Topological dynamics and differential equations

Abstract

By reviewing our previous works on lifting dynamics in skew-product semi-flows and also the work of Johnson on almost periodic Floquet theory, we show several significant applications of the abstract theory of topological dynamics to the qualitative study of non-autonomous differential equations. The paper also contains some detailed discussions on a conjecture of Johnson.

Almost periodic dynamics of perturbed infinite-dimensional dynamical systems

“…Particularly, the ω-limit sets of almost periodic equations have been studied by the authors of [7,8,9,10,11].…”

Attractivity and Bifurcation for Nonautonomous Dynamical Systems

“…The notion of nonautonomous dynamical system has emerged in the late 1990s as an abstraction of both continuous skew product flows (see, e.g., MILLER [119] and SELL [164,165,166]) and random dynamical systems (see, e.g., the monograph ARNOLD [5]). The definition is given as follows (see also the conference proceedings COLONIUS & KLOEDEN & SIEGMUND [51]).…”

“…This can be avoided for a special class of right hand sides f by considering the Bebutov flow on the hull of f (see, e.g., BEBUTOV [26] and SELL [166]). …”

“…Markus discusses properties of nonautonomous ω-limit sets and generalizes the THEOREM OF POINCARÉ & BENDIXSON (see, e.g., PALIS & DE MELO [123] and HIRSCH & SMALE [80,Chapter 11]) to asymptotically autonomous planar systems. His work has stimulated the qualitative theory of nonautonomous differential equations (see, e.g., SELL [164,165,166]). Further fundamental work on asymptotically autonomous systems was achieved by STRAUSS & YORKE [177], ARTSTEIN [10,11], THIEME [178] and MISCHAIKOW & SMITH & THIEME [120] in the context of differential equations (see also KATO & MARTYNYUK & SHESTAKOV [89]); for difference equations, we refer to SCHÖNEFUSS [162].…”

“…The notion of nonautonomous dynamical system was created in the 1990s from the studies of both topological skew product flows and random dynamical systems. The theory of topological skew product flows was founded in the late 1960s by George R. Sell and Richard K. Miller (see [119,164,165,166]), and the notion of the random dynamical system is based on research by Baxendale, Bismut, Elworthy, Ikeda, Kunita, Watanabe and many others (see [25,32,64,82,99]). Further progress in this field was achieved by Ludwig Arnold and his "Bremen Group".…”