Summer 2011

Seminar on Bifurcation Theory (Verzweigungstheorie)

Prof. Dr. Bernold Fiedler, Dr. Stefan Liebscher

Friday, 14:15, room 140, Arnimallee 7, rear building

Inhalt

Based on (and going beyond) the course Dynamical Systems II, we want to study the behaviour of parameter dependent dynamical systems. Particular emphasis is on homoclinic bifurcation and the Newhouse phenomenon. This constitutes on of the "paths into chaos".

Aufbauend auf der Vorlesung Differentialgleichungen II aus dem vergangenen Semester wollen wir das Verhalten parameterabhängiger dynamischer Systeme studieren. Dabei wollen wir uns insbesondere auf homokline Verzweigungen als einen der "Wege ins Chaos" konzentrieren.

References

• J. Palis & F. Takens: Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Univ. Press, 1993.

Target audience

Students of semesters 6-8, students of the BMS (talks can be given in German and/or English)

Prerequisites

Analysis I-III, Dynamical Systems I-II

attendance of Dynamical Systems III this semester recommended

Prospects

Bachelor, Master and Diploma theses

Topics

Introduction & basic concepts [1-2 talks]

• References: Palis & Takens, ch. 0 (possibly 1, 2)
• Scope: limit sets, attractors, nonwandering sets, chain recurrence, basic sets, continuation of basic sets, hyperbolicity, structural stability, transversality, Axiom A, Anosov systems, Morse-Smale systems, chaos

Cascades of homoclinic bifurcations & scaling [1-2 talks]

• References: Palis & Takens, ch. 3
• Scope: quadratic tangency, unfolding, cascades of tangencies, period doubling, cascades of period doublings, scaling, quadratic maps

Dynamically defined Cantor sets [1 talk]

• References: Palis & Takens, ch. 4.1, A.2
• Scope: Cantor sets, dynamic definition, stable foliations, Markov partitions, Markov partitions for 2-d Cantor sets, bounded distortion, self-similarity

Global properties of Cantor sets [1-2 talks]

• References: Palis & Takens, ch. 4.2
• Scope: Hausdorff dimension, limiting capacity, thickness, denseness, measure, gap lemma, relations

Local properties of Cantor sets [1 talk]

• References: Palis & Takens, ch. 4.3
• Scope: local Hausdorff dimension, local limiting capacity, local thickness, local denseness, distance of Cantor sets, conjugacy, continuous dependence of Hausdorff dimension, limiting capacity, thickness, denseness

Homoclinic bifurcation [1-2 talks]

• References: Palis & Takens, ch. 5
• Scope: bifurcating family on S², bifurcating set of small measure

Newhouse phenomenon [1-2 talks]

• References: Palis & Takens, ch. 6
• Scope: dense set of diffeomorphisms with homoclinic tangencies, residual set of diffeomorphisms with infinitely many periodic sinks