysteresis oscillator between two stable limit cycles, if the rotation gain is positive. For either case, an analogous 2-dimensional “branched papersheet flow” exists. Both flows are determined by a single-variable discrete dynamical system of the Lorenz-Li-Yorke type (as a cross-section), as well as by an equivalent new map. Numerical simulations of two abstract reaction systems giving rise to non-idealized (that is, truly 3-dimensional) flows of either type are presented. The corresponding discrete dynamical systems (Poincare maps) are 2-dimensional now, having the form of a flattened hairpin (“horseshoe”) in the simplest case. Thus, two actual examples for 3-dimensional flows suspended by a horseshoe diffeomorphism seem to have been found. One contains just a single functionally effective nonlinearity. Real systems of either type may be found in physics, chemistry, biochemistry, biophysics and economy.