Piecewise Implicit Differential Systems

Abstract

In this article we deal with non-smooth dynamical systems expressed by a piecewise first order implicit differential equations of the form

$$\begin{aligned} \dot{x}=1,\quad \left( \dot{y}\right) ^2=\left\{ \begin{array}{lll} g_1(x,y) \quad \text{ if }\quad \varphi (x,y)\ge 0 \\ g_2(x,y) \quad \text{ if }\quad \varphi (x,y)\le 0 \end{array},\right. \end{aligned}$$

where \(g_1,g_2,\varphi :U\rightarrow \mathbb {R}\) are smooth functions and \(U\subseteq \mathbb {R}^2\) 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

$$\begin{aligned} \dot{x}= f(x,y,\varepsilon ) ,\quad (\varepsilon \dot{ y})^2=g ( x,y,\varepsilon ) \end{aligned}$$

arise when the Sotomayor–Teixeira regularization is applied with \((x, y) \in U\) , \(\varepsilon \ge 0\), and fg smooth in all variables.

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Acknowledgments

The authors are partially supported by CAPES, CNPq-Brazil, FAPESP and FP7-PEOPLE-2012-IRSES 318999.

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Correspondence to Bruno D. Lopes.

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Lopes, B.D., da Silva, P.R. & Teixeira, M.A. Piecewise Implicit Differential Systems. J Dyn Diff Equat 29, 1519–1537 (2017). https://doi.org/10.1007/s10884-016-9538-2

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Keywords

  • Non-smooth dynamical system
  • Implicit differential equation
  • Singular perturbation
  • Sliding vector fields