Compound windows of the Hénon-map

Abstract

For the two-parameter second-order Hénon map, the shapes and locations of the periodic windows–continua of parameter values for which solutions x 0 , x 1 , … can be stably periodic, embedded in larger regions where chaotic solutions or solutions of other periods prevail–are found by a random searching procedure and displayed graphically. Many windows have a typical shape, consisting of a central “body” from which four narrow “antennae” extend. Such windows, to be called compound windows, are often arranged in bands, to be called window streets, that are made up largely of small detected but poorly resolved compound windows. For each fundamental subwindow–the portion of a window where a fundamental period prevails–a stability measure U is introduced; where the solution is stable, | U | < 1 . Curves of constant U are found by numerical integration. Along one line in parameter space the Hénon-map reduces to the one-parameter first-order logistic map, and two antennae from each compound window intersect this line. The curves where U = 1 and U = − 1 that bound either antenna are close together within these intersections, but, as either curve with U = − 1 leaves the line, it diverges from the curve where U = 1 , crosses the other curve where U = − 1 , and nears the other curve where U = 1 , forming another antenna. The region bounded by the numerically determined curves coincides with the subwindow as found by random searching. A fourth-degree equation for an idealized curve of constant U is established. Points in parameter space producing periodic solutions where x 0 = x m = 0 , for given values of m , are found to lie on Cantor sets of curves that closely fit the window streets. Points producing solutions where x 0 = x m = 0 and satisfying a third condition, approximating the condition that x n be bounded as n → − ∞ , lie on curves, to be called street curves of order m , that approximate individual members of the Cantor set and individual window streets. Compound windows of period m + m ′ tend to occur near the intersections of street curves of orders m and m ′ . Some exceptions to what appear to be fairly general results are noted. The exceptions render it difficult to establish general theorems.