Coupled Maps: Local Bifurcations of Fixed Points Initiate Global Phase Portrait Changes

We present two map examples such that bifurcations of their fixed point which is embedded in a topologically transitive invariant chaotic set can generate global map phase portrait changes. To be more precise, we consider two coupled map families such that the family maps all have the same fixed point which is nested within the same topologically transitive invariant set which is nested in turn within the same invariant subspace. We prove in such a case that these point bifurcations which are transversal to the invariant subspace generate two periodic of period 2 points in a neighbourhood of the given point and besides can simultaneously give rise to orbits that are homoclinic to the periodic points. These orbits appear suddenly and consist of points of transversal intersections of stable manifolds and unstable ones built up at the periodic points. Therefore, at a moment immediately just after the bifurcation, a countable set of periodic points and, moreover, a whole large invariant topologically transitive set appear in a neighbourhood of the invariant set. Thus, in the case under study, a local bifurcation of fixed point initiates a global one of phase portrait of map.