Fabio Dercole |
General information
Time schedule
20 hours distributed among
Programme
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.
Detailed programme
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
using BVPs.
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
ODEs.
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.
Teaching Material
Books
Kuznetsov, Yu.A. Elements of Applied Bifurcation Theory, Springer-Verlag, 3rd ed., 2004 (Chap. 10 in particular).
Lecture notes
Computer labs
Further reading
Exam
The following rules are valid for both Master (laurea specialistica) and Ph.D. students:
Student hours
On appointment or by email.
Notice
Pagina a cura di Fabio Dercole