Generalized Hénon Difference Equations with Delay

Charles Conley once said his goal was to reveal the discrete in the continuous. The idea here of using discrete cohomology to elicit the behavior of continuous dynamical systems was central to his program. We combine this idea with our idea of “expanders” to investigate a difference equation of the form xn = F (xn−1, . . . , xn−m) when F has a special form. Recall that the equation xn = q(xn−1) is chaotic for continuous real-valued q that satisfies q(0) < 0, q(1/2) > 1, and q(1) < 0. For such a q, it is also easy to analyze xn = q(xn−k) where k > 1. But when a small perturbation g(xn−1, . . . xn−m) is added, the equation xn = q(xn−k) + g(xn−1, . . . , xn−m) (where 1 < k < m) is far harder to analyze and appears to require degree theory of some sort. We use k-dimensional cohomology to show that this equation has a 2-shift in the dynamics when g is sufficiently small.