Kennedy, Judy A.; Yorke, James A. Generalized Hénon difference equations with delay. (English) Zbl 1058.39014 Zesz. Nauk. Uniw. Jagiell. 1269, Univ. Iagell. Acta Math. 41, 9-28 (2003). 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. Cited in 3 Documents 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 Keywords:chaos; sensitivity to initial conditions; discrete cohomology; dynamical systems; Hénon difference equations with delay PDFBibTeX XMLCite \textit{J. A. Kennedy} and \textit{J. A. Yorke}, Zesz. Nauk. Uniw. Jagiell., Univ. Iagell. Acta Math. 1269(41), 9--28 (2003; Zbl 1058.39014) Full Text: EuDML