Four Limit Cycles of Discontinuous Piecewise Differential Systems with Nilpotent Saddles Separated by a Straight Line

• Imane Benabdallah ^[1] ; Rebiha Benterki ^[1]
  1. [1] University Mohamed El Bachir El Ibrahimi of Bordj Bou Arréridj
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
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• Resumen

  • The study of discontinuous piecewise differential systems has attracted increasing attention in recent decades as a result of their applications to a variety of physical phenomena. The role of limit cycles in the study of any planar differential system is well known, but finding the maximum number of limit cycles that a class of planar differential systems can have is one of the most difficult tasks in the qualitative theory of planar differential systems. Thus, in this work, we solve the extinction of Hilbert problem for all classes of discontinuous piecewise differential systems with Hamiltonian nilpotent saddles, separated by the straight line x = 0. Firstly, we study the discontinuous piecewise differential system formed by linear center and one of the six Hamiltonian nilpotent saddles, we provide that these systems can have at most one limit cycle. Secondly, we consider all possible discontinuous piecewise differential systems formed by Hamiltonian nilpotent saddles in each piece, where we show that these 21 classes of piecewise differential systems can have at most four limit cycles.

    We have strengthened our result by giving examples for each case.

• Referencias bibliográficas
  • 1. Andronov, A., Vitt, A., Khaikin, S.: Theory of Oscillations. Pergamon Press, Oxford (1966)
  • 2. Belousov, B.H.: A periodic reaction and its mechanism, Collection of short hapers on radiation medicine for 1958. Med. Publ, Moscow (1959)
  • 3. Belfar, A., Benterki, R., Llibre, J.: Limit cycles of planar discontinuous piecewise linear Hamiltonian systems without equilibrium points...
  • 4. Benterki, R., Damene, L., Llibre, J.: The Limit Cycles of Discontinuous Piecewise Linear Differential Systems Formed by Centers and Separated...
  • 5. Benterki, R., Llibre, J.: The limit cycles of discontinuous piecewise linear differential systems formed by centers and separated by irreducible...
  • 6. Benterki, R., Llibre, L.: On the limit cycles of discontinuous piecewise linear differential systems formed by centers and separated by...
  • 7. Benterki, R., Llibre, L.: Crossing limit cycles of planar piecewise linear Hamiltonian systems without Equilibrium Points. Mathematics....
  • 8. Benterki, R., Llibre, J.: The solution of the second part of the 16th Hilbert problem for nine families of discontinuous piecewise differential...
  • 9. Braga, D.C., Mello, L.F.: Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane....
  • 10. Buzzi, C., Pessoa, C., Torregrosa, J.: Piecewise linear perturbations of a linear center. Discrete Contin. Dyn. Syst. 33, 3915–3936 (2013)
  • 11. Corbera, M., Valls, C.: Hamiltonian nilpotent saddles of linear plus cubic homogeneous polynomial vector fields. Nonlinear Analysis: Real...
  • 12. Esteban, M., Valls, C., Llibre, J.: The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated...
  • 13. Giannakopoulos, F., Pliete, K.: Planar systems of piecewise linear differential equations with a line of discontinuity. Nonlinearity....
  • 14. Han, M., Zhang, W.: On hopf bifurcation in non-smooth planar systems. J. Differential Equations. 248(9), 2399–2416 (2010)
  • 15. Huan, S.M., Yang, X.S.: On the number of limit cycles in general planar piecewise linear systems. Disc. Cont. Dyn. Syst. 32(6), 2147–2164...
  • 16. Jimenez, J.J., Llibre, J., Medrado, J.C.: Crossing limit cycles for a class of piecewise linear differential centers separated by a conic....
  • 17. Jimenez, J.J., Llibre, J., Medrado, J.C.: Crossing limit cycles for piecewise linear differential centers separated by a reducible cubic...
  • 18. Li, J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13, 47–106...
  • 19. Li, L.: Three crossing limit cycles in planar piecewise linear systems with saddle-focus type. Electron. J. Qual. Theory Differ. Equ....
  • 20. Llibre, J., Novaes, D.D., Teixeira, M.A.: Maximum number of limit cycles for certain piecewise linear dynamical systems. Nonlinear Dyn....
  • 21. Llibre, J., Ponce, E.: Three nested limit cycles in discontinuous piecewise linear differential systems with two zones. Dyn. Cont. Disc....
  • 22. Llibre, J., Teixeira, M.A.: Piecewise linear differential systems with only centers can create limit cycles? Nonlinear Dynam. 91, 249–255...
  • 23. Lum, R., Chua, L.O.: Global properties of continuous piecewise-linear vector fields. Part II: Simplest symmetric case in R2. Internat....
  • 24. Van Der Pol, B.: A theory of the amplitude of free and forced triode vibrations. Radio Review (later Wireless World). 1, 701–710 (1920)
  • 25. Van Der Pol, B.: On relaxation-oscillations. The London, Edinburgh and Dublin ppil. Mag. and J. of Sci. 2, 978–992 (1926)