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2020 | Buch

Almost Periodic Motions of Dynamical Systems Kapitel lesen Erstes Kapitel lesen

Nonautonomous Dynamics

Nonlinear Oscillations and Global Attractors

verfasst von: Prof. David N. Cheban

Verlag: Springer International Publishing

Buchreihe : Springer Monographs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book emphasizes those topological methods (of dynamical systems) and theories that are useful in the study of different classes of nonautonomous evolutionary equations. The content is developed over six chapters, providing a thorough introduction to the techniques used in the Chapters III-VI described by Chapter I-II.

The author gives a systematic treatment of the basic mathematical theory and constructive methods for Nonautonomous Dynamics. They show how these diverse topics are connected to other important parts of mathematics, including Topology, Functional Analysis and Qualitative Theory of Differential/Difference Equations. Throughout the book a nice balance is maintained between rigorous mathematics and applications (ordinary differential/difference equations, functional differential equations and partial difference equations).

The primary readership includes graduate and PhD students and researchers in in the field of dynamical systems and their applications (control theory, economic dynamics, mathematical theory of climate, population dynamics, oscillation theory etc).

Inhaltsverzeichnis

Frontmatter
Chapter 1. Almost Periodic Motions of Dynamical Systems
Abstract
In the first chapter, for our examination of semigroup dynamical systems, we introduce and study different kinds of Poisson stability of motions and their comparability by the nature of the recurrence relations: Bohr/Levitan almost periodicity, almost automorphy, Bebutov almost recurrence, Birkhoff recurrence, pseudorecurrence, and other types of Poisson stability and the relationships between them.
David N. Cheban
Chapter 2. Compact Global Attractors
Abstract
The second chapter of this book is dedicated to the study of different kinds of dissipativity for dynamical systems (both autonomous and nonautonomous): point, compact, local, bounded, and weak. Criteria for point, compact, and local dissipativity are given. It is shown that for dynamical systems in locally compact spaces, the three types of dissipativity are equivalent. Examples are given showing that in the general case, the notions of point, compact, and local dissipativity are different. The notion of Levinson’s center (the maximal compact invariant set), which is an important characteristic of compact dissipative systems, is introduced.
David N. Cheban
Chapter 3. Analytical Dissipative Systems
Abstract
One of the most studied classes of nonlinear ODEs is the class of \(\mathbb C\)-analytic differential equations, i.e., the equations
$$ \frac{dz}{dt}=f(t, z) , $$
where the right-hand side f is a holomorphic function with respect to a complex variable \( z \in \mathbb C ^{d} \).
David N. Cheban
Chapter 4. Almost Periodic Solutions of Linear Differential Equations
Abstract
The well-known Favard’s theorem states that the linear differential equation
$$\begin{aligned} x^{\prime }=A(t)x+f(t) \end{aligned}$$
with Bohr almost periodic coefficients admits at least one Bohr almost periodic solution if it has a bounded solution.
David N. Cheban
Chapter 5. Almost Periodic Solutions of Monotone Differential Equations
Abstract
This chapter is devoted to studying the problem of the existence of Bohr/Levitan almost periodic, almost automorphic, and Poisson stable solutions of monotone differential equations.
David N. Cheban
Chapter 6. Gradientlike Dynamical Systems
Abstract
In this chapter we give a complete description of the structure of compact global (forward) attractors for nonautonomous perturbations of autonomous gradientlike dynamical systems under the assumption that the original autonomous system has a finite number of hyperbolic stationary solutions. We prove that the perturbed nonautonomous (in particular \(\tau \)-periodic, quasiperiodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff) system has exactly the same number of invariant sections (in particular, perturbed systems have the same number of \(\tau \)-periodic, quasiperiodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff) solutions. It is shown that a compact global (forward) attractor of a nonautonomous perturbed system coincides with the union of unstable manifolds of this finite number of invariant sections.
David N. Cheban
Backmatter
Metadaten
Titel
Nonautonomous Dynamics
verfasst von
Prof. David N. Cheban
Copyright-Jahr
2020
Electronic ISBN
978-3-030-34292-0
Print ISBN
978-3-030-34291-3
DOI
https://doi.org/10.1007/978-3-030-34292-0

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