×

Generalized Hénon difference equations with delay. (English) Zbl 1058.39014

Summary: 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 \(x_n= F(x_{n-1},\dots,x_{n-m})\) when \(F\) has a special form. Recall that the equation \(x_n=q(x_{n-1})\) 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 \(x_n=q(x_{n-k})\) where \(k>1\). But when a small perturbation \(g(x_{n-1}, \dots,x_{n-m})\) is added, the equation \[ x_n= q(x_{n-k})+g(x_{n-1},\dots, x_{n,m}) \] (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.

MSC:

39A12 Discrete version of topics in analysis
37B10 Symbolic dynamics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57N65 Algebraic topology of manifolds
PDFBibTeX XMLCite
Full Text: EuDML