Piecewise Implicit Differential Systems

• Published in: Journal of dynamics and differential equations 2017-12, Vol.29 (4), p.1519-1537
• Main Authors: Lopes, Bruno D., da Silva, Paulo R., Teixeira, Marco A.
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Differential equations; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Regularization; Singular perturbation; Singularity (mathematics)
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-016-9538-2

In this article we deal with non-smooth dynamical systems expressed by a piecewise first order implicit differential equations of the form x ˙ = 1 , y ˙ 2 = g 1 ( x , y ) if φ ( x , y ) ≥ 0 g 2 ( x , y ) if φ ( x , y ) ≤ 0 , where g 1 , g 2 , φ : U → R are smooth functions and U ⊆ R 2 is an open set. The main concern is to study sliding modes of such systems around some typical singularities. The novelty of our approach is that some singular perturbation problems of the form x ˙ = f ( x , y , ε ) , ( ε y ˙ ) 2 = g ( x , y , ε ) arise when the Sotomayor–Teixeira regularization is applied with ( x , y ) ∈ U , ε ≥ 0 , and f ,  g smooth in all variables.