Spectral Theory of Nonautonomous Dynamical Systems and Applications

The main challenge in the study of nonautonomous phenomena is to understand the very complicated dynamical behaviour both as a scientific and mathematical problem. 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, random dynamical systems, finite-time dynamics and control systems. The book provides new insights in many aspects of the qualitative theory of nonautonomous dynamical systems including the spectral theory, the linearization theory, the bifurcation theory.

The book first introduces several important spectral theorem for nonautonomous differential equations including the Lyapunov spectrum, Sacker-Sell spectrum and finite-time spectrum. The author also establishes the smooth linearization and partial linearization for nonautonomous differential equations in application part. Then the second part recalls the multiplicative ergodic theorem for random dynamical systems and discusses several explicit formulas in computing the Lyapunov spectrum for random dynamical systems generated by linear stochastic differential equations and random difference equations with random delay. In the end, the Pitchfork bifurcation and Hopf bifurcation with additive noise are investigated in terms of change of the sign of Lyapunov exponents and loss of topological equivalence.

This book might be appealing to researchers and graduate students in the field of dynamical systems, stochastic differential equations, ergodic theory.

Provides a very complete picture of the spectral theory for nonautonomous dynamical systems Provides new insights in the theory of linearization for nonautonomous dynamical systems Presents some important ideas and results to develop a theory on stochastic bifurcation which is not well-established

Autorentext

Thai Son Doan received the B.Sc. degree in mathematics from Hanoi University of Science, Viet Nam in 2006 and received the Ph.D degree in mathematics from Technical University Dresden, Germany in 2009. He is a senior researcher at the Institute of Mathematics, Vietnam Academy of Science and Technology. He was a research fellow at Imperial College London, United Kingdom from 2011 to 2015, and a research fellow at Hokkaido University, Japan from 2015 to 2017. He has held several prestigious research fellowships including Marie-Curie Intra European Fellowships (2013-2015) and Marie Sklodowska-Curie Fellowships (2022-2024) at Imperial College London, United Kingdom and Japan Society for the Promotion of Science Postdoctoral Fellowships for Research (2015-2017).

Thai Son Doan's research interest covers nonautonomous dynamical systems. He has published more than 50 publications in this area. Many of his publications are in the leading journals in the field of differentialequations & dynamical system and control theory.

Inhalt

chapter 1 spectral theory of nonautonomous differential equations.- chapter 2 linearization for nonautonomous differential equations.- chapter 3 spectral theory for random dynamical systems.- chapter 4 genericity of lyapunov spectrum of random dynamical systems.- chapter 5 pitchfork and hopf bifurcation under additive noise.