20 hours distributed among
This course presents modern numerical methods and software for bifurcation analysis of finite-dimensional dynamical systems generated by parameter-dependent smooth autonomous ordinary differential equations (ODEs) and iterated maps. The main problems are: How to continue equilibria and periodic orbits with respect to a parameter? How to compute stability boundaries of equilibria and periodic orbits (limit cycles) in the parameter space? How to predict qualitative changes in system's behavior (bifurcations) occurring at these boundaries? Only the most efficient methods will be described, which are based on projection and bordering techniques and employ boundary value problems (BVPs). All developed methods will be illustrated using the latest version of MATCONT and cl_MATCONT_for_maps.
Lecture 1, May, Tue 5:
Continuation problems. Numerical continuation of equilibria and
limit cycles of ODEs.
Finite-dimensional continuation problems. Limit and branch points. Moor-Penrose continuation. Continuation of equilibria of ODEs with respect to one parameter. Continuation of solutions to boundary-value problems (BVPs). Discretization via the orthogonal collocation. Continuation of limit cycles in one-parameter families of ODEs.
Computer lab 1, May, Wed 6: Numerical simulation of ODEs with MATCONT.
Lecture 2, May, Thu 7:
Equilibrium bifurcations of ODEs and their numerical analysis.
Detection and normal form analysis of codim 1 bifurcations of equilibria (i.e. fold and Andronov-Hopf) in one-parameter families of ODEs. Bialternate matrix product. Projection techniques to compute critical normal forms. Continuation of codim 1 equilibrium bifurcations in two parameters. Bordering methods to setup minimally-extented defining systems. Detection of codim 2 bifurcations.
Computer lab 2, May, Mon 11: One-parameter bifurcation analysis of equilibria with MATCONT.
Lecture 3, May, Tue 12:
Bifurcations of limit cycles of ODEs and their numerical analysis
Detection and normal form analysis of codim 1 bifurcations of limit cycles (i.e. cycle-fold, period-doubling, and torus) in one-parameter families of ODEs. Periodic normalization. Continuation of codim 1 bifurcations cycle bifurcations in two control parameters. Detection of codim 2 bifurcations of limit cycles.
Computer lab 3, May, Wed 13: One-parameter bifurcation analysis of limit cycles with MATCONT.
Lecture 4, May, Tue 19:
Numerical local bifurcation analysis of iterated maps.
Continuation of fixed points and cycles of iterated maps with respect to one parameter. Detection and normal form analysis of codim 1 bifurcations of fixed points (i.e. fold, flip, and Neimark-Sacker) in one-parameter families of iterated maps. Continuation of codim 1 bifurcations in two parameters. Detection of codim 2 bifurcations.
Computer lab 4, May, Wed 20: Two-parameter bifurcation analysis of equilibria and limit cycles with MATCONT.
Lecture 5, May, Tue 26:
Numerical continuation of connecting orbits of iterated maps and
Orbits connecting fixed points of maps. Orbits connecting equilibria of ODEs. Truncated defining systems with projection boundary conditions to continue orbits connecting fixed and equilibrium points. Contiuation of equilibrium-to-cycle and cycle-to-cycle connecting orbits in three-dimensional ODEs.
Computer lab 5, May, Wed 27: Iteration and bifurcation analysis of maps with cl_MATCONT_for_maps.
Kuznetsov, Yu.A. Elements of Applied Bifurcation Theory, Springer-Verlag, 3rd ed., 2004 (Chap. 10 in particular).
The following rules are valid for both Master (laurea specialistica) and Ph.D. students:
On appointment or by email.
Pagina a cura di Fabio Dercole