Final Report Summary - QTFRDS (Qualitative Theory of finite-time and random dynamical systems)
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, which is partially due to the failure of instruments to cope with volatility, 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. One of the most basic question is to detect the critical point when the dynamical behaviour of systems before and after this point are different. All of the results in this project aim to find the mathematical answer of this question, i.e. to develop the stochastic bifurcation theory. Precisely, the main results in this project are:
The first main result of this project is to establish the characterization of stochastic bifurcation in terms of the break of topological equivalence. This work can be considered as the first fundamental step to build the whole theory of stochastic bifurcation.
The second main result of this project is to contribute to the normal form theory of nonautonomous differential equations. This work enable us to find the most simples form of the systems under an equivalent relation and hence is very important for understanding bifurcation phenomena of nonautonomous systems.
In summary, the result of the project has been published in the high level international peer-review journal articles:
+ Doan Thai Son, Martin Rasmussen and Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to mean-square attractor. Discrete and Continuous Dynamical Systems Series B 20 (2015), no. 3, 875-887.
+ Nguyen Dinh Cong, Doan Thai Son and Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete and Continuous Dynamical Systems Series B 20 (2015), no. 3, 861-874.
+ Patrick Bonckaert, Peter De Maesschalck, Doan Thai Son and Stefan Siegmund. Partial linearization for planar nonautonomous differential equations. Journal Differential Equations 258 (2015), no. 5, 1618-1652.
+ Doan Thai Son, Martin Rasmussen and Kennth Palmer. The Bohl spectrum for nonautonomous differential equations. arXiv:1507.01230
+ Mark Callaway, Thai Son Doan, Jeroen S.W. Lamb, Martin Rasmussen. The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise. . arXiv:1310.6166.
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, which is partially due to the failure of instruments to cope with volatility, 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. One of the most basic question is to detect the critical point when the dynamical behaviour of systems before and after this point are different. All of the results in this project aim to find the mathematical answer of this question, i.e. to develop the stochastic bifurcation theory. Precisely, the main results in this project are:
The first main result of this project is to establish the characterization of stochastic bifurcation in terms of the break of topological equivalence. This work can be considered as the first fundamental step to build the whole theory of stochastic bifurcation.
The second main result of this project is to contribute to the normal form theory of nonautonomous differential equations. This work enable us to find the most simples form of the systems under an equivalent relation and hence is very important for understanding bifurcation phenomena of nonautonomous systems.
In summary, the result of the project has been published in the high level international peer-review journal articles:
+ Doan Thai Son, Martin Rasmussen and Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to mean-square attractor. Discrete and Continuous Dynamical Systems Series B 20 (2015), no. 3, 875-887.
+ Nguyen Dinh Cong, Doan Thai Son and Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete and Continuous Dynamical Systems Series B 20 (2015), no. 3, 861-874.
+ Patrick Bonckaert, Peter De Maesschalck, Doan Thai Son and Stefan Siegmund. Partial linearization for planar nonautonomous differential equations. Journal Differential Equations 258 (2015), no. 5, 1618-1652.
+ Doan Thai Son, Martin Rasmussen and Kennth Palmer. The Bohl spectrum for nonautonomous differential equations. arXiv:1507.01230
+ Mark Callaway, Thai Son Doan, Jeroen S.W. Lamb, Martin Rasmussen. The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise. . arXiv:1310.6166.