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Sliding Cycles of Regularized Piecewise Linear Visible–Invisible Twofolds

  • Renato Huzak [1] ; Kristian Uldall Kristiansen [2]
    1. [1] University of Hasselt

      University of Hasselt

      Arrondissement Hasselt, Bélgica

    2. [2] Technical University of Denmark

      Technical University of Denmark

      Dinamarca

  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº Extra 1, 2024
  • Idioma: inglés
  • DOI: 10.1007/s12346-024-01111-y
  • Enlaces
  • Resumen

    • The goal of this paper is to study the number of sliding limit cycles of regularized piecewise linear visible–invisible twofolds using the notion of slow divergence integral. We focus on limit cycles produced by canard cycles located in the half-plane with an invisible fold point. We prove that the integral has at most 1 zero counting multiplicity (when it is not identically zero). This will imply that the canard cycles can produce at most 2 limit cycles. Moreover, we detect regions in the parameter space with 2 limit cycles.

  • Referencias bibliográficas
    • 1. Álvarez, M.J., Coll, B., De Maesschalck, P., Prohens, R.: Asymptotic lower bounds on Hilbert numbers using canard cycles. J. Differ. Equ....
    • 2. Berger, E.J.: Friction modeling for dynamic system simulation. Appl. Mech. Rev. 55(6), 535–577 (2002)
    • 3. Bonet-Reves, C., Larrosa, J., M-Seara, T.: Regularization around a generic codimension one fold–fold singularity. J. Differ. Equ. 265(5),...
    • 4. Bossolini, E., Brøns, M., Kristiansen, K.U.: A stiction oscillator with canards: on piecewise smooth nonuniqueness and its resolution by...
    • 5. Braga, D.C., Mello, L.F.: Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane....
    • 6. Carmona, V., Fernández-Sánchez, F.: Integral characterization for Poincaré half-maps in planar linear systems. J. Differ. Equ. 305, 319–346...
    • 7. Carmona, V., Fernández-Sánchez, F., Garcia-Medina, E., Novaes, D.: Properties of Poincaré half-maps for planar linear systems and some...
    • 8. Carmona, V., Fernández-Sánchez, F., Novaes, D.D.: A succinct characterization of period annuli in planar piecewise linear differential...
    • 9. Carmona, V., Fernández-Sánchez, F., Novaes, D.D.: Uniform upper bound for the number of limit cycles of planar piecewise linear differential...
    • 10. Carmona, V., Fernández-Sánchez, F., Novaes, D.D.: Uniqueness and stability of limit cycles in planar piecewise linear differential systems...
    • 11. Carmona, V., Fernández-Sánchez, Fernando, Novaes, D.D.: A new simple proof for Lum–Chua’s conjecture. Nonlinear Anal. Hybrid Syst. 40,...
    • 12. Caubergh, M.: Hilbert’s sixteenth problem for polynomial Liénard equations. Qual. Theory Dyn. Syst. 11(1), 3–18 (2012)
    • 13. Cheesman, N.D., Hogan, S.J., Uldall Kristiansen, K.: The Painlevé paradox in three dimensions: resolution with regularization. Proc. R....
    • 14. De Maesschalck, P., Dumortier, F.: Time analysis and entry–exit relation near planar turning points. J. Differ. Equ. 215(2), 225–267 (2005)
    • 15. De Maesschalck, P., Dumortier, F.: Canard cycles in the presence of slow dynamics with singularities. Proc. R. Soc. Edinb. Sect. A 138(2),...
    • 16. De Maesschalck, P., Dumortier, F., Roussarie, R.: Canard cycles—from birth to transition, volume 73 of Ergebnisse der Mathematik und ihrer...
    • 17. De Maesschalck, P., Huzak, R.: Slow divergence integrals in classical Liénard equations near centers. J. Dyn. Differ. Equ. 27(1), 177–185...
    • 18. di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer, London...
    • 19. Dumortier, F., Panazzolo, D., Roussarie, R.: More limit cycles than expected in Liénard equations. Proc. Am. Math. Soc. 135(6), 1895–1904...
    • 20. Dumortier, F., Roussarie, F.: Canard cycles and center manifolds. Mem. Am. Math. Soc. 121(577), x+100 (1996). (With an appendix by...
    • 21. Dumortier, F., Roussarie, R.: Canard cycles with two breaking parameters. Discrete Contin. Dyn. Syst. 17(4), 787–806 (2007)
    • 22. Dumortier, F., Roussarie, R., Rousseau, C.: Hilbert’s 16th problem for quadratic vector fields. J. Differ. Equ. 110(1), 86–133 (1994)
    • 23. Esteban, M., Llibre, J., Valls, C.: The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated...
    • 24. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Mathematics and its Applications, Kluwer Academic Publishers,...
    • 25. Freire, E., Ponce, E., Rodrigo, F., Torres, F.: Bifurcation sets of continuous piecewise linear systems with two zones. Int. J. Bifurc....
    • 26. Freire, E., Ponce, E., Torres, F.: Canonical discontinuous planar piecewise linear systems. SIAM J. Appl. Dyn. Syst. 11(1), 181–211 (2012)
    • 27. Freire, E., Ponce, E., Torres, F.: Planar filippov systems with maximal crossing set and piecewise linear focus dynamics. Springer Proc....
    • 28. Gasull, A., Torregrosa, J., Zhang, X.: Piecewise linear differential systems with an algebraic line of separation. Electron. J. Differ....
    • 29. Guardia, M., Seara, T.M., Teixeira, M.A.: Generic bifurcations of low codimension of planar Filippov systems. J. Differ. Equ. 250(4),...
    • 30. Han, M., Zhang, W.: On Hopf bifurcation in non-smooth planar systems. J. Differ. Equ. 248(9), 2399– 2416 (2010)
    • 31. Huan, S.M., Yang, X.S.: On the number of limit cycles in general planar piecewise linear systems. Discrete Contin. Dyn. Syst. 32(6), 2147–2164...
    • 32. Huan, S.M., Yang, X.S.: Existence of limit cycles in general planar piecewise linear systems of saddle– saddle dynamics. Nonlinear Anal....
    • 33. Huan, S.M., Yang, X.S.: On the number of limit cycles in general planar piecewise linear systems of node–node types. J. Math. Anal. Appl....
    • 34. Huzak, R., De Maesschalck, P., Dumortier, F.: Limit cycles in slow–fast codimension 3 saddle and elliptic bifurcations. J. Differ. Equ....
    • 35. Huzak, R., Uldall Kristiansen, K.: General results on sliding cycles in regularized piecewise linear systems (in progress)
    • 36. Huzak, R., Uldall Kristiansen, K.: The number of limit cycles for regularized piecewise polynomial systems is unbounded. J. Differ. Equ....
    • 37. Huzak, R., Uldall Kristiansen, K., Radunovi´c, G.: Slow divergence integral in regularized piecewise smooth systems (2023). (submitted)
    • 38. Jelbart, S., Kristiansen, K.U., Wechselberger, M.: Singularly perturbed boundary-equilibrium bifurcations. Nonlinearity 34(11), 7371–7414...
    • 39. Jelbart, S., Kristiansen, K.U., Wechselberger, M.: Singularly perturbed boundary-focus bifurcations. J. Differ. Equ. 296, 412–492 (2021)
    • 40. Kristiansen, K.U., Hogan, S.J.: Resolution of the piecewise smooth visible–invisible two-fold singularity in R3 using regularization and...
    • 41. Kristiansen, K.U.: The regularized visible fold revisited. J. Nonlinear Sci. 30(6), 2463–2511 (2020)
    • 42. Kristiansen, K.U., Hogan, S.J.: Regularizations of two-fold bifurcations in planar piecewise smooth systems using blowup. SIAM J. Appl....
    • 43. Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174(2), 312–368 (2001)
    • 44. Kuznetsov, Yu.A., Rinaldi, S., Gragnani, A.: One parameter bifurcations in planar Filippov systems. Int. J. Bifurc. Chaos 13, 2157–2188...
    • 45. Li, S., Liu, S., Llibre, J.: The planar discontinuous piecewise linear refracting systems have at most one limit cycle. Nonlinear Anal....
    • 46. Li, S., Llibre, J.: Phase portraits of planar piecewise linear refracting systems: focus-saddle case. Nonlinear Anal. Real World Appl....
    • 47. Li, T., Llibre, J.: On the 16th Hilbert problem for discontinuous piecewise polynomial Hamiltonian systems. J. Dyn. Differ. Equ. 35, 1–16...
    • 48. Llibre, J., Ordóñez, M., Ponce, E.: On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without...
    • 49. Llibre, J., Ponce, E.: Three nested limit cycles in discontinuous piecewise linear differential systems with two zones. Dyn. Contin. Discrete...
    • 50. Llibre, J., Teixeira, M.A., Torregrosa, J.: Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential...
    • 51. Llibre, J., Teruel, A.E.: Introduction to the Qualitative Theory of Differential Systems. Planar, Symmetric and Continuous Piecewise Linear...
    • 52. Lum, R.: Global properties of continuous piecewise linear vector-fields. 1. Simplest case in r2. Int. J. Circuit Theory Appl. 19(3), 251–307...
    • 53. Medrado, J.C., Torregrosa, J.: Uniqueness of limit cycles for sewing planar piecewise linear systems. J. Math. Anal. Appl. 431(1), 529–544...
    • 54. Simpson, D.J.W.: A general framework for boundary equilibrium bifurcations of Filippov systems. Chaos 28(10), 103114 (2018)
    • 55. Smale, S.: Mathematical problems for the next century. In: Mathematics: Frontiers and Perspectives, pp. 271–294. American Mathematical...
    • 56. Sotomayor, J., Teixeira, M.A.: Regularization of discontinuous vector fields. In: Proceedings of the International Conference on Differential...
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