The birth of a cusp: The unfolding of a ‘boundary catastrophe’

Maier, Robert S.; Stein, Daniel L.
February 2000
AIP Conference Proceedings;2000, Vol. 502 Issue 1, p26
Academic Journal
We study a phenomenon, analogous to a phase transition, that takes place in the theory of noise-activated transitions between attractors. Large fluctuations away from any attractor tend to be concentrated along certain optimal trajectories, and the flow field of optimal trajectories may contain focal points, or cusps. Optimal trajectories may be viewed as Hamiltonian trajectories, and a cusp may be viewed as a cusp catastrophe in the Lagrangian manifold that the optimal trajectories trace out. As the parameters of a noise-driven system are changed, a cusp may emerge from a saddle point of the underlying deterministic dynamics. This corresponds to a cusp catastrophe being formed at the boundary of the Lagrangian manifold, and moving inward. It is possible to find a nonpolynomial normal form that unfolds the corresponding ‘boundary catastrophe,’ in a space of higher dimensionality. Just as the quartic normal form for a cusp catastrophe resembles the scaling form for a classical (Ginzburg-Landau) phase transition, so the normal form for a boundary catastrophe resembles the scaling form for a nonclassical phase transition. Both have continuously varying exponents. © 2000 American Institute of Physics.


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