- 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

- lec01. Wed Sep 6
- lec02. Fri Sep 8
- lec03. Mon Sep 11
- lec04. Wed Sep 13
- lec05. Fri Sep 15
- lec06. Mon Sep 18
- lec07. Fri Sep 22
- lec08. Mon Sep 25
- lec09. Wed Sep 27
- lec10. Fri Sep 29
- lec11. Wed Oct 4
- lec12. Fri Oct 6
- lec13. Wed Oct 11
- lec14. Thu Oct 12
- lec15. Fri Oct 13
- lec16. Mon Oct 16
- lec17. Wed Oct 18
- lec18. Fri Oct 20
- lec19. Mon Oct 23
- lec20. Wed Oct 25
- lec21. Fri Oct 27
- lec22. Mon Oct 30
- lec23. Wed Nov 1
- lec24. Fri Nov 3
- lec25. Mon Nov 6
- lec26. Wed Nov 8
- lec27. Fri Nov 10
- lec28. Fri Nov 17
- lec29. Mon Nov 20
- lec30. Wed Nov 22
- lec31. Fri Nov 24
- lec32. Mon Nov 27
- lec33. Wed Nov 29
- lec34. Fri Dec 1
- lec35. Mon Dec 4
- lec36. Wed Dec 6

- A. Review of (Multivariable) Differential Calculus for Wed Sep 6

- 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;