We will use the basic tools of analysis, geometry, topology and ODE theory to study the qualitative behavior of dynamical systems, including many concrete examples. The solutions of an ordinary differential equation determine a continuous-time dynamical system (a flow) on a state space, which is usually a domain in Euclidean space or a manifold. One starts to obtain a qualitative understanding of the flow by finding simple solutions (equilibrium points, periodic orbits) and connections between them. Sometimes one can reduce the problem to the study of a discrete-time dynamical system. for example by setting up a Poincare section near a known periodic solution. Discrete-time systems (iterated mappings) have become and important research topic in their own right. Fixed points and periodic orbits are the simplest "invariant sets" (subsets of the state space left invariant by the flow). More complicated invariant sets also occur. Showing that they exist and studying them uses a wide variety of techniques from all parts of mathematics. Different kinds of examples, such as gradient systems or Hamiltonian systems, require their own special techniques. Topics will include as many of the following as time permits:
Review Equilibrium points and periodic solutions: local analysis, existence theorems, index theory, special results for Hamiltonian and gradient systems. Homoclinic points: transverse homoclinic and heteroclinic points, Melnikov's integral, Smale horseshoe model for homoclinic chaos, symbolic dynamics Bifurcations: saddle-node, Hopf and period-doubling bifurcations. Lyapunov center theorem Dynamics of circle maps and area-preserving maps. Introduction to ergodic theory.
None required.
There will be 4 or 5 homework assignments. No exams. The link leads PDF versions of the homework assignments so far.
| Homework | 100 % |
"Ordinary Differential Equations", by V.I. Arnold "Dynamical Systems: Stability, Symbolic Dynamics and Chaos", by Clark Robinson "Introduction to Chaotic Dynamical Systems", by R. Devaney "Mathematical Methods of Classical Mechanics", by V.I. Arnold "Math. Aspects of Classical and Celestial Mechanics", by Arnold, Koslov and Neistadt "Lectures on Celestial Mechanics", by Siegel and Moser