Persistence of periodic solutions from discontinuous planar piecewise linear Hamiltonian differential systems with three zones
Tóm tắt
In this paper, we study the number of limit cycles that can bifurcate from a period annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines. More precisely, we consider the case where the period annulus, bounded by a heteroclinic orbit or homoclinic loop, is obtained from a real center of the central subsystem, i.e. the system defined between the two parallel lines, and two real saddles of the others subsystems. Denoting by H(n) the number of limit cycles that can bifurcate from this period annulus by polynomial perturbations of degree n, we prove that if the period annulus is bounded by a heteroclinic orbit then
$$H(1)\ge 2$$
,
$$H(2)\ge 3$$
and
$$H(3)\ge 5$$
. Now, if the period annulus is bounded by a homoclinic loop then
$$H(1)\ge 3$$
,
$$H(2)\ge 4$$
and
$$H(3)\ge 7$$
. For this, we study the number of zeros of a Melnikov function for piecewise Hamiltonian system.
Tài liệu tham khảo
Buzzi, C.A., Pessoa, C., Torregrosa, J.: Piecewise linear perturbations of a linear center. Discrete Contin. Dyn. Syst. 33, 3915–3936 (2013)
Braga, D.C., Mello, L.F.: Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane. Nonlinear Dyn. 73, 1283–1288 (2013)
Chua, L.O., Lin, G.: Canonical realization of Chua’s circuit family. IEEE Trans. Circuits Syst. 37, 885–902 (1990)
di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer, New York (2008)
FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)
Fonseca, A. F., Llibre, J., Mello, L. F.: Limit cycles in planar piecewise linear Hamiltonian systems with three zones without equilibrium point. Int. J. Bifurc. Chaos Appl. Sci. Eng. 30, 2050157, 8 pp (2020)
Freire, E., Ponce, E., Rodrigo, F., Torres, F.: Bifurcation sets of continuous piecewise linear systems with two zones. Int. J. Bifurc. Chaos Appl. Sci. Eng. 8, 2073–2097 (1998)
Freire, E., Ponce, E., Torres, F.: The discontinuous matching of two planar linear foci can have three nested crossing limit cycles. Publ. Math. 2014, 221–253 (2014)
Hilbert, D.: Mathematical problems, M. Newton. Trans. Bull. Am. Math. Soc. 8, 437–479 (1902)
Hu, N., Du, Z.: Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems. Commun. Nonlinear Sci. Numer. Simul. 18, 3436–3448 (2013)
Li, Z., Liu, X.: Limit cycles in the discontinuous planar piecewise linear systems with three zones. Qual. Theory Dyn. Syst. 20, 79 (2021)
Lima, M., Pessoa, C., Pereira, W.: Limit cycles bifurcating from a periodic annulus in continuous piecewise linear differential systems with three zones. Int. J. Bifurc. Chaos Appl. Sci. Eng. 27, 1750022 (2017)
Liu, X., Han, M.: Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 5, 1–12 (2010)
Llibre, J., Novaes, D., Teixeira, M.A.: On the birth of limit cycles for non-smooth dynamical systems. Bull. Sci. Math. 139, 229–244 (2015)
Llibre, J., Teruel, E.: Introduction to the Qualitative Theory of Differential Systems. Planar, Symmetric and Continuous Piecewise Linear Systems, Birkhauser (2014)
Llibre, J., Ponce, E., Valls, C.: Uniqueness and non-uniqueness of limit cycles for piecewise linear differential systems with three zones and no symmetry. J. Nonlinear Sci. 25, 861–887 (2015)
Llibre, J., Teixeira, M.A.: Limit cycles for m-piecewise discontinuous polynomial Liénard differential equations. Z. Angew. Math. Phys. 66, 51–66 (2015)
Llibre, J., Teixeira, M.A.: Piecewise linear differential systems with only centers can create limit cycles? Nonlinear Dyn. 91, 249–255 (2018)
McKean, H.P., Jr.: Nagumo’s equation. Adv. Math. 4, 209–223 (1970)
Nagumo, J.S., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2071 (1962)
Wang, Y., Han, M., Constantinesn, D.: On the limit cycles of perturbed discontinuous planar systems with 4 switching lines. Chaos Soliton Fract. 83, 158–177 (2016)
Xiong, Y., Han, M.: Limit cycle bifurcations in discontinuous planar systems with multiple lines. J. Appl. Anal. Comput. 10, 361–377 (2020)
Xiong, Y., Wang, C.: Limit cycle bifurcations of planar piecewise differential systems with three zones. Nonlinear Anal. Real World Appl. 61, 103333 (2021)
Yang, J.: On the number of limit cycles by perturbing a piecewise smooth Hamilton system with two straight lines of separation. J. Appl. Anal. Comput. 6, 2362–2380 (2020)
Zhang, X., Xiong, Y., Zhang, Y.: The number of limit cycles by perturbing a piecewise linear system with three zones. Commun. Pure Appl. Anal. 21, 1833–1855 (2022)