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Nonautonomous Bifurcation Theory

Final Report Summary - NBT (Nonautonomous bifurcation theory)

The central goal of this research project is the development of the bifurcation theory of nonautonomous (i.e. time-dependent, random or control) systems beyond the traditional setting.

The following main objectives have been set up:
(1) Characterization of bifurcations of low-dimensional nonautonomous dynamical systems.
(2) Study of higher-dimensional bifurcation scenarios.
(3) Development of a bifurcation theory for random systems with bounded noise.
(4) Development of a concept of topological equivalence for nonautonomous dynamical systems; application to nonautonomous normal form theory.

The following main results have been achieved:

(1) Classification of bifurcations in one-dimensional nonautonomous differential equations: analogues of the classical bifurcation patterns.
(2) Development of a numerical scheme to detect controlled heteroclinic orbits; application to a model for ship roll-motion, and to a pendulum coupled to an oscillator.
(3) Development of a concept of exponential attractivity and bifurcation for finite-time systems; application to characterization the corresponding domain of attractions via an analogue of Borg's criterion.
(4) Characterization of the almost periodicity of variational equations of almost periodic equations.
(5) Characterization of discontinuous bifurcations in random systems systems with bounded noise as a collision process of attractors and repellers.
(6) Development of a numerical scheme to approximate invariant sets in random systems with bounded noise; application to the perturbed Henon map.
(7) Development of a concept of topological equivalence for nonautonomous differential equations; extension of conjugacies in the vicinity of hyperbolic solutions to global conjugacies.