Proof Of Chaos Research Articles

Detecting invariant expanding cones for generating word sets to identify chaos in piecewise-linear maps

We show how to formally identify chaotic attractors in continuous, piecewise-linear maps on . For such a map f, this is achieved by constructing three objects. First, is trapping region for f. Second, is a finite set of words that encodes the forward orbits of all points in . Finally, is an invariant expanding cone for derivatives of compositions of f formed by the words in . The existence of , , and C implies f has a topological attractor with a positive Lyapunov exponent. We develop an algorithm that identifies these objects for two-dimensional homeomorphisms comprised of two affine pieces. The main effort is in the explicit construction of and C. Their existence is equated to a set of computable conditions in a general way. This results in a computer-assisted proof of chaos throughout a relatively large region of parameter space. We also observe how the failure of C to be expanding can coincide with a bifurcation of f. Lyapunov exponents are evaluated using one-sided directional derivatives so that forward orbits that intersect a switching manifold (where f is not differentiable) can be included in the analysis.

Estimates of topological entropy of continuous maps with applications

We present a simple theory on topological entropy of the continuous maps defined on a compact metric space, and establish some inequalities of topological entropy. As an application of the results of this paper, we give a new simple proof of chaos in the so-calledN-buffer switched flow networks.

Hysteresis in the trade cycle

The Kaldor model—one of the first nonlinear models of macroeconomics—is modified to incorporate a Preisach nonlinearity. The new dynamical system thus created shows highly complicated behaviour. This paper presents a rigorous (computer aided) proof of chaos in this new model, and of the existence of unstable periodic orbits of all minimal periods p > 57 .

A NEW AND SIMPLE CHAOS TOY

We present two new, and perhaps the simplest yet, mechanical chaos demonstrations. They are both designed based on a recipe of competing nonlinear oscillations. One of these devices is simple enough that using the provided description, it can be built using a bicycle wheel, a piece of wood routed with an elliptical track, and a ball bearing. We provide a thorough Lagrangian mechanics based derivation of equations of motion, and a proof of chaos based on showing the existence of an embedded Smale horseshoe using Melnikov's method. We conclude with discussion of a future application.

CHAOTIC PULSATION IN HUMAN CAPILLARY VESSELS AND ITS DEPENDENCE ON MENTAL AND PHYSICAL CONDITIONS

We found a chaotic pulsation in a finger’s capillary vessels in both normal subjects and psychiatric patients, as well as cardiac chaos. A proof of chaos was made by the reconstruction of the dynamics in phase space and the calculation of the Lyapunov exponents. From the aspect of chaotic information processing, we give a measure of the information storage capacity of the observed chaos. We also found a difference in the topology of capillary and cardiac chaos, and a difference in their dependence on the subject’s conditions.

Analytic proof of chaos in Leggett's equations for3He

The Leggett equations, assumed to govern the magnetization for superfluid3He, are proved to exhibit “deterministic chaos.” As a corollary, this explains why Maki and Ebisawa (1976) were unable to find a complete set of integrals of the motion for the B phase in the presence of a constant magnetic field.