STABILITY OF ORDER: AN EXAMPLE OF HORSESHOES "NEAR" A LINEAR MAP

Abstract

We have been motivated by a question of "anticontrol" of chaos [Schiff et al., 1994], in which recent examples [Chen & Lai, 1997] of controlling nonchaotic maps to chaos have required large perturbations. Can this be done without such brute force? In this paper, we present an example in which a family of maps, Gε, numerically displays a transverse homoclinic point, and hence a horseshoe and chaos, for a fixed value of the parameter ε. We show that these maps converge pointwise to a linear map. Furthermore, a simple scaling conjugacy is shown for a family of maps which even shows geometric similarity of all relevant structures. This is in seeming contradiction to well-known structural stability results concerning horseshoes, but careful consideration reveals that these theorems require convergence in a uniform topology in function space. We show that no such convergence is possible for our family of maps, since it is impossible to find a finite radius disk which contains all of the horseshoes Λε for every ε. Thus, there is no contradiction. Our example may be considered to be a new kind of bifurcation route to chaos by horseshoes, in which rather than creating/destroying a horseshoe by creating/destroying transverse homoclinic points, the horseshoe is sent/brought to/from infinity.