Prerequisites

Bachelor courses on ODEs and/or Numerical Analysis, e.g. based on

• Hirsch, M.W., Smale, S., and Devaney, R.L. "Differential Equations, Dynamical Systems, and an Introduction to Chaos". Academic Press, 2013
• Süli, E. and Mayers, D.F.. "An Introduction to Numerical Analysis". Cambridge University Press, Cambridge, 2003.

Some knowledge about bifurcations of dynamical systems, e.g.

• Meiss, J.D.. Differential Dynamical Systems, SIAM, Philadelphia, 2017 [Chapter 8]

will be an advantage but is not required.

Aim/Description

This course presents numerical methods and software for bifurcation analysis of finite-dimensional dynamical systems generated by smooth autonomous ordinary differential equations (ODEs) and iterated maps. After completion of the course, the student will be able to perform rather complete analysis of ODEs and maps depending on two control parameters by combining analytical and numerical tools.

Organization: 2 hrs lectures per week + 1h computer lab.

The lectures will cover

• Basic Newton-like methods to solve systems of nonlinear equations
• Continuation methods to compute implicitly-defined curves in the n-dimensional space
• Techniques to continue equilibria and periodic orbits (cycles) of ODEs and fixed points of maps in one control parameter
• Methods to detect and continue in two parameters all generic local bifurcations of equilibria and fixed points, i.e. fold, Hopf, period-doubling, and Neimark-Sacker bifurcations, and to detect their higher degeneracies
• Methods to detect and continue in two control parameters all generic local bifurcations of cycles in ODEs (i.e. fold, period-doubling, and torus bifurcations) with detection of the higher degeneracies
• Relevant normal form techniques combined with the center manifold reduction, including periodic normal forms for bifurcation of cycles
• Continuation methods for homoclinic orbits of ODEs and maps, including initialization by homotopy.

Necessary results from the Bifurcation Theory of smooth dynamical systems will be reviewed. Modern methods based on projection and bordering techniques, as well as on the bialternate matrix product, will be presented and compared with other approaches.

The course includes exercises with sophisticated computer tools, in particular using the latest versions of the interactive MATLAB bifurcation software MATCONT. It is assumed that all participants have own laptops with a recent MATLAB installed.

Lecturer

Yuri Kuznetsov (UU)