Alternation of phases of regular and chaotic dynamics is called intermittency. One can see intermittency near tangent bifurcation of window of regular dynamics. Fig.1. shows that for c values when an attracting point merges with repelling one and loses its stability iterations are regular and diverge slowly while x passes through narrow channel. It is assumed that after every laminar phase iterations go into remote regions (where dynamic is chaotic) and then return into the regular corridor (re-injection). The first two pictures show the regular phase in which iterations diverge slowly from the point x = 0 for parameter values near the tangent bifurcation point c_{•} = -1.75 . |
On complex parameter plane tangent bifurcations and intermittency take place near the cusp of every miniature M-set. For tiny M-set with period-3 intermitency takes place at Im(c) = 0, Re(c) > -1.75 . M-sets m7, m8 and m50 correspond to periodic orbits with periods 7, 8, 50. But there is dense set of tiny M-sets and periodic cycles along the ray. |
[1] J.Hanssen, W.Wilcox Lyapunov Exponents for the Intermittent Transition to Chaos