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Qualitative Theory of finite-time and random dynamical systems (QTFRDS)
Start date: Mar 1, 2013, End date: Mar 14, 2015 PROJECT  FINISHED 

This research project aims at developing the qualitative theory of nonautonomous (i.e. time-dependent, random or control) systems in new directions beyond the traditional setting which are highly relevant in the applied science, but surprisingly almost unexplored.The theory of nonautonomous dynamical systems has experienced a renewed and steadily growing interest in the last twenty years, stimulated also by synergetic effects of disciplines which have developed relatively independent for some time such as topological skew product flows, random dynamical systems, finite-time dynamics, and control systems. The importance of nonautonomous dynamical systems is illustrated by the fact that the technological and economical development of our society has generated the need to deal with very complex systems that require an accurate level of understanding. The crisis of the financial markets and weather phenomena associated to climate change such as El Nino, are examples of dynamical processes with a deep economic impact that require sophisticated models to take nonautonomous influences into account.The main challenge in the study of nonautonomous phenomena is to understand the often very complicated dynamical behaviour both as a scientific and mathematical problem. The central aim of this research project is to develop insights and tools in finite-time and random qualitative theory from a mathematical viewpoint which are relevant and have a potentially high impact on the applied sciences. Building upon my success I had during the graduate years from 2006 to 2009 and postdoc since 2009, the proposal contains the following research directions:(i) Invariant manifold theory of finite-time dynamical systems,(ii) Bifurcation theory of finite-time dynamical systems,(iii) Bifurcation theory of random dynamical systems,(iv) Normal form theory of random dynamical systems.
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