On the Poincaré–Bendixson Index Theorem for a Class of Piecewise Linear Differential Systems

• Ke Li ^[2] ; Shimin Li ^[1]
  1. [1] Hangzhou Normal University

     Hangzhou Normal University

     China

     
  2. [2] Hunan University of Information Technology
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 1, 2024
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Poincaré–Bendixson index theorem showed that: if planar smooth differential systems have only finitely number of singular points inside a limit cycle, then the sum of the indices at these singular points is 1. Hence there is at least one singular point lie inside a limit cycle. These results are useful in practice since its provide information about the existence and location of limit cycles. It is also well known that planar smooth linear system cannot have limit cycles. While in a recent paper Llibre and Teixeira (Nonl Dyn 88:157–164, 2017), Llibre and Teixeira constructed a planar piecewise linear differential systems formed by two linear differential systems separated by the straight line : {(x, y)|x = 0}, such that both linear differential have no singular points, neither real nor virtual, but it can have a limit cycle. This paper revisit these piecewise linear differential systems by regularization process. Our results show that there are three −singular points inside the limit cycle, and the sum of the indices at these singular points is 1. Thus the Poincaré-Bendixson index theorem is also valid for such piecewise linear differential systems.

• Referencias bibliográficas
  • 1. Bendixson, I.: Sur les courbes definies par des equations differentielles. Acta Math. 24, 14–22 (1913)
  • 2. Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise–Smooth dynamical systems: theory and applications. In: Applied Mathematical...
  • 3. Buzzi, C.A., Carvalho, T.de, Euzébio, R.D.: On Poincaré–Bendixson theorem and non-trival minimal sets in planar nonsmooth vector fields....
  • 4. Buzzi C.A., Carvalho T.de, Silva P.R. da.: Closed poly-trajectories and Poincaré index of non-smooth vector fields on the plane. J. Dyn....
  • 5. Carmona, V., Sanchez, F.F., Novaes, D.D.: Uniform upper bound for the number of limit cycles of planar piecewise linear differential systems...
  • 6. da Cruz, L.P.C., Torregrosa, J.: A Bendixon–Dulac theorem for some piecewise systems. Nonlinearity 33, 2455–2480 (2020)
  • 7. Dumortier, F., Llibre, J., Artés, J.: Qualitative Theory of Planar Differential Systems. Universitext, Springer, New York (2006)
  • 8. Euzebio, R.D., Gouveia, M.R.A.: Poincaré recurrence theorem for non-smooth vector fields. Z. Angew. Math. Phys. 68, 40 (2017)
  • 9. Freire, E., Ponce, E., Torres, F.: Canonical discontinuous planar piecewise linear systems. SIAM J. Appl. Dyn. Syst. 11, 181–211 (2012)
  • 10. Guardia, M., Seara, T.M., Teixeira, M.A.: Generic bifurcations of low codimension of planar Filippov systems. J. Diff. Equat. 250, 1967–2023...
  • 11. Li, J.: Hilbert’s 16th problem and bifurcation of planar polynomial vector fields. Int. J. Bifur. Chaos 13, 47–106 (2003)
  • 12. Li, F., Jin, Y., Tian, Y., Yu, P.: Integrability and linearizability of cubic Z2 systems with non-resonant singular points. J. Diff. Equ....
  • 13. Li, F., Liu, Y., Liu, Y., Yu, P.: Bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z2-equivariant cubic...
  • 14. Li, S., Llibre, J.: Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line. J....
  • 15. Li, S., Liu, C., Llibre, J.: The planar discontinuous piecewise linear refracting systems have at most one limit cycle. Nonlinear Anal....
  • 16. Llibre, J., Teixeira, M.A.: Piecewise linear differential systems without equilibria produce limit cycles? Nonl. Dyn. 88, 157–164 (2017)
  • 17. Poincare, H.: Memoire sur les courbes definies par une equation differentiells. J. Math. Pura Appl. 7, 375–422 (1881)
  • 18. Teixeira, M.A., daSilva, P.R.: Regularization and singular perturbation techniques for non-smooth systems. Physica D 241, 1948–1955 (2012)
  • 19. Zhang, Z., Ding, T., Huang, W., Dong, Z.: Qualitative Theory of Differential Equation. Transl. Math. Monogr. Providence, Rhode Island...