STABLE PIECEWISE POLYNOMIAL VECTOR FIELDS
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Data
2012-09-22
Autores
Pessoa, Claudio [UNESP]
Sotomayor, Jorge
Título da Revista
ISSN da Revista
Título de Volume
Editor
Texas State Univ
Resumo
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.
Descrição
Palavras-chave
Structural stability, piecewise vector fields, compactification.
Como citar
Electronic Journal of Differential Equations. San Marcos: Texas State Univ, p. 15, 2012.