Fabio Dercole

Numerical bifurcation analysis of dynamical systems
Milano Leonardo, II sem. (081669, 2.5 crediti)

General information

• Time schedule
• Programme
• Teaching Material
• Exam
• Student hours
• Notice

Time schedule

20 hours distributed among

• Mon. 14.15-16.15 (lab, room D3.2)
• Tue. 8.15-10.15 (lecture, room D2.6)
• Wed. 12.15-14.15 (lab, room D3.2)
• Thu. 8.15-10.15 (lecture, room D2.2)
• Start: May, Tue. 5, 2009 (the precise schedule will be posted as soon as available)
• Venue: Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano.

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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.

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Teaching Material
 

Books

Kuznetsov, Yu.A. Elements of Applied Bifurcation Theory, Springer-Verlag, 3rd ed., 2004 (Chap. 10 in particular).

Lecture notes

• Lecture 1
• Lecture 2
• Lecture 3
• Lecture 4
• Lecture 5

Computer labs

• Computer lab 1
• Computer lab 2
• Computer lab 3
• Computer lab 4
• Computer lab 5

Further reading

• Beyn, W.-J., Champneys, A., Doedel, E., Govaerts, W., Kuznetsov, Yu.A., and Sandstede, B. Numerical continuation, and computation of normal forms. In: B. Fiedler(ed.) Handbook of Dynamical Systems, v.2, Elsevier Science, North-Holland, 2002, pp. 149-219 (pdf)

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Exam

The following rules are valid for both Master (laurea specialistica) and Ph.D. students:

• The exam consists in solving at least three of the five examination problems proposed at the end of the lecture notes on computer labs.
• Three correct solutions are needed to pass the exam with the minimum mark.
• Each solutions must be reported in written form, with the solving procedure and the obtained results in clear evidence. A single pdf document containing at least three proposed solutions must be produced.
• The report must be sent by email to Professor Kuznetsov ( kuznet@math.uu.nl ) at any time, but once sent there is no possibility for updates or supplements of any form. We encourage students to send the report in the first week of June, 2009, while Professor Kuznetsov is still at Politecnico di Milano.
• The mark will be communicated by email. If the mark is negative or refused, the student must repeat the exam by solving a new set of problems to be assigned by Professor Kuznetsov.

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Student hours

On appointment or by email.

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Notice

• Periodically check this page for further information.

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Pagina a cura di Fabio Dercole