Nonautonomous bifurcation patterns for one-dimensional differential equations. (English) Zbl 1125.34032

Rasmussen, Martin

J. Differ. Equations 234, No. 1, 267-288 (2007).

In order to establish an adequate and coherent theory of bifurcations of non-autonomous dynamical systems, the author introduces special notions of past and future attractivity and the same of past and future repulsivity in a general setting, inspired by the previous notions of pullback, forward and uniform attractors. These notions are applied to one-dimensional problems obtaining non-autonomous bifurcation patterns for two canonical examples, extensions of patterns considered in the theory of autonomous dynamical systems: transcritical and pitchfork bifurcation patterns. This is done by studying the following two one-dimensional differential equations put in canonical form:
\[ \dot{x} = a(t, \alpha)x + b(t, \alpha)x^{2} + r(t, x, \alpha), \]
\[ \dot{x} = a(t, \alpha)x + b(t, \alpha)x^{3} + r(t, x, \alpha), \]
where the one-dimensional parameter \(\alpha\) and the variable \(x\) belong to an unbounded interval of \(\mathbb{R}\). Under some very technical conditions on the linearised problems and on \(a(t, \alpha)\), \(b(t, \alpha)\) and \(r(t, x, \alpha)\), he obtains such patterns. What is interesting is the comparative study made by the author with respect to results on the same equations obtained by J.-A. Langa, J.-C. Robinson and A. Suárez [J. Differ. Equations 221, No. 1, 1–35 (2006; Zbl 1096.34026)] where other notions of attractivity are used.
The paper under review contributes to a unified approach to bifurcation patters in non-autonomous differential and difference equations.

Reviewer: Francisco Balibrea (Murcia)

MSC:

34D05	Asymptotic properties of solutions to ordinary differential equations
34C23	Bifurcation theory for ordinary differential equations
37C60	Nonautonomous smooth dynamical systems
37G99	Local and nonlocal bifurcation theory for dynamical systems

Keywords:

non-autonomous differential equation; attractive solution; attractor; repulsive solution; repeller; Non-autonomous bifurcation theory

Citations:

Zbl 1096.34026

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