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






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