Details

Autor(en) / Beteiligte

• Broer, H
• Hasselblatt, B
• Takens, F

Titel

Handbook of dynamical systems [electronic resource]. Volume 3

Link zum Volltext

• Elsevier Books AutoHoldings

Beschreibungen/Notizen

• Description based upon print version of record.
• Includes bibliographical references and indexes.
• Front cover; Half title page; Title page; Copyright page; Dedication page; Preface; List of Contributors; Contents; Chapter 1. Preliminaries of Dynamical Systems Theory; 1. General definition of a dynamical system; 2. Transversality and generic properties; 3. Generic properties which are not based on transversality: the Closing Lemma; 4. Generic local bifurcations; 5. Structural stability and moduli; References; Chapter 2. Prevalence; 1. Introduction; 2. Linear prevalence; 3. Nonlinear prevalence; 4. Other notions of genericity; References
• Chapter 3. Local Invariant Manifolds and Normal Forms1. Introduction; 2. Construction of invariant manifolds: the graph transform; 3. Invariant foliations; 4. Linearizations and partial linearizations; 5. Normal forms; 6. Liapunov-Schmidt reduction; References; Chapter 4. Complex Exponential Dynamics; 1. Introduction; 2. Basic notions; 3. Quadratic dynamics; 4. Exponential dynamics; 5. Cantor bouquets; 6. Indecomposable continua; 7. The parameter plane; 8. Untangling hairs; 9. Back to polynomials; 10. Other families of maps; References; Chapter 5. Some Applications of Moser's Twist Theorem
• 1. Background: the action-angle variables, the generating functions2. Basic statements of KAM theory; 3. A variational approach to Moser's twist theorem; 4. Applications; 5. Arnold diffusion; References; Chapter 6. KAM Theory: Quasi-periodicity in Dynamical Systems; 1. Introduction; 2. Complex linearization; 3. KAM Theory for circle and annulus maps; 4. KAM Theory for flows; 5. Further developments in KAM Theory; 6. Quasi-periodic bifurcations: dissipative setting; 7. Quasi-periodic bifurcation theory in other settings; 8. Further Hamiltonian KAM Theory; 9. Whitney smooth bundles of KAM tori
• 10. ConclusionAcknowledgments; References; Chapter 7. Reconstruction Theory and Nonlinear Time Series Analysis; 1. Introduction; 2. An experimental example: the dripping tap; 3. The reconstruction theorem; 4. The reconstruction theorem and nonlinear time series analysis: discrimination between deterministic and random time series; 5. Stationarity and reconstruction measures; 6. Correlation dimensions and entropies; 7. Numerical estimation of correlation integrals and the corresponding dimensions and entropies
• 8. Classical time series analysis, the analysis in terms of correlation integrals, and predictability9. Miscellaneous subjects; References; Chapter 8. Homoclinic and Heteroclinic Bifurcations in Vector Fields; 1. Introduction; 2. Homoclinic and heteroclinic orbits, and their geometry; 3. Analytical and geometric approaches; 4. Phenomena; 5. Catalogue of homoclinic and heteroclinic bifurcations; 6. Related topics; References; Author Index; Subject Index
• In this volume, the authors present a collection of surveys on various aspects of the theory of bifurcations of differentiable dynamical systems and related topics. By selecting these subjects, they focus on those developments from which research will be active in the coming years. The surveys are intended to educate the reader on the recent literature on the following subjects: transversality and generic properties like the various forms of the so-called Kupka-Smale theorem, the Closing Lemma and generic local bifurcations of functions (so-called catastrophe theory) and generic local bifur
• English