A Global Map of Local Bifurcations

During last decades, vast amount of publications has been devoted both to theoretical and to computational aspects of the bifurcation theory (Nicolis, 1995; Kuznetsov, 1995; Seydel, 1994). However, most of the works have been concentrated on the phenomena occuring in the vicinity of the bifurcation points. The aim of this article is to demonstrate some properties of the 3D continuous dynamical systems possessing periodic attractors. The emphasis is on the behavior of the systems during their evolution from one local bifurcation point to another. The properties under discussion are independent of the number of bifurcation parameters. We will start with introducing the Floquet’ multipliers (FMs), and deriving their relationship to the largest Lyapunov exponent (LLE) and the generalized winding number (GWN) in Section 2. To illustrate the ideas presented in this article, a map of local bifurcations will be constructed in Section 3. The map can be called global, because it encompasses all local bifurcations of the above systems, however, the bifurcations with codimension higher than one are indistinguishable from each other on the map. Using this map, the structure of the manifolds of the periodic solutions in the neighborhood of degenerate bifurcation points will be revealed (Section 4). A derivation of the analytical approximations to the LLE and GWN in period doubling (PD) cascade will be discussed in Section 5 and the conclusions will be drawn in Section 6.