Global attractors of non-autonomous dynamical and control systems / by David N Cheban (State University of Moldova, Moldova).

Format:

Book

Author/Creator:

Cheban, David N., author.

Series:

Interdisciplinary mathematical sciences ; v. 18.

Interdisciplinary mathematical sciences, 1793-1355 ; volume 18

Language:

English

Subjects (All):

Attractors (Mathematics).

Differentiable dynamical systems.

Differential equations.

Physical Description:

1 online resource (616 pages).

Edition:

Second edition.

Place of Publication:

New Jersey : World Scientific, [2015]

Language Note:

English

Summary:

The study of attractors of dynamical systems occupies an important position in the modern qualitative theory of differential equations. This engaging volume presents an authoritative overview of both autonomous and non-autonomous dynamical systems, including the global compact attractor. From an in-depth introduction to the different types of dissipativity and attraction, the book takes a comprehensive look at the connections between them, and critically discusses applications of general results to different classes of differential equations. The new Chapters 15–17 added to this edition include some results concerning Control Dynamical Systems - the global attractors, asymptotic stability of switched systems, absolute asymptotic stability of differential/difference equations and inclusions - published in the works of author in recent years.

Contents:

1. Stability of autonomous dynamical systems. 1.1. Some notions, notations and facts from theory of dynamical systems. 1.2. Limit properties of dynamical systems. 1.3. Center of Levinson. 1.4. Dissipative systems on the local compact spaces. 1.5. Criteria of compact dissipativity. 1.6. Local dissipative systems. 1.7. Global attractors. 1.8. On a problem of J. Hale. 1.9. Connectedness of the Levinson's center. 1.10. Weak attractors and center of Levinson. 1.11. Birkhoff's center. 1.12. Asymptotic stability

2. Non-autonomous dissipative dynamical systems. 2.1. On the stability of Levinson's center. 2.2. The positively stable systems. 2.3. Behavior of dissipative dynamical systems under homomorphisms. 2.4. Non-autonomous dynamical systems with convergence. 2.5. Tests for convergence. 2.6. Global attractors of non-autonomous dynamical systems. 2.7. Global attractor of cocycles. 2.8. Global attractors of non-autonomous dynamical system with minimal base. 2.9. Homogeneous dynamical systems. 2.10. Power-law asymptotic of homogeneous systems. 2.11. Linear systems

3. Analytic dissipative systems. 3.1. Skew-product dynamical systems and cocycles. 3.2. [symbol]-analytic systems. 3.3. Converse of Lyapunov's theorem for [symbol]-analytic systems. 3.4. On the structure of compact attracting sets of [symbol]-analytic systems. 3.5. Dynamical systems in spaces of sections. 3.6. Quasi-periodic solutions. 3.7. The analogy of Cameron-Johnson's theorem. 3.8. Almost periodic solutions of the weak nonlinear dissipative systems

4. The structure of the Levinson center of system with the condition of the hyperbolicity. 4.1. The chain recurrent motions. 4.2. The spectral decomposition of the Levinson's center. 4.3. One-dimensional systems with hyperbolic center. 4.4. The dissipative cascades. 4.5. The periodic dissipative systems

5. Method of Lyapunov functions. 5.1. Criteria of dissipativity in term of Lyapunov functions. 5.2. Some criteria of dissipativity of differential equations. 5.3. Theorem of Barbashin-Krasovskii for non-autonomous dynamical systems. 5.4. Equations with convergence. 5.5. Dissipativity and convergence of some equations of 2nd and 3rd order. 5.6. Construction of Lyapunov function for homogeneous systems. 5.7. Differentiable homogeneous systems. 5.8. Global attractors of quasi-homogeneous systems

6. Dissipativity of some classes of equations. 6.1. Difference equations. 6.2. Equations with impulse. 6.3. Convergent periodic equations with impulse. 6.4. Asymptotic stability of linear functional differential equations. 6.5. Convergence of monotone evolutionary equations. 6.6. Global attractors of non-autonomous Lorenz systems

7. Upper semi-continuity of attractors. 7.1. Introduction. 7.2. Maximal compact invariant sets. 7.3. Upper semi-continuity. 7.4. Connectedness. 7.5. Applications

8. The relationship between pullback, forward and global attractors. 8.1. Pullback, forward and global attractors. 8.2. Asymptotic stability in [symbol]-condensing semi-dynamical systems. 8.3. Uniform pullback attractors and global attractors. 8.4. Examples of uniform pullback attractors

9. Pullback attractors of [symbol]-analytic systems. 9.1. [symbol]-analytic cocycles. 9.2. Some general facts about non-autonomous dynamical systems. 9.3. Positively uniformly stable cocycles. 9.4. The compact global pullback attractors of [symbol]-analytic cocycles with compact base. 9.5. The uniform dissipative cocycles with noncompact base. 9.6. The compact and local dissipative cocycles with noncompact base. 9.7. Applications

10. Pullback attractors under discretization. 10.1. Non-autonomous dynamical systems and pullback attractors. 10.2. Non-autonomous quasi-linear differential equation. 10.3. Cocycle property. 10.4. Main result. 10.5. Singleton set-valued pullback attractor case

11. Global attractors of non-autonomous Navier-Stokes equations. 11.1. Non-autonomous Navier-Stokes equations. 11.2. Attractors of non-autonomous dynamical systems. 11.3. Almost periodic and recurrent solutions of non-autonomous Navier-Stokes equations. 11.4. Uniform averaging for a finite interval. 11.5. The global averaging principle for Navier-Stokes equations

12. Global attractors of V-monotone dynamical systems. 12.1. Global attractors of V-monotone NDS. 12.2. On the structure of Levinson center of V-monotone NDS. 12.3. Almost periodic solutions of V-monotone systems. 12.4. Pullback attractors of V-monotone NDS. 12.5. Applications

13. Linear almost periodic dynamical systems. 13.1. Bounded motions of linear systems. 13.2. Bounded solutions of linear equations. 13.3. Finite-dimensional systems. 13.4. Relationship between different types of stability. 13.5. Linear [symbol]-condensing systems. 13.6. Exponential stable systems. 13.7. Linear system with a minimal base. 13.8. Some classes of uniformly exponentially stable equations. 13.9. Linear periodic systems

14. Triangular maps. 14.1. Triangular maps and non-autonomous dynamical systems. 14.2. Linear non-autonomous dynamical systems. 14.3. Quasi-linear non-autonomous dynamical systems. 14.4. Global attractors of quasi-linear triangular systems. 14.5. Almost periodic and recurrent solutions. 14.6. Pseudo-recurrent solutions. 14.7. Chaos in triangular maps

15. Compact global attractors of control systems. 15.1. Upper semi-continuous invariant sections of NDS. 15.2. Discrete inclusions, collages and cocycles. 15.3. Attractors of skew-product systems, collages and cocycles. 15.4. Chaotic attractors of discrete control systems. 15.5. General case. 15.6. Some applications. 15.7. Global attractors of switched systems with continuous time

16. Asymptotic stability of switched systems. 16.1. Switched dynamical systems. 16.2. Asymptotic stability of switched systems with continuous time. 16.3. Some test of asymptotic stability. 16.4. Asymptotic stability of homogeneous switched systems. 16.5. Slow homogeneous switched systems. 16.6. Asymptotic stability of linear switched systems. 16.7. Asymptotic stability of gradient switched systems. 16.8. Asymptotic stability of discrete nonlinear switched systems. 16.9. Homogeneous switched systems with discrete time. 16.10. Asymptotic stability of discrete gradient switched systems

17. Absolute asymptotic stability of differential (difference) equations and inclusions. 17.1. Discrete linear inclusions and cocycles. 17.2. Dynamical system of translations, set-valued cocycles and NDS. 17.3. Non-stationary discrete linear inclusions. 17.4. Absolute asymptotic stability of discrete linear inclusions in Banach spaces. 17.5. Asymptotically compact discrete linear inclusions. 17.6. Absolute asymptotic stability of finite-dimensional discrete linear inclusions. 17.7. Absolute asymptotic stability of discrete linear inclusions in Banach spaces. 17.8. Asymptotic stability of switched systems.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references and index.

Description based on print version record.

ISBN:

981-4619-83-3