Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 80, pp. 1-16.
Title: Cyclic approximation to stasis
Authors: Stewart D. Johnson (Williams College, Williamstown, MA, USA)
Jordan Rodu (Williams College, Williamstown, MA, USA)
Abstract:
Neighborhoods of points in $\mathbb{R}^n$ where a positive linear
combination of $C^1$ vector fields sum to zero contain,
generically, cyclic trajectories that switch between the vector
fields. Such points are called stasis points, and the
approximating switching cycle can be chosen so that the timing of
the switches exactly matches the positive linear weighting. In the
case of two vector fields, the stasis points form one-dimensional
$C^1$ manifolds containing nearby families of two-cycles. The
generic case of two flows in $\mathbb{R}^3$ can be diffeomorphed
to a standard form with cubic curves as trajectories.
Submitted July 27, 2007. Published June 24, 2009.
Math Subject Classifications: 37C10, 37C27.
Key Words: Two-cycles; stasis points; switching systems;
piecewise smooth; relaxed controls.