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Practical Bifurcation Theory

John J. TysonVirginia Polytechnic Institute

& Virginia Bioinformatics Institute

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References

• Strogatz, Nonlinear Dynamics and Chaos (Addison Wesley)

• Kuznetsov, Elements of Applied Bifurcation Theory (Springer)

• XPP-AUT www.math.pitt.edu/~bard/xpp

• Oscill8 http://oscill8.sourceforge.net

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The Dynamical Perspective

Molec Genetics Biochemistry Cell Biology

Kinetic Equations

State Space, Vector Field

Attractors, Transients, Repellors

Bifurcation Diagrams

Molecular Mechanism

Signal-Response Curves

The Curse ofParameter

Space

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Generic Bifurcations ofDynamical Systems

•Saddle-Node•Hopf•Cyclic Fold•Saddle Loop•Saddle-Node Invariant Circle

Rene Thom (1972), “StructuralStability and Morphogenesis”

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Torus

Heteroclinic

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Torus?

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Trajectories for the Fold-Hopf bifurcationsas illustrated in the Kuznetsov’s book.

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