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1. Dynamics
and bifurcations
-Topological
equivalence of dynamical systems
-Bifurcations
and bifurcation diagrams. Classifications. The codimension
of a bifurcation. Structural stability
-Topological
normal forms for bifurcations
-Versal deformations of matrices
-Center
manifolds.
-Some models
from biology: Van der Pol,
Hodgkin-Huxley, FitzHugh-Nagumo, predator-prey.
2. Bifurcations of fold singularities
-Normal form
of the fold bifurcation. Fold bifurcation theorem
-Computation
of the center manifold
-Pitchfork
bifurcation
-Cusp
bifurcation
3. Hopf bifurcations
-Normal form of the Hopf bifurcation. Hopf
bifurcation theorem
-Computation of the first Liapunov coefficient
-Computation of the center
manifold
-Bautin
(generalized Hopf) bifurcation. Normal form. An
approximation of the parameters curve corresponding to fold bifurcation of
cycles.
4. Homoclinic
bifurcations
-
Homoclinic bifurcations in planar systems. Andronov-Leontovich theorem. Saddle-node homoclinic bifurcation. Double homoclinic
bifurcation
- Homoclinic
bifurcations in n-dimensional dynamical systems, n>2. Shilnikov
theorems.
5. Bogdanov-Takens bifurcation
-Bogdanov
normal form
-Takens
normal form. Topological equivalence between Bogdanov
normal form and Takens normal form
-An approximation of the
parameters curve corresponding to homoclinic
bifurcation
6. Heteroclinic
bifurcations
7. Global bifurcation diagram
8. Numerical analysis of bifurcations
- The software WINPP
(XPPAUT)
- Phase dynamics using
WINPP
- Parametric portrait using the package
LOCBIF from WINPP
Applications.
Bibliography
- Arrowsmith, D. K., Place, C. M.
An introduction to dynamical
systems, Cambridge University Press, Cambridge, 1990.
- Chow, S.N., Li, C., Wang, D.- Normal
forms and bifurcations of planar vector fields, Cambridge Univ.
Press, Cambridge, 1994.
- Chow, S.N., Hale, J.-Methods
of bifurcation theory, Springer, New-York, 1982.
- Dumortier, F., Roussarie, R., Sotomayor,
J., Zoladek, H. - Bifurcations of planar vector fields, nilpotent singularities
and abelian integrals, Springer, Berlin,
1991.
- Georgescu, A., Moroianu, M., Oprea, I. –Bifurcation theory. Principles and
applications, Pitesti Univ. Press, Pitesti, 1999 (Romanian).
- Giurgiteanu, N.- Computational economical and biological dynamics-DIECBI,
Europa, Craiova, 1997 (Romanian).
- Guckenheimer, J., Holmes, P. –Nonlinear oscillations, dynamical
systems and bifurcations of vector fields, Springer, New-York, 1983.
- Hale, J.K., Kocak, H.-Dynamics and bifurcations,
Springer, New York, 1991.
- Kuznetsov, Yu. - Elements of applied bifurcation
theory, Springer, New York, 1995.
- Murray, J.D.-Mathematical biology, Springer, Berlin, 1993.
- Rocsoreanu, C., Georgescu, A., Giurgiteanu,
N.-The FitzHugh-Nagumo
model. Bifurcation and dynamics, Kluwer
Academic Publishers, Dordrecht, 2000.
- Rocsoreanu, C.-Bifurcations of continuous dynamical
systems. Applications in economics and biology, Universitaria,
Craiova, 2006 (Romanian).
- Tu, P. - Dynamical
systems. An introduction with applications in economics and biology,
Springer, Berlin, 1994.
- Ermentrout, B. XPPAUT, http://www.math.pitt.edu/xpp/xpp.html.
- Ermentrout, B. - Simulating,
analyzing and animating dynamical systems: a guide to xppaut for researches and students, SIAM, 2002.
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