On the Dynamics of the Maximin Flow
Moody Chu
Department of Mathematics
North Carolina State University
Abstract
In a complex system, such as the molecular dynamics, chemical kinetics, nucleation mechanism, or even the Lagrangian of a constrained convex programming problem, the presence of a saddle point often represents that a transition of events has occurred. Determining the locations of saddle points in the con guration space and the way they a ect the transition provide critical information about the underlying complex system. This paper proposes a dynamical system approach to explore this problem. In addition to being capable of nding saddle points, the ow exhibits some intriguing behavior nearby a saddle point, which is demonstrated by graphic examples in various settings. Maximin ows also arise naturally from complex-valued di erential equations over analytic vector elds due to the Cauchy-Riemann equations. The maximin ow can be cast as a gradient ow in the Kre in space under inde nite inner product, whence the Lojasiewicz gradient inequality can be generalized. It is proved that a solution trajectory has nite arc length and, hence, converges to a singleton saddle point.
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