Regularity results for real analytic homotopies

Summary In this paper, we study two main features of the homotopy curves which we follow when we use the homotopy method for solving the zeros of analytic maps. First, we prove that near the solution the curve behaves nicely. Secondly, we prove that the set of starting points which give smooth homotopy curves is open and dense. The second property is of particular importance in computer implementation.This research was supported by the National Science Foundation under Grants MCS 78-02420 (Li) and MCS-7818858 (Mallet-Paret) and MCS-7818221 (Yorke), by the Army Research office under Grant DAAG-29-80-C-0040 (Li and Yorke)     

A scaling law: How an attractor's volume depends on noise level

We investigate the meaning of the dimension of a strange attractor for systems with noise. More specifically, we investigate the effect of adding noise of magnitude ε to a deterministic system with D degrees of freedom. If the attractor has dimension d and d < D, then its volume is zero. The addition of noise may be an important physical probe for experimental situations, useful for determining how much of the observed phenomena in a system is due to noise already present. When the noise is added, the attractor A_ε has positive volume. We conjecture that the generalized volume of A_ε is proportional to ε_{D ? d} for ε near 0 and show this relationship is valid in several cases. For chaotic attractors there are a variety of ways of defining d and the generalized volume definition must be chosen accordingly.     

Families of periodic orbits: Virtual periods and global continuability

For a differential equation depending on a parameter, there have been numerous investigations of the continuation of periodic orbits as the parameter is varied. Mallet-Paret and Yorke investigated in generic situations how connected components of orbits must terminate. Here we extend the theory to the general case, dropping genericity assumptions.     

Period doubling cascades of attractors: A prerequisite for horseshoes

This paper shows that if a horseshoe is created in a natural manner as a parameter is varied, then the process of creation involves the appearance of attracting periodic orbits of all periods. Furthermore, each of these orbits will period double repeatedly, with those periods going to infinity.In memory of Charles ConleyResearch partially supported by the National Science FoundationResearch partially supported by the Air Force Office of Scientific ResearchThe paper was written while Dr. Alligood was visiting Michigan State University and the University of Maryland     

Exterior dimension of fat fractals

Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which we call the exterior dimension. In addition, it is shown that the exterior dimension is related to the “uncertainty exponent” previously used in studies of fractal basin boundaries, and it is shown how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.