James L. Kaplan

The liapunov dimension of strange attractors

Abstract Many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange. In this paper the question of estimating the dimension of the attractor is addressed. While more general conjectures are made here, particular attention is paid to the idea that if the Jacobian determinant of a map is greater than one and a ball is mapped into itself, then generically, the attractor will have positive two-dimensional measure, and most of this paper is devoted to presenting cases with such Jacobians for which the attractors are proved to have non-empty interior.

Preturbulence: A Regime Observed in a Fluid Flow Model of Lorenz*

This paper studies a forced, dissipative system of three ordinary differential equations. The behavior of this system, first studied by Lorenz, has been interpreted as providing a mathematical mechanism for understanding turbulence. It is demonstrated that prior to the onset of chaotic behavior there exists a preturbulent state where turbulent orbits exist but represent a set of measure zero of initial conditions. The methodology of the paper is to postulate the short term behavior of the system, as observed numerically, to establish rigorously the behavior of particular orbits for all future time. Chaotic behavior first occurs when a parameter exceeds some critical value which is the first value for which the system possesses a homoclinic orbit. The arguments are similar to Smales “horseshoe”.

A periodicity threshold theorem for epidemics and population growth

Abstract Some infectious diseases have an incidence which is periodic in time. Contact rates may vary greatly during a year due to seasonal factors. In this paper we study the scalar delay integral equation x(t)= ʃ t t-τ f (s,x(s))ds , where f(t,x) is a continuous function which is periodic in t, f(t,0)=0, and where τ is a positive constant. This equation can be interpreted as a model for some infectious diseases with periodic contact rate, or as a growth equation for a single species population when the birth rate varies seasonally. The principal result is a threshold theorem: it is proved that if τ is small enough, all nonnegative solutions approach 0 as t→∞, whereas if τ is large enough, there exists a positive periodic solution with period equal to the period of f. Estimates on the critical size of τ are given in terms of ∂f ∂x (t,0) . A numerical example is given.

The Lyapunov dimension of a nowhere differentiable attracting torus

The fractal dimension of an attracting torus T k in × T k is shown to be almost always equal to the Lyapunov dimension as predicted by a previous conjecture. The cases studied here can have several Lyapunov numbers greater than 1 and several less than 1

Singular Perturbations of Two-Point Boundary Value Problems Arising in Optimal Control

This paper considers a two-point boundary value problem which arises from an application of the Pontryagin maximal principle to some underlying optimal control problem. The system depends singularly upon a small parameter,

THE ONSET OF CHAOS IN A FLUID FLOW MODEL OF LORENZ

\varepsilon

Nonassociative, real algebras and quadratic differential equations

. It is assumed that there exists a continuous solution of the system when

CHAOTIC BEHAVIOR IN DYNAMICAL SYSTEMS

\varepsilon = 0

A Perturbation Method for the Solution of an Optimal Control Problem Involving Bang–Bang Control

, known as the reduced solution. Conditions are given under which there exists an “outer solution”, and “left and right boundary-layer solutions” whose sum constitutes a solution of the system which degenerates uniformly on compact sets to the reduced solution. The principal tool used in the proof is a Banach space implicit function theorem.

TOWARD A UNIFICATION OF ORDINARY DIFFERENTIAL EQUATIONS WITH NONLINEAR SEMI-GROUP THEORY

There have been many attempts in the scientific literature to provide a mathematical explanation o f the nature of turbulence in fluids, We take the most common approach, arguing that turbulence is a phenomenon of systems of differential equations. The behavior o f the fluid is represented as the solution of a system of ordinary differential equations and the system is assumed to depend on a parameter r; hence