Kathleen T. Alligood

Explosions of chaotic sets

Abstract Large-scale invariant sets such as chaotic attractors undergo bifurcations as a parameter is varied. These bifurcations include sudden changes in the size and/or type of the set. An explosion is a bifurcation in which new recurrent points suddenly appear at a non-zero distance from any pre-existing recurrent points. We discuss the following. In a generic one-parameter family of dissipative invertible maps of the plane there are only four known mechanisms through which an explosion can occur: (1) a saddle-node bifurcation isolated from other recurrent points, (2) a saddle-node bifurcation embedded in the set of recurrent points, (3) outer homoclinic tangencies, and (4) outer heteroclinic tangencies. (The term “outer tangency” refers to a particular configuration of the stable and unstable manifolds at tangency.) In particular, we examine different types of tangencies of stable and unstable manifolds from orbits of pre-existing invariant sets. This leads to a general theory that unites phenomena such as crises, basin boundary metamorphoses, explosions of chaotic saddles, etc. We illustrate this theory with numerical examples.

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.

Metamorphoses: Sudden jumps in basin boundaries

In some invertible maps of the plane that depend on a parameter, boundaries of basins of attraction are extremely sensitive to small changes in the parameter. A basin boundary can jump suddenly, and, as it does, change from being smooth to fractal. Such changes are calledbasin boundary metamorphoses. We prove (under certain non-degeneracy assumptions) that a metamorphosis occurs when the stable and unstable manifolds of a periodic saddle on the boundary undergo a homoclinic tangency.

One-Dimensional Maps

THE FUNCTION f(x) = 2x is a rule that assigns to each number x a number twice as large. This is a simple mathematical model. We might imagine that x denotes the population of bacteria in a laboratory culture and that f(x) denotes the population one hour later. Then the rule expresses the fact that the population doubles every hour. If the culture has an initial population of 10,000 bacteria, then after one hour there will be f(10,000) = 20,000 bacteria, after two hours there will be f(f(l0,000)) = 40,000 bacteria, and so on.

Families of periodic orbits: Virtual periods and global continuability

Abstract 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.

Rotation numbers of periodic orbits in the Hénon map

For invertible, area-contracting maps of the plane, it is common for a basin of attraction to have a fractal basin boundary. Certain periodic orbits on the basin boundary are distinguished by being accessible (by a path) from the interior of the basin. A numerical study is made of the accessible periodic orbits for the Hénon family of maps. Theoretical results on rotary homoclinic tangencies are given, which describe the appearance of the accessible saddles, and organize them in a natural way according to the continued fractions expansions of their rotation numbers.

Explosions: global bifurcations at heteroclinic tangencies

We investigate bifurcations in the chain recurrent set for a particular class of one-parameter families of diffeomorphisms in the plane. We give necessary and sufficient conditions for a discontinuous change in the chain recurrent set to occur at a point of heteroclinic tangency. These are also necessary and sufficient conditions for an � -explosion to occur at that point.

Why period-doubling cascades occur: periodic orbit creation followed by stability shedding

Abstract Period-doubling cascades of attractors are often observed in low-dimensional systems prior to the onset of chaotic behavior. We investigate conditions which guarantee that some kinds of cascades must exist.

Chaos in Differential Equations

In the late 1950s, a meteorologist at MIT named Edward Lorenz acquired a Royal-McBee LGP-30 computer. It was the size of a refrigerator carton and contained 16KB of internal memory in a thicket of vacuum tubes. It could calculate at the rate of 60 multiplications per second. For the time, it was a staggering cache of computational power to be assigned to a single scientist.

Stable Manifolds and Crises

WE INTRODUCED the subject of stable and unstable manifolds for saddles of planar maps in Chapter 2. There we emphasized that Poincare used properties of these sets to predict when systems would contain complicated dynamics. He showed that if the stable and unstable manifolds crossed, there was behavior that we now call chaos. For a saddle fixed point in the plane, these “manifolds” are curves that can be highly convoluted. In general, we cannot hope to describe the manifolds with simple formulas, and we need to investigate properties that do not depend on this knowledge. Recall that for an invertible map of the plane and a fixed point saddle p, the stable manifold of p is the set of initial points whose forward orbits (under iteration by the map) converge to p, and the unstable manifold of p is the set whose backward orbits (under iteration by the inverse of the map) converge to p.