Two-Step Transition in Nonautonomous Bifurcations: An Explanation

Abstract

Dedictated to Ludwig Arnold on the occasion of his 65th birthdayWe consider the two-step bifurcation scenario which has been studied by L. Arnold and his co-workers. We formulate a "continuous case" and a "measurable case" of the scenario, and present results and conjectures regarding sufficient conditions that it take place.

Negatively Invariant Sets and Entire Solutions

“…There are at present few general, theoretical results about bifurcations in nonautonomous dynamical systems, e.g., [7,8,10,11,14,15,16]. The above results allow us to make a preliminary investigation to show that what could be considered to be a bifurcation has occurred.…”

A Biography of Russell A. Johnson

“…Such a generic property was shown by Russell with Yi [48] to play an important role in studying generic, intermittent Hopf bifurcations from invariant tori when a family of spectral intervals cross zero as the parameters vary. This study was later extended by Russell [38] and also by Russell with Kloeden and Pavani [61] to random, two-step bifurcations. In two recent works [5,13] of Russell with Fabbri and Zampogni, the generic property of exponential dichotomy for trace-free, quasiperiodic linear systems is shown to be closely related to the density of positive Lyapunov exponents.…”

Attractivity and Bifurcation for Nonautonomous Dynamical Systems

“…[66] and FABBRI & JOHNSON [65], for one-dimensional nonautonomous differential equations with strictly ergodic time dependence (e.g., quasi-periodic equations are of this type), attractor-repeller bifurcations are considered. Bifurcations of attractors and repellers are also studied in JOHNSON & KLOEDEN & PAVANI [84] and JOHNSON [83] for deterministic counterparts of the Two-Step-BifurcationPattern. These considerations are based on the studies of Ludwig Arnold and his coworkers in the context of stochastic differential equations (see ARNOLD [5]).…”

“…also Subsection 2.2.3). Nonautonomous bifurcation theory is a new branch which has been developed quite independently for topological skew product flows (see Fabbri, Johnson, Kloeden, Mantellini [66,84,85,86]; cf. also Subsection 2.4.2) and random dynamical systems (see Arnold, Sri Namachchivaya, Schenk-Hoppé [6,8,155,175]; cf.…”