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Description
This book focuses on dissipative Caputo fractional differential equations (FDEs) with an autonomous vector field. The introduction of Caputo FDEs in the 1960s allowed initial value problems to be handled more naturally and the asymptotic behaviour of models based on them to be investigated by researchers. More recently, mathematically defined dynamical systems generated by Caputo FDEs and their attractors have been introduced.
Dissipative Caputo FDEs have vector fields which satisfy a dissipativity property. For ordinary differential equations (ODEs) it follows from such a property that an absorbing set exists which contains all the long-term dynamical behaviour of the system such as the existence of an attractor. The situation is more complicated for Caputo FDEs, since these are essentially integral equations, and the dissipative inequalities cannot be so easily exploited. Moreover, such integral equations are essentially nonautonomous due to the form of the kernel in the integral equations, even when the vector field is "autonomous," i.e., does not depend explicitly on time.
The book is based on recent results of the three coauthors in various combinations with each other and with their other coauthors, in particular Nguyen Dinh Cong and Hieu Trinh. The main aim is to develop and present a theory of dynamical systems and their attractors for Caputo FDEs.
Preface.- Introduction.- Useful Inequalities.- Existence and Uniqueness of Solutions.- Linear Systems of Caputo FDEs.- Asymptotic Stability and instability.- Lyapunov Functions.- Non-Intersecting Solutions.- Caputo Dynamical Systems.- Dissipative Caputo FDEs.- Attractors of Scalar Caputo FDEs.- Attractors of Caputo Semi-Groups.- References.
Peter E. Kloeden is retired Chair of Applied and Instrumental Mathematics is the Goethe University Frankfurt and is now a visiting researcher at the University of Tübingen. His research interests include analysis and numerics of random and nonautonomous systems and their applications. He is a Fellow of SIAM and was awarded the W. T. and Idalia Reid Prizefrom SIAM in 2006. In 2014, he received a Thousand Talents Award from the government of China. He served as co-editor-in-chief of Discrete and Continuous Dynamical Systems, Series B. He is the co-author of best selling books on stochastic numerics, numerical dynamics, nonautonomous dynamical systems, lattice systems, fuzzy metric spaces, etc.
