MATH 552
Introduction to Dynamical Systems
Session 2023W Term 1 (Sep - Dec 2023)
Course instructor's page
Last update: 2023-12-04
(
lec36
)
- Course Canvas page (for syllabus, completed lecture notes, appendices, homework, zoom links, recorded lectures, etc.).
- Recommended textbook (not required): Y. A. Kuznetsov,
Elements of Applied Bifurcation Theory,
Springer (2004, 3rd ed.).
- See the course Canvas page for more references and links.
Lectures: Mon Wed Fri, 10:00--10:50 Vancouver BC Canada local time, on the course Canvas page (via Zoom).
Instructor: Wayne Nagata
Office: online
Office hours: TBA on the course Canvas page (via Zoom), or by appointment
Email: nagata at math dot ubc dot ca
Preliminary Lecture Notes
Preliminary versions of lecture notes (before annotation) should be posted before each lecture, here:
Completed versions (after annotation) will be posted following each lecture, on the course Canvas page.
Appendices
See the course Canvas page for more appendices.
Learning Outcomes
By the end of the course, a student should be able to:
- prove basic properties of linear flows and maps;
- for a given matrix, find the real normal form and the corresponding linear change of variables;
- for a matrix in real normal form, find the exponential of the matrix explicitly;
- for a matrix in real normal form, find the integer power of the matrix explicitly;
- determine the qualitative (especially asymptotic) behaviour of a linear dynamical system;
- determine the existence of invariant stable, unstable or centre subspaces for a linear dynamical system;
- find Floquet multipliers for a periodic linear homogeneous system of ODEs;
- prove basic properties of families of systems of ODEs, flows and maps;
- prove basic properties of topological equivalence, topological conjugacy, smooth equivalence, orbital equivlence;
- perform a "linearized stability analysis" for a nonlinear dynamical system: linearize at an equilibrium (for a flow), at a fixed point (for a map) or at a cycle (for a flow or for a map), and decide whether linearization is sufficient or not to determine local topological behaviour;
- analyze a flow with a cycle, using a Poincaré map;
- analyze a nonautonomous periodically forced ODE, using a "global" Poincaré map;
- prove basic perturbation results, using the implicit function theorem;
- determine the existence of invariant stable or unstable manifolds for a hyperbolic equilibrium, fixed point or cycle;
- determine local or global properties of a flow for a Hamiltonian system;
- determine local or global properties of a flow using a (local or global) Lyapunov function;
- prove basic properties of local bifurcations in families of 1- and 2-dimensional vector fields or 1-dimensional maps (e.g. using the implicit function theorem);
- analyze local bifurcations in families of 1- and 2-dimensional vector fields or 1-dimensional maps;
- calculate Poincaré normal forms for vector fields and maps;
- determine local properties of a flow or map by reduction to a local centre manifold;
- use centre manifold and normal form theory to analyze local bifurcations in n-dimensional systems;
- analyze a homoclinic bifurcation in a family of 2-dimensional vector fields;
- use a Melnikov integral to determine the presence (or absence) of homoclinic solutions;
Homework Assignments
See the course Canvas page.