Bifurcation phenomena in ordinary differential equations (USMS 2009, 17-18 August)

Lecturer: Yuri A. KuznetsovPhasePortrait.pdf

We will provide a catalogue of various dynamical regimes (equilibrium, periodic,
quasiperiodic, chaotic) in systems of smooth ordinary differential equations
(ODEs) and their qualitative changes under parameter variations (called 'bifurcations')
such as saddle-node, Hopf, period-doubling, torus, and homoclinic bifurcations. The exposition
will include an overview 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. An integral part of the course is a computer practicum
where the students will have a possibility to simulate various ODEs interactively,
and to perform their bifurcation analysis with modern software tools.


[1] V.I. Arnold "Geometrical Methods in the Theory of Ordinary Differntial Equations", Springer-Verlag, 1983
[2] Yu.A. Kuznetsov "Elements of Applied Bifurcation Theory", 3rd ed., Springer, 2004
[3] L.P. Shilnikov,  A.L. Shilnikov, D.V. Turaev, and L. Chua "Methods of Qualitative Theory in Nonlinear Dynamics", Parts I and II, World Scientific, 1998 and 2001
[4]  V.I. Arnold, V.S. Afraimovich, Yu.S. Ilyashenko, and L.P. Shilnikov  "Bifurcation Theory", Dynamical Systems V. Encyclopaedia ofr Mathematical Sciences", Springer-Verlag, 1994

MatCont (
link), dfield5/6/7, pplane5/6/7 (MATLAB5/6/7 functions /Java applets)

Lecture topics
Practicum notes
17 August 2009
Planar ODEs:
Solutions of planar autonomous ODE systems. Orbits and phase portraits.
Equilibria and cycles. Homo- and heteroclinic orbits to equilibria.
Classification of equilibria, cycles, and homoclinic orbits. Poincaré return maps.
Poincare-Bendixson Theorem. Dulac criteria.
Planar Hamiltonian systems and their dissipative perturbations.
Equivalence of planar ODEs and their structural stability.

18 August 2009

practicum 1
19 August 2009
One-parameter bifurcations of planar ODEs:
Bifurcations and their codimension.
Fold (saddle-node) and Andronov-Hopf bifurcations of equilibria and their normal forms.
Fold bifurcation of cycles and the normal form for its Poincaré return map.
Saddle homoclinic and heteroclinic bifurcations.
Bifurcation of a homoclninc orbit to a saddle-node.
practicum 2
21 August 2009
Two-parameter bifurcations of planar ODEs:
Curves of fold and Andronov-Hopf bifurcations in the parameter plane.
Local codim 2 bifurcations (cusp, Bogdanov-Takens, and Bautin) and their normal forms.
Some global codim 2 bifurcations (triple cycle, neutral saddle homoclinic orbit, noncentral homoclininc orbit to a saddle-node, saddle heteroclinic cycle).
practicum 3
24 August 2009
Some bifurcations of n-dimensional ODEs:
Equilibria, cycles, invariant tori, and chaotic invariant sets of n-dimensional ODEs.
Center-manifold reduction for bifurcations of equilibria and cycles.
Codim 1 bifurcations of equilibria (fold and Andronov-Hopf) in n-dimensional systems. Normal form coefficients.
Remarks on multidimensional codim 2 equilibrium bifurcations (fold-Hopf and double Hopf).
Codim 1 bifurcations of cycles (fold, period-doubling, and Neimark-Sacker) and the normal forms for their Poincaré return maps.
Codim 1 bifurcations of saddle homoclinic orbits. Shilnikov's Theorems.
Bifurcations of homoclinic orbits to the saddle-node and saddle-saddle equilibria.

25 August 2009

practicum 4

Last updated: August 24, 2009