Piecewise Implicit Differential Systems


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


  1. 1.

    Arnold, V.I.: Geometrical methods in the theory of ordinary differential equations, 2nd edn. Springer, New york (1987)

    Google Scholar 

  2. 2.

    Banerjee, S., Verghese, G.C.: Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos, and Nonlinear Control. IEEE Press, Piscataway (2001)

    Book  Google Scholar 

  3. 3.

    Begely, C.J., Virgin, L.N.: Grazing bifurcations and basins of attraction in an impact-friction oscillator. Physica D 130, 43–57 (1999)

    Article  MATH  Google Scholar 

  4. 4.

    Brogliato, B.B., Heemels, W.P.M.H.: The complementarity class of hybrid dynamical systems. Eur. J. Control. 9, 311–319 (2003)

    MATH  Google Scholar 

  5. 5.

    Bruce, J.W., Tari, F.: Duality and implicit differential equations. Nonlinearity 13, 791–811 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Buzzi, C., Silva, P.R., Teixeira, M.A.: A Singular approach to discontinuous vector fields on the plane. J. Differ. Equ. 231, 633–655 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Davydov, A.A.: Qualitative theory of control systems. Translation of mathematical monographs, vol. 141. American Mathematical Society, Providence (1994)

    MATH  Google Scholar 

  8. 8.

    Davydov, A.A., Ishikawa, G., Izumiya, S., Sun, W.Z.: Generic singularities of implicit systems of first order differential equations on the plane. Jpn. J. Math. 3, 93–119 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-smooth dynamical systems. Theory and aplications. Applied Mathematical Science, vol. 163. Springer, London (2008)

    MATH  Google Scholar 

  10. 10.

    Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 53–98 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Filippov, A.F.: Differential equations with discontinuous right-hand sides, Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers, Dordrecht (1988)

    Book  Google Scholar 

  12. 12.

    Fossas, E., Hogan, S.J., Seara, T.M.: Two-parameter bifurcation curves in power electronic converters. Int. J. Bifur. Chaos Appl. Sci. Eng. 19, 349–357 (2009)

    Article  MathSciNet  Google Scholar 

  13. 13.

    Llibre, J., Silva, P.R., Teixeira, M.A.: Regularization of discontinuous vector fields via singular perturbation. J. Dyn. Differ. Equ. 19(2), 309–331 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Llibre, J., Silva, P.R., Teixeira, M.A.: Sliding vector fields via slow fast systems. Bull. Belg. Math. Soc. Simon Stevin 15, 851–869 (2008)

    MATH  MathSciNet  Google Scholar 

  15. 15.

    Llibre, J., Silva, P.R., Teixeira, M.A.: Study of singularities in non smooth dynamical systems via singular perturbation. SIAM J. Appl. Dyn. Syst. 8, 508–526 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Llibre, J., Teixeira, M.A.: Regularization of discontinuous vector fields in dimension three. Discret. Contin. Dyn. Syst. 3, 235–241 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

    Sotomayor, J., Teixeira, M.A.: Regularization of discontinuous vector fields. In: International conference on differential equations, vol. 95, pp. 207–223. Equadiff, Lisboa (1996)

  18. 18.

    Takens, F.: Implicit differential equations; some open problems. Singularités d’applications différentiables (Sém., Plans-sur-Bex. Lecture Notes in Math, vol. 535, pp. 237–253. Springer, Berlin (1976)

    Google Scholar 

Download references


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

Author information



Corresponding author

Correspondence to Bruno D. Lopes.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation


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