52,219 research outputs found
On existence and uniqueness of the carrying simplex for competitive dynamical systems
Certain dynamical models of competition have a unique invariant hypersurface
to whichevery nonzero tractory is asymptotic, having simple geometry and
topology.Comment: Submitted to Journal of Biological Dynamics. 13 page
Common zeros of inward vector fields on surfaces
A vector field X on a manifold M with possibly nonempty boundary is inward if
it generates a unique local semiflow . A compact relatively open set K
in the zero set of X is a block. The Poincar\'e-Hopf index is generalized to an
index for blocks that may meet the boundary. A block with nonzero index is
essential.
Let X, Y be inward vector fields on surface M such that and let K be an essential block of zeros for X. Among the main results are
that Y has a zero in K if X and are analytic, or Y is and
preserves area. Applications are made to actions of Lie algebras and groups
Zero sets of Lie algebras of analytic vector fields on real and complex 2-manifolds
Let X be an analytic vector field on a real or complex 2-manifold, and K a
compact set of zeros of X whose fixed point index is not zero. Let A denote the
Lie algebra of analytic vector fields Y on M such that at every point of M the
values of X and [X,Y] are linearly dependent. Then the vector fields in A have
a common zero in K.
Application: Let G be a connected Lie group having a 1-dimensional normal
subgroup. Then every action of G on M has a fixed point.Comment: 22 page
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