Degree theory and symmetric equations assisted by GAP system : with a special focus on systems with hysteresis / Zalman Balanov, [and four others].

Format:

Book

Author/Creator:

Balanov, Zalman, 1959- author.

Krawcewicz, Wiesław, author.

Rachinskii, Dmitrii, author.

Series:

Mathematical Surveys and Monographs

Mathematical Surveys and Monographs ; v.286

Subjects (All):

Topological degree--Data processing.

Topological degree.

Symmetry (Mathematics)--Data processing.

Symmetry (Mathematics).

Hysteresis--Data processing.

Hysteresis.

Bifurcation theory--Data processing.

Bifurcation theory.

Physical Description:

1 online resource (671 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2025]

Summary:

Symmetries are a common feature of real-world phenomena in many fields, including physics, biology, materials science, and engineering. They can help understand the behavior of a system and optimize engineering designs. Nonlinear effects such as delays, nonsmoothness, and hysteresis can have a significant impact on the dynamics and contribute to the increased complexity of symmetric systems. The goal of this book is to provide a complete theoretical and practical manual for studying a large class of dynamical problems with symmetries using degree theory methods. To study the impact of symmetries on the occurrence of periodic solutions in dynamical systems, special variants of the Brouwer degree, the Brouwer equivariant degree, and the twisted equivariant degree are developed to predict patterns, regularities, and symmetries of solutions. Applications to specific dynamical systems and examples are supported by a software package integrated with the GAP system, which provides assistance in the group-theoretic computations involved in equivariant analysis. This book is intended for readers with a basic knowledge of analysis and algebra, including researchers in pure and applied mathematical analysis, graduate students, and scientists interested in areas involving mathematical modeling of symmetric phenomena. The text is self-contained, and the necessary background material is provided in the appendices.

Contents:

Cover

Title page

Preface

Acknowledgments

Chapter 1. Introduction

1.1. Brouwer degree and symmetric equations

1.2. Equivariant degree

1.2.1. Equivariant Brouwer degree

1.2.2. Twisted equivariant degree

1.3. Application scheme

1.3.1. Applications of the equivariant Brouwer degree

1.3.2. Applications of twisted equivariant degree

1.4. Content of the book

Part 1. Brouwer Equivariant Degree and Applications

Chapter 2. Local Brouwer Degree

2.1. Properties of local Brouwer degree

2.2. Degree of linear isomorphisms and analytic formula for degree

2.3. Existence and uniqueness of degree

2.4. Brouwer degree on manifolds

2.5. Concept of a crossing number

2.6. Additional useful properties of degree

2.7. Degree of maps between spheres

2.8. Leray-Schauder extension of Brouwer degree

2.9. Problems for Chapter 2

Chapter 3. Equivariant Brouwer Degree

3.1. Equivariant jargon and three basic concepts

3.2. Definition of Brouwer -equivariant degree

3.3. Burnside ring: definition

3.4. Burnside ring: examples

3.5. Properties of the Brouwer equivariant degree

3.6. Computations of equivariant degree for linear -isomorphisms: reduction to basic degrees

3.7. Properties of basic degrees

3.8. Leray-Schauder -equivariant degree

3.9. Problems for Chapter 3

Chapter 4. Subharmonic Solutions to Reversible Difference Equations

4.1. Introduction

4.2. Main results: formulations and underlying ideas

4.3. Step 1: setting problem (4.1) as an operator equation

4.4. Step 2: a priori bounds

4.5. Step 3: -isotypic decomposition of ℋ

4.6. Step 4: computational formula

4.7. Step 5: proof of Theorem 4.3

4.8. Step 6: proof of Theorem 4.5

4.9. Problems for Chapter 4

Chapter 5. Periodic Solutions to -Reversible Continuous Time Systems with Multiple Delays.

5.1. Introduction

5.2. Autonomous systems

5.2.1. Statement of the problem

5.2.2. Normalization of period and a priori bounds

5.2.3. Operator reformulation in functional spaces

5.2.4. Abstract equivariant degree based result

5.2.5. Computation of \gdeg(\scrA, (\scrE)): reduction to basic degrees

5.2.6. Maximal orbit types in products of basic -degrees

5.2.7. Main results and examples

5.3. Non-autonomous systems

5.3.1. Statement of the problem

5.3.2. A priori bounds

5.3.3. Setting system (5.56) in functional spaces

5.3.4. Computation of and main results

5.3.5. Examples

5.4. Effect of domain for autonomous systems

5.4.1. Statement of the problem

5.4.2. A priori bounds and -touching

5.4.3. Operator reformulation and related homotopies

5.4.4. Main result and example

5.5. Problems for Chapter 5

Chapter 6. Equivariant Bifurcation of Periodic Solutions with Fixed Period

6.1. Introduction

6.2. Abstract equivariant bifurcation

6.2.1. Kuratowski's lemma

6.2.2. Local Krasnosel'skii's bifurcation theorem

6.2.3. Global continuation and Rabinowitz alternative

6.3. Bifurcation of subharmonic solutions

6.3.1. Local result

6.3.2. Global result and examples

6.4. Bifurcation of periodic solutions in symmetric -reversible systems

6.4.1. Local equivariant bifurcation in (6.27)

6.4.2. Global equivariant bifurcation in (6.29)

6.5. Problems for Chapter 6

Chapter 7. Non-Radial Solutions to Coupled Semilinear Elliptic Systems on a Disc

7.1. Introduction

7.2. Functional spaces reformulation and a priori bounds

7.3. Existence results

7.4. Semilinear elliptic systems on a disc with additional symmetries

7.4.1. Symmetrically interacting systems

7.4.2. Equivariant setting in functional spaces

7.4.3. Main results

7.4.4. Example.

7.5. Problems for Chapter 7

Part 2. Twisted Equivariant Degree and Applications

Chapter 8. Local ¹-Equivariant Degree

8.1. Properties of local ¹-equivariant degree

8.2. Paradigmatic examples

8.3. Proof of Theorem 8.1

8.4. Computational formulas for the ¹-equivariant degree

8.5. First application: bifurcation of relative equilibria

8.6. Problems for Chapter 8

Chapter 9. Local Twisted Equivariant Degree

9.1. From ℤ-module ₁( ¹) to (Γ)-module ₁^{ }(Γ× ¹)

9.2. Example: ( ₆)-module ₁^{ }( ₆× ¹)

9.3. Definition of twisted -equivariant degree

9.4. Properties of the twisted -equivariant degree

9.5. Basic degrees and related computational formulas

9.5.1. Basic maps

9.5.2. Computational formula for twisted degree

9.6. Leray-Schauder -equivariant twisted degree

9.7. Problems for Chapter 9

Chapter 10. Two Parameter -Equivariant Bifurcation

10.1. Abstract results for two parameter bifurcation

10.2. Local bifurcation

10.3. Global bifurcation

10.4. Bifurcation from a cluster of critical points

10.5. Computation of local bifurcation invariant

10.6. Problems for Chapter 10

Chapter 11. Hopf Bifurcation

11.1. Hopf bifurcation in symmetric ODEs

11.1.1. Reformulation of (11.4) in functional spaces

11.1.2. Local Hopf bifurcation results

11.1.3. Crossing numbers for (11.4)

11.1.4. Computation of non-zero coefficients in (11.17)

11.1.5. Global Hopf bifurcation result

11.2. Special cases of local Hopf bifurcation

11.2.1. Change of stability and Hopf bifurcation

11.2.2. Simple symmetric configurations of identical oscillators

11.3. Hopf bifurcation in interval systems

11.3.1. Interval polynomials, Kharitonov's theorem and Descartes' criterion

11.3.2. Interval systems

11.3.3. Examples

11.4. Symmetric dynamical systems with memory.

11.4.1. Reformulation of (11.60) as a two-parameter bifurcation problem

11.4.2. Equivariant bifurcation invariant

11.5. Problems for Chapter 11

Chapter 12. Hopf Bifurcation of Relative Periodic Solutions

12.1. Some bibliographical remarks

12.2. Γ× ¹-symmetric systems of FDEs

12.2.1. Notation and statement of the problem

12.2.2. Symmetric bifurcation of relative equilibria from an equilibrium

12.2.3. Hopf bifurcation from a relative equilibrium

12.2.4. Proof of Theorem 12.9

12.3. DDE model of a symmetric configuration of passively mode-locked semiconductor lasers

12.3.1. Mathematical model

12.3.2. _{ }-configuration of identical semiconductor lasers

12.3.3. Bifurcation of symmetric relative equilibria

12.3.4. Bifurcation of relative periodic solutions

12.4. Problems for Chapter 12

Chapter 13. Global Hopf Bifurcation of Differential Equations with Threshold Type State-Dependent Delay by Qingwen Hu

13.1. Some bibliographical remarks

13.2. Local Hopf bifurcation

13.3. Global Hopf bifurcation

13.4. Mechanical model with discrete and distributed delays

13.5. A priori bounds of periodic solutions

13.6. Bounds of periods and critical points

13.6.1. Case ℎ&gt

0

13.6.2. Case ℎ&lt

13.7. Global Hopf bifurcation of turning processes

13.8. Concluding remarks

13.9. Problems for Chapter 13

Chapter 14. Hysteresis Models as Rate-Independent Operators

14.1. Hysteresis operators

14.1.1. Hysteresis operators: definition and properties

14.1.2. Arbitrary time intervals

14.1.3. Classification

14.1.4. Bibliography

14.2. Preisach model

14.2.1. Opening remarks

14.2.2. Non-ideal relay

14.2.3. Preisach operator as a superposition of relays

14.2.4. Geometric description of dynamics of state.

14.2.5. Constructive formal definition of input-state-output operators

14.2.6. Monocyclic property and congruency of hysteresis loops

14.2.7. Memory structure of the Preisach model

14.2.8. Proof of Lemma 14.8

14.3. Periodic solutions to differential systems with Preisach operator

14.3.1. Closed systems with hysteresis operators

14.3.2. Initial value problem

14.3.3. Periodic solutions

14.3.4. Hopf bifurcation problem

14.3.5. Inverse Preisach operator

14.4. Local equivariant Hopf bifurcation theorem

14.5. Prandtl-Ishlinskii model

14.5.1. Stop hysteron

14.5.2. Prandtl-Ishlinskii operator

14.5.3. The inverse operator

14.6. Problems for Chapter 14

Chapter 15. Hopf Bifurcations in Systems of Symmetrically Coupled Oscillators with Hysteretic Elements

15.1. Hopf bifurcation in a van der Pol oscillator with a hysteretic element

15.2. Hopf bifurcation in ₄-symmetric cube-like configuration of van der Pol oscillators with ferromagnetic elements

15.2.1. Electrical circuit model

15.2.2. ₄-equivariant Hopf bifurcation

15.2.3. Γ-equivariant system of coupled van der Pol oscillators with hysteresis

15.2.4. Proof of Theorem 15.2

15.3. Hopf bifurcation of relative periodic solutions to symmetric systems with hysteresis

15.3.1. Relative equilibria

15.3.2. Relative periodic solutions

15.3.3. Hopf bifurcation of relative periodic solutions from a relative equilibrium

15.3.4. Γ× ¹-equivariant Hopf bifurcation of relative periodic solutions from a relative equilibrium

15.3.5. An ¹× ¹-equivariant electro-mechanical oscillator

15.3.6. A symmetrically coupled system of electro-mechanical oscillators

15.3.7. A symmetrically coupled system of hysteretic electro-mechanical oscillators

15.3.8. ₈× ¹-equivariant Hopf bifurcation of relative periodic solutions.

15.4. Branches of periodic solutions of systems with hysteresis (without parameters).

Notes:

Includes bibliographical references and index.

Description based on publisher supplied metadata and other sources.

Description based on print version record.

ISBN:

9781470479619

1470479613