THIRD SEMESTER

CM623: Bifurcation Theory with Applications to Biology

 

 

Prof. Carmen Rocşoreanu

 

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

 

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  2. Chow, S.N., Li, C., Wang, D.- Normal forms and bifurcations of planar vector fields, Cambridge Univ. Press, Cambridge, 1994.
  3. Chow, S.N., Hale, J.-Methods of bifurcation theory, Springer, New-York, 1982.
  4. Dumortier, F., Roussarie, R., Sotomayor, J., Zoladek, H. - Bifurcations of planar vector fields, nilpotent singularities and abelian integrals, Springer, Berlin, 1991.
  5. Georgescu, A., Moroianu, M., Oprea, I. Bifurcation theory. Principles and applications, Pitesti Univ. Press, Pitesti, 1999 (Romanian).
  6. Giurgiteanu, N.- Computational economical and biological dynamics-DIECBI, Europa, Craiova, 1997 (Romanian).
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  9. Kuznetsov, Yu. - Elements of applied bifurcation theory, Springer, New York, 1995.
  10. Murray, J.D.-Mathematical biology, Springer, Berlin, 1993.
  11. Rocsoreanu, C., Georgescu, A., Giurgiteanu, N.-The FitzHugh-Nagumo model. Bifurcation and dynamics, Kluwer Academic Publishers, Dordrecht, 2000.
  12. Rocsoreanu, C.-Bifurcations of continuous dynamical systems. Applications in economics and biology, Universitaria, Craiova, 2006 (Romanian).
  13. Tu, P. - Dynamical systems. An introduction with applications in economics and biology, Springer, Berlin, 1994.
  14. Ermentrout, B. XPPAUT, http://www.math.pitt.edu/xpp/xpp.html.
  15. Ermentrout, B. - Simulating, analyzing and animating dynamical systems: a guide to xppaut for researches and students, SIAM, 2002.