Electron. J. Diff. Eqns., Vol. 2009(2009), No. 80, pp. 1-16.

Cyclic approximation to stasis

Stewart D. Johnson, Jordan Rodu

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.

Show me the PDF file (625 KB), TEX file, and other files for this article.

Stewart Johnson
Bronfman Science Center, Williams College
Williamstown, MA 01267, USA
email: sjohnson@williams.edu
Jordan Rodu
Bronfman Science Center, Williams College
Williamstown, MA 01267, USA
email: jordan.rodu@gmail.com

Return to the EJDE web page