## Equations for strongly time-dependent and random systems

Highly time-dependent and random systems are important in nature and engineering. The mathematics needed to describe them deserves more attention.

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In mathematics, a non-autonomous system is a system of differential equations that explicitly depends on the independent variable, which is often time. They are often relevant in applied science, but have been little explored.

The QTFRDS (Qualitative theory of finite-time and random dynamical systems) project aimed at developing the qualitative theory of non-autonomous (i.e. time-dependent, random or control) dynamical systems. The theory has experienced a renewed and steadily growing interest in the last 20 years, stimulated by the synergetic effects of disciplines that have developed relatively independently. These include topological skew product flows, random dynamical systems, finite-time dynamics and control systems.

Technological and economic developments have generated the need to deal with very complex systems. The crises of financial markets, partially due to the failure of models to cope with volatility, and weather phenomena associated with climate change, such as El Nino, are examples of dynamical processes with a deep economic impact that require better mathematics to take non-autonomous influences into account.

The main challenge in the study of non-autonomous phenomena is to understand the often very complicated dynamical behaviour both as a scientific and mathematical problem. One of the most basic questions is to detect critical points where the dynamical behaviour of systems changes. This project aimed to find the mathematical answer to this question – i.e. to develop the stochastic bifurcation theory.

A first significant result was the characterisation of stochastic bifurcation in terms of breaks of topological equivalence. This is a fundamental step towards a new theory of stochastic bifurcation.

The second main result of this project has been contributions to the normal form theory of non-autonomous differential equations. These permit the simplest form of the systems under an equivalent relation to be identified and will be important for the understanding of bifurcation phenomena of non-autonomous systems.

The QTFRDS (Qualitative theory of finite-time and random dynamical systems) project aimed at developing the qualitative theory of non-autonomous (i.e. time-dependent, random or control) dynamical systems. The theory has experienced a renewed and steadily growing interest in the last 20 years, stimulated by the synergetic effects of disciplines that have developed relatively independently. These include topological skew product flows, random dynamical systems, finite-time dynamics and control systems.

Technological and economic developments have generated the need to deal with very complex systems. The crises of financial markets, partially due to the failure of models to cope with volatility, and weather phenomena associated with climate change, such as El Nino, are examples of dynamical processes with a deep economic impact that require better mathematics to take non-autonomous influences into account.

The main challenge in the study of non-autonomous phenomena is to understand the often very complicated dynamical behaviour both as a scientific and mathematical problem. One of the most basic questions is to detect critical points where the dynamical behaviour of systems changes. This project aimed to find the mathematical answer to this question – i.e. to develop the stochastic bifurcation theory.

A first significant result was the characterisation of stochastic bifurcation in terms of breaks of topological equivalence. This is a fundamental step towards a new theory of stochastic bifurcation.

The second main result of this project has been contributions to the normal form theory of non-autonomous differential equations. These permit the simplest form of the systems under an equivalent relation to be identified and will be important for the understanding of bifurcation phenomena of non-autonomous systems.