Dynamical Systems: Models of Chaotic Dynamics

Lecturer

Dr Andy Hammerlindl, Monash University

Course Overview

Dynamics is the study of systems that change in time. In some cases, the long-term behaviour is simple, such as a system settling down to an equilibrium or a regular periodic motion. In other cases, the system is chaotic. This does not mean, however, that we cannot analyze the system or determine its properties. In fact, many highly chaotic dynamical systems are very well understood.

In this course, we cover dynamical systems theory, starting with Poincar√©’s foundational work showing that no chaos is possible for flows on the plane, and going up to recent advances and discoveries. A large focus of the course will be on known models of chaotic dynamics and the geometric and topological objects associated with their behaviour. This includes horseshoes, geometric Lorenz attractors, Anosov systems, homoclinic tangencies, heterodimensional cycles, and blenders. We will also cover classification results that show how well these models relate to dynamical systems that arise in the physical world.

Assessment

Assessment will be by regular assignments during the summer school as well as a final exam.

Prerequisites

To take this course, students should be familiar with the following concepts:

  • epsilon-delta proofs
  • Cauchy sequences
  • open, closed, compact and connected subsets of R^n
  • multivariable calculus, and
  • linear algebra

Experience with point-set topology, analysis, and differential geometry will be helpful, but not strictly necessary. If you are unsure if you have the necessary background for the course, please email Andy Hammerlindl to get more information.

Resources and Recommended Texts

David Ruelle – Elements of Differentiable Dynamics and Bifurcation Theory
Michael Shub – Global Stability of Dynamical Systems
Norton, Douglas E. The fundamental theorem of dynamical systems. Comment. Math. Univ. Carolin. 36 (1995), no. 3, 585–597

Dynamical Systems: Models of Chaotic Dynamics

 Biography

Dr Andy Hammerlindl, Monash University

Andy is a Lecturer at Monash University who is originally from Canada. He has held previous positions in Sydney, Australia and Rio de Janeiro, Brazil. His research is in Dynamical Systems, which studies the long-term behaviour of systems which change in time. A large part of his research involves determining the types of chaotic behaviour which can occur in 3-dimensional manifolds. His work uses a combination of topology, geometry, foliations, and measure theory to establish new results.

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