| blok 2 | time | place | lecture | tuesday 11:00 - 12:45 | MIN 0.09 | lecture | thursday 15:15 - 17:00 | BBG 069 | exercises | tuesday 9:00 - 10:45 | MIN 0.09 | exercises | thursday 13:15 - 15:00 | BBG 385 |
ECTS : 7.5 credit points
The course is a gentle introduction to the modern theory of
nonlinear ordinary differential equations (ODEs) and the dynamical
systems theory in general. This theory links topology, analysis and
algebra together.
Many notions, results and methods from the dynamical systems theory
are widely used in the mathematical modelling of the behavior of
many physical, biological and social systems.
We provide a catalogue of various dynamical regimes (equilibrium, periodic, quasiperiodic, chaotic) in systems of smooth ordinary differential equations (ODEs) and their qualitative changes (called 'bifurcations') under parameter variations, such as the saddle-node, Hopf, period-doubling, torus and homoclinic bifurcations. The exposition includes an overview (in most cases without proofs) of all local bifurcations possible in generic ODEs depending on one and two parameters, as well as some global bifurcations involving limit cycles and homoclinic orbits. This allows to get insight into modern methods to study ODEs: normal forms, center manifold reduction, return maps, perturbation of Hamiltonian systems.
This course develops some geometric intuition about orbit structure and its rearrangements in systems of nonlinear ODEs depending on parameters. This enables to identify by analytical techniques and numerical simulations the appearance of equilibria, periodic and quasi-periodic motions, period-doubling cascades and homoclinic bifurcations in concrete ODEs, with examples from ecology and engineering.
After the course one is able to
- perform the phase-plane analysis using zero-isoclines and
Poincaré-Bendixson-Dulac theorems for planar systems;
- locate and analyze fold and Hopf bifurcations of equilibria in
simple 2D and 3D systems depending on one parameter;
- produce two-parameter bifurcation diagrams for equilibria in
planar systems and predict on this basis the existence and
bifurcations of limit cycles in such systems;
- simulate planar and 3D ODEs using the standard interactive
software and relate observations to bifurcation theory;
Mathematics is best learned by doing, so next to the two lectures there are every week two exercise sessions to simulate various ODEs on a computer and perform the bifurcation analysis by combining analytical and software tools. It is assumed that everyone uses her/his own laptop with MATLAB installed, including MATLAB Compiler and Symbolic Math toolboxes (free for UU students, see https://students.uu.nl/gratis-software). After each exercise session a compulsory home assignment is given, to be handed in a week later at the beginning of the exercise session.
At the end of the course each student presents a given topic. The final grade then is composed from 40% for a written essay on this topic, 20% for the oral presentation and 40% for homework exercises.
Yu.A. Kuznetsov "Four Lectures on Bifurcation Phenomena in ODEs" (on-line notes: L1.pdf, L2.pdf, L3.pdf, L4.pdf)
Yu.A. Kuznetsov and N. Neirynck, three tutorials (TUTORIAL I, TUTORIAL II, TUTORIAL III) to learn how to use MatCont to simulate multidimensional ODEs, on continuation of equilibria and to learn how to perform one-parameter analysis of equilibria and cycles in MatCont.
A lecture on practical computation of the normal form coefficients for codim 2 bifurcations of equilibria. (CODIM2.pdf)
A MAPLE session to Ex.2 of Practicum 2 (P2-EX2.pdf)
Software: MatCont (MatCont7p4.zip) pplane (MATLAB function, Java applet, MATLAB App)
12.11. Exercises (pdf, ps). Solutions of autonomous ODE systems. Flow box theorem on straightening out a vector field, equilibria, periodic orbits (cycles) and connecting orbits (homo- and heteroclinic orbits), Poincaré-Bendixson Theorem, orbits and phase portraits, planar Hamiltonian systems.
14.11. Exercises (pdf, ps). Equivalence of planar ODEs (smooth, orbital, topological), Hartman-Grobman Theorem, Pontryagin and Bendixson-Dulac criteria.
19.11. Exercises (pdf, ps). Proof of the Bendixson-Dulac criterium, index of equilibria, hyperbolic cycles and their stability, systems with families of cycles: integrable and reversible planar systems, homoclinic orbits, Melnikov theory on dissipative perturbations.
21.11. Exercises (pdf, ps). One-parameter local bifurcations of planar ODEs. Bifurcations and their codimension, fold (saddle-node) bifurcation of equilibria and its normal form.
26.11. Exercises (pdf, ps). Hopf bifurcation of equilibria and its normal form, computation of the first Lyapunov coefficient for planar ODEs.
28.11. Exercises (pdf, ps). One-parameter global bifurcations of planar ODEs. Fold bifurcation of cycles and the normal form for its Poincaré return mapping, saddle homoclinic and heteroclinic bifurcations, bifurcation of an orbit homoclinic to a saddle-node, structural stability of planar ODEs.
3.12. Exercises (pdf, ps). Two-parameter local bifurcations of planar ODE. Curves of fold and Hopf bifurcations in the parameter plane, codim 2 bifurcations of equilibria (cusp, Bautin and Bogdanov-Takens) and their normal forms.
5.12. Exercises (pdf, ps). Some two-parameter global bifurcations of planar ODEs. Triple cycle, neutral saddle homoclinic orbit, non-central homoclininc orbit to a saddle-node, saddle with two homoclinic orbits, saddle heteroclinic cycle.
10.12. Exercises (pdf, ps). Local one-parameter bifurcations of n-dimensional ODEs. Equilibria, cycles, invariant tori and chaotic invariant sets of n-dimensional ODEs, center-manifold reduction for bifurcations of equilibria, codim 1 bifurcations of equilibria (fold and Hopf) in n-dimensional systems and practical computation of their normal form coefficients. Assignment of individual examination topics.
12.12.
To become acquainted with the MATLAB bifurcation toolbox MatCont
install your personal copy of MatCont.
Start MATLAB and type "mex -setup" to control that C compiler is
installed.
Change the current directory in MATLAB to the created MatCont
directory and type "matcont" to start the software.
Follow TUTORIAL I to learn how to use MatCont to simulate
multidimensional ODEs and make Additional Problems C and D listed at
the end of the tutorial text; problem D is to be handed in.
Some semi-local one-parameter bifurcations of n-dimensional ODEs.
Center-manifold reduction for bifurcations of limit cycles,
codim 1 bifurcations of cycles (fold, period-doubling and
Neimark-Sacker) and the normal forms for their Poincaré
return mappings.
17.12. Exercises, but no lecture.
19.12. Exercises, but no lecture.
7.1.
Follow Section 3 of TUTORIAL III to learn how to perform
one-parameter analysis of equilibria and cycles in MatCont.
Consult, if necessary, TUTORIAL II.
Additional Problem A in TUTORIAL III is to be handed in.
Hint: study the Hopf bifurcation at mu=0 analytically and then
investigate numerically the bifurcating cycle.
Codim 1 bifurcations of saddle homoclinic orbits.
Shilnikov's Theorems, bifurcations of homoclinic orbits to the
saddle-node and saddle-saddle equilibria.
9.1. Exercises (pdf, ps). Two-parameter local bifurcations of n-dimensional ODEs. Curves of fold and Hopf bifurcations in the parameter plane, multidimensional codim 2 equilibrium bifurcations (fold-Hopf and double Hopf), practical computation of the normal form coefficients for codim 2 bifurcations of equilibria.
14.1. Last exercise session, last lecture.
16.1. Presentations of theoretical topics (TBA)
21.1. Presentations of theoretical topics (TBA)
23.1. Presentations of theoretical topics (TBA)