# MATH 552

## 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.