This book is an introduction to the qualitative theory of dynamical systems on manifolds of low dimension (on the circle and on surfaces). Along with classical results, it reflects the most significant achievements in this area obtained in recent times by Russian and foreign mathematicians whose work has not yet appeared in the monographic literature. The main stress here is put on global problems in the qualitative theory of flows on surfaces.
Despite the fact that flows on surfaces have the same local structure as flows on the plane, they have many global properties intrinsic to multidimensional systems. This is connected mainly with the existence of nontrivial recurrent trajectories for such flows. The investigation of dynamical systems on surfaces is therefore a natural stage in the transition to multidimensional dynamical systems.
The reader of this book need be familiar only with basic courses in differential equations and smooth manifolds. All the main definitions and concepts required for understanding the contents are given in the text.
The results expounded can be used for investigating mathematical models of mechanical, physical, and other systems (billiards in polygons, the dynamics of a spinning top with nonholonomic constraints, the structure of liquid crystals, etc.).
In our opinion the book should be useful not only to mathematicians in all areas, but also to specialists with a mathematical background who are studying dynamical processes: mechanical engineers, physicists, biologists, and so on.
Readership: Graduate students and researchers working in dynamical systems and differential equations, as well as specialists with a mathematical background who are studying dynamical processes: mechanical engineers, physicists, biologists, etc.
Author(s): G. R. Belitsky, and E. V. Zhuzhoma S. Kh. Aranson
Series: Translations of Mathematical Monographs, Vol. 153
Publisher: American Mathematical Society
Year: 1996
Language: English
Pages: C, xiii+325, B
Cover
Front Matter
S Title
Introduction to the Qualitative Theory of Dynamical Systems on Surfaces
© 1996 by the American Mathematical Society
ISBN 0-8218-0369-7
QA614.82.A7313 1996 514'.74-dc20
LCCN 96019197
Contents
Foreword
CHAPTER 1 Dynamical Systems on Surfaces
§1. Flows and vector fields
1.1. Definitions and examples.
1.2. Connection between flows and vector fields.
1.3. Vector fields and systems of differential equations
1.4. Diffeomorphisms of vector fields.
§2. Main ways of specifying flows on surfaces
2.1. The projection method
2.2. Systems of differential equations in local charts
2.3. Specification of a flow with the help of a universal covering
2.4. Specification of a flow with the help of a branched covering.
2.4.1. Definition of a branched covering
2.4.2. Covering flow
2.4.3. Construction of transitive flows.
2.5. The pasting method.
2.6. Suspensions
2.6.1. The suspension over a homeomorphism of the circle
2.6.2. The suspension over an exchange of open intervals
2.7. Whitney's theorem
2.7.1. The theorem on continuous dependence on the initial conditions
2.7.2. The rectification theorem
2.7.3. Orientability. We consider two orientable arcs ab and cd lying on the manifold M
§3. Examples of flows with limit set of Cantor type
3.1. The example of Denjoy.
3.2. Cherry flows
3.3. An example of a flow on the sphere.
§4. The Poincare index theory
4.1. Contact-free segments and cycles
4.2. The index of a nondegenerate cycle in a simply connected domain.
4.3. The index of an isolated equilibrium state
4.4. The Euler characteristic and the Poincare index
4.5. Connection between the index and the orientability of foliations.
4.6. An example of a foliation that is locally but not globally orientable.
Remark. About a result of El'sgol'ts
CHAPTER 2 Structure of Limit Sets
§1. Initial concepts and results
1.1. The long flow tube theorem, and construction of a contact-free cycle.
1.2. The Poincare mapping.
1.3. The limit sets.
1.4. Minimal sets.
1.5. Nonwandering points
§2. The theorems of Maier and Cherry
2.1. Definitions of recurrence
2.2. The absence of nontrivial recurrent semitra jectories on certain surfaces.
2.3. The Cherry theorem on the closure of a recurrent semitrajectory.
2.4. The Mater criterion for recurrence.
2.5. The Mater estimate for the number of independent nontrivial recurrent semitrajectories.
§3. The Poincare-Bendixson theory
3.1. The Poincare-Bendixson theorem.
3.2. Bendixson extensions
3.3. Separatrices of an equilibrium state
3.4. The Bendixson theorem on equilibrium states.
3.5. One-sided contours.
3.6. Lemmas on the Poincare mapping.
3.7. Description of quasiminimal sets
3.8. Catalogue of limit sets.
3.9. Catalogue of minimal sets
§4. Quasiminimal sets
4.1. An estimate of the number of quasiminimal sets
4.2. A family of special contact-free cycles.
4.3. Partition of a contact-free cycle
4.4. The Gardiner types of partition elements
4.5. The structure theorem.
CHAPTER 3 Topological Structure of a Flow
§1. Basic concepts of the qualitative theory
1.1. Topological and smooth equivalence.
1.2. Invariants
1.3. Classification
§2. Decomposition of a flow
2.1. Characteristic curves of a quasiminimal set
2.2. Periodic elements of a partition.
2.3. Criterion for a flow to be irreducible.
2.4. Decomposition of a flow into irreducible flows and flows without nontrivial recurrent semitrajectories
2.5. The Levitt decomposition
§3. The structure of an irreducible flow
3.1. Blowing-down and blowing-up operations
3.2. Irreducible flows on the torus
§4. Flows without nontrivial recurrent trajectories
4.1. Singular trajectories.
4.2. Cells
4.3. Topology of cells
4.4. Structure of a flow in cells
4.5. Smooth models
4.6. Morse-Smale flows
4.7. Cells of Morse-Smale flows.
§5. The space of flows
5.1. The metric in the space of flows.
5.2. The concepts of structural stability and the degree of structural instability.
5.3. The space of structurally stable flows
5.4. Flows of the first degree of structural instability
5.5. On denseness of flows of the first degree of structural instability in the space of structurally unstable flows.
CHAPTER 4 Local Structure of Dynamical Systems
§1. Dynamical systems on the line
1.1. Linearization of a diffeomorphism
1.2. Lemmas on functional equations.
1.3. Proof of Theorem 1.1.
1.4. Flows on the line.
§2. Topological linearization on the plane
2.1. Formulation of the theorem
2.2. Proof of the theorem.
§3. Invariant curves of local diffeomorphisms
3.1. Invariant curves of a node
3.2. Invariant curves of a saddle point.
§4. C1-linearization on the plane
§5. Formal transformations
5.1. Formal mappings
5.2. Conjugacy of formal mappings
5.3. Formal vector fields and flows.
§6. Smooth normal forms
6.1. Normal forms with flat residual
6.2. Smooth normal forms of a node
6.3. Smooth normalization in a neighborhood of a saddle point
6.4. The Sternberg-Chern theorem.
6.5. The smoothness class as an obstacle to smooth normalization.
§7. Local normal forms of two-dimensional flows
7.1. Topological and C1-linearization
7.2. Invariant curves of a flow.
7.3. Smooth normal forms.
7.4. The correspondence mapping at a saddle point
§8. Normal forms in a neighborhood of an equilibrium state (survey and comments)
CHAPTER 5 Transformations of the Circle
§ 1. The Poincare rotation number
1.1. Definitions and notation
1.2. Invariance of the rotation number
1.3. Continuous dependence of the rotation number on a parameter
1.4. The rotation number of a homeomorphism of the circle.
§2. Transformations with irrational rotation number
2.1. Transformations semiconjugate to a rotation
2.2. A criterion for being conjugate to a rotation.
2.3. Limit sets
2.4. Classification of transitive homeomorphisms.
2.5. Classification of Denjoy homeomorphisms
2.6. Classification of Cherry transformations
§3. Structurally stable diffeomorphisms
3.1. The C'-topology
3.2. Main definitions
3.3. Instability of an irrational rotation number
3.4. Openness and denseness of the set of weakly structurally stable diffeomorphisms.
3.5. Classification of weakly structurally stable diffeomorphisms
§4. The connection between smoothness properties and topological properties of transformations of the circle
4.1. Continued fractions
4.2. The order of the points on the circle
4.3. The theorem of Denjoy.
4.4. The theorem of Yoccoz
4.6. The Herman index of smooth conjugacy to a rotation
§5. Smooth classification of structurally stable diffeomorphisms
5.1. Pasting cocycles
5.2. C°'a-conjuga
5.3. Smooth classification
5.4. Corollaries
5.5. Conjugacy of flows
5.6. Inclusion of a difeomorphism in a flow
5.7. Comments
CHAPTER 6 Classification of Flows on Surfaces
§1. Topological classification of irreducible flows on the torus
1.1. Preliminary facts
1.2. Curvilinear rays.
1.3. Asymptotic directions
1.4. The Poincare rotation number
1.5. The rotation orbit
1.6. Classification of minimal flows
1.7. Classification of Denjoy flows
§2. The homotopy rotation class
2.1. Lobachevsky geometry and uniformization
2.2. The axes of hyperbolic isometries
2.3. Asymptotic directions.
2.4. Arithmetic properties of the homotopy rotation class
2.5. The homotopy rotation class of a nontrivial recurrent semitrajectory.
2.6. The connection between quasiminimal sets and geodesic laminations.
2.7. Accessible points of the absolute
2.8. Classification of accessible irrational points
2.9. The orbit of a homotopy rotation class.
§3. Topological equivalence of transitive flows
3.1. Homotopic contact-free cycles
3.2. Auxiliary results.
3.3. Construction of a fundamental domain
3.4. Necessary and sufficient conditions for topological equivalence of transitive flows
§4. Classification of nontrivial minimal sets
4.1. Special and basic trajectories
4.2. The canonical set.
4.3. Topological equivalence of minimal sets.
4.4. Realization of nontrivial minimal sets by geodesic curves.
§5. Topological equivalence of flows without nontrivial recurrent trajectories
5.1. Schemes of semicells
5.2. Schemes of spiral cells.
5.3. The orbit complex
5.4. Neighborhoods of limit singular trajectories
5.5. Main theorems
CHAPTER 7 Relation Between Smoothness Properties and Topological Properties of Flows
§1. Connection between smoothness of a flow and the existence of a nontrivial minimal set
1.1. The theorems of Denjoy and Schwartz
1.2. The theorem of Neumann.
1.3. The theorem of Gutierrez
§2. The problem of Cherry
2.1. Gray and black cells.
2.2. The Poincare mapping in a neighborhood of a structurally stable saddle.
2.3. Sufficient conditions for the absence of gray cells.
2.4. Cherry flows with gray cells.
Back Matter
Bibliography
Back Cover
