1. Topological
equivalence of dynamical systems. Bifurcations and bifurcation diagrams.
Classifications. The codimension of a bifurcation. Structural stability.
Topological normal forms for bifurcations. Center manifolds. Some models
from biology and economics: Van der Pol, Hodgkin-Huxley, FitzHugh-Nagumo, predator-prey.
2. Codimension
one bifurcations in continuous-time dynamical systems
- Saddle-node bifurcation. The
normal form derivation. Computation of the center manifold.
- Hopf bifurcation. The normal form
derivation. Computation of the center manifold.
-
Homoclinic bifurcation. Andronov-Leontovich theorem. Shilnikov theorems.
Melnikov integral.
-
Saddle-node homoclinic bifurcation.
-
Breaking saddles connection
bifurcation.
-
Bifurcations of limit cycles: fold
bifurcation, flip bifurcation, Neimark-Sacker bifurcation.
3. Codimension two bifurcations in
continuous-time dynamical systems
- Bogdanov-Takens bifurcation. Normal
form. An approximation of the parameters curve corresponding to homoclinic
bifurcation.
-
Bautin bifurcation. Normal form.
An approximation of the parameters curve corresponding to fold bifurcation of
cycles.
-
Cusp bifurcation.
-
Fold-Hopf bifurcation
-
Hopf-Hopf bifurcation.
-
Other codimension two bifurcations.
4. Numerical analysis of bifurcations. The software XPP. Applications.
Bibliography
1.
Chow, S.N., Li, C., Wang, D., Normal
forms and bifurcations of planar vector fields, Cambridge Univ. Press, Cambridge,
1994.
2.
Chow, S.N., Hale, J., Methods
of bifurcation theory, Springer, New-York, 1982.
3.
Dumortier, F., Roussarie, R., Sotomayor,
J., Zoladek, H., Bifurcations of planar vector fields, nilpotent
singularities and abelian integrals, Springer, Berlin, 1991.
4.
Georgescu, A., Moroianu, M., Oprea,
I.,
Bifurcation theory. Principles and applications, Pitesti Univ. Press, Pitesti, 1999
(Romanian).
5.
Giurgiteanu, N., Computational
economical and biological dynamics-DIECBI, Craiova, 1997. (Romanian)
6.
Guckenheimer, J., Holmes, P., Nonlinear
oscillations, dynamical systems and bifurcations of vector fields,
Springer, New-York, 1983.
7.
Hale, J.K. and Kocak, H., Dynamics
and bifurcations, Springer, New York, 1991.
8.
Kuznetsov, Yu., Elements of
applied bifurcation theory, Springer, New
York, 1995.
9.
Murray, J.D., Mathematical Biology, Springer, Berlin, 1993.
10.
Rocsoreanu, C., Georgescu, A., Giurgiteanu,
N., The FitzHugh-Nagumo model. Bifurcation and dynamics, Kluwer Academic
Publishers, Dordrecht, 2000.
11.
Tu, P., Dynamical systems. An
introduction with applications in economics and biology, Springer, Berlin, 1994.