STABLE PIECEWISE POLYNOMIAL VECTOR FIELDS

Author(s)

Pessoa, Claudio ; Sotomayor, Jorge

• Document type: Journal article
• Source: Electronic Journal of Differential Equations; v. N/A, p. 15-pg., 2012-09-22.

Abstract

Let N = {y > 0} and S = {y < 0} be the semi-planes of R-2 having as common boundary the line D = {y = 0}. Let X and Y be polynomial vector fields defined in N and S, respectively, leading to a discontinuous piecewise polynomial vector field Z = (X, Y). This work pursues the stability and the transition analysis of solutions of Z between N and S, started by Filippov (1988) and Kozlova (1984) and reformulated by Sotomayor-Teixeira (1995) in terms of the regularization method. This method consists in analyzing a one parameter family of continuous vector fields Z(epsilon), defined by averaging X and Y. This family approaches Z when the parameter goes to zero. The results of Sotomayor-Teixeira and Sotomayor-Machado (2002) providing conditions on (X, Y) for the regularized vector fields to be structurally stable on planar compact connected regions are extended to discontinuous piecewise polynomial vector fields on R-2. Pertinent genericity results for vector fields satisfying the above stability conditions are also extended to the present case. A procedure for the study of discontinuous piecewise vector fields at infinity through a compactification is proposed here. (AU)

• FAPESP's process: 08/02841-4 - Topology, geometry and ergodic theory of dynamical systems
• Grantee: Jorge Manuel Sotomayor Tello
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 11/13152-8 - Qualitative theory of polynomial vector fields
• Grantee: Cláudio Gomes Pessoa
• Support Opportunities: Regular Research Grants