The semiflow obtained by integrating the projection onto a submanifold with corners of Euclidean space of a smooth vector field

Abstract

We study the semiflow on a submanifold with corners M of Euclidean Space Rn obtained as follows. If a smooth vector field X is given on a neighbourhood of M in Rn we project it at each point of M onto the tangent cone to M at the point and integrate the resulting inner vector field X(M) on M: such systems arise in mathematical economics, mathematical biology and in the theory of electrical networks. We obtain an existence-uniqueness result and construct a device, the iteration, with which to study the local behaviour of trajectories, in particular in relation to the smooth flows obtained by projecting X onto individual strata of M. We investigate the relation between the iteration, right hand time derivatives of the trajectories, and generalisations of the classical tangency sets, establish a canonical form for intersections of the last and establish their generic properties. We investigate the local geometry of the semiflow and show that in most cases the classical theory has no simple generalisation to these systems, but using an ad hoc equivalence relation which respects the natural stratification of M we show that some significant local geometric results can be established. We show that if a condition involving the absence of infinite order tangencies is satisfied at a point then the number of stratum jumps made by the trajectories on a neighbourhood of this point is uniformly bounded, and we use this to show that the semiflow obtained from a residual subset of polynomial vector fields with M an orthant (this context includes the biological models) is in our strong stratum preserving sense locally stable near points x where X(M)(x) is non-vanishing. We consider briefly the global geometry of these systems, and in particular obtain a result with significant implications for the piece-wise linear systems occurring in mathematical biology which inspired the study.

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