Abstract
In this paper, we study Poincaré bifurcation of limit cycles from a piecewise linear Hamiltonian system with a center at the origin and a homoclinic loop round the origin. By using the Melnikov function method, we give an estimation of the number of limit cycles which bifurcate from the period annulus between the center and the homoclinic loop under the piecewise polynomial perturbations of degree n. This result confirms a conjecture.
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References
Filippov, A.F.: Differential Equations with Discontinuous Right-hand Sides. Kluwer Academic Publishers, Dordrecht (1988)
Bernaedo, M., Budd, C.J., Champneys, A.R., et al.: Piecewise-smooth dynamical systems: theory and applications. Applied Mathematical Sciences, 163, Springer (2008)
Luo, A.C.J.: Discontinuous Dynamical Systems. Springer, Berlin (2012)
Chen, X., Huang, L.: A Filippov system describing the effect of prey refuge use on a ratio-dependent predator-prey model. J. Math. Anal. Appl. 428, 817–837 (2015)
Liu, X., Han, M.: Bifurcation of limit cycles by perturbating piecewise Hamiltonian systems. Int. J. Bifurcat. Chaos. 20(5), 1379–1390 (2010)
Han, M., Sheng, L.: Bifurcation of limit cycles in piecewise smooth system via Melnikov function. J. Appl. Anal. Comput. 5(4), 809–815 (2015)
Han, M.: On the maximal number of periodic solution of piecewise smooth periodic equations by average method. J. Appl. Anal. Comput. 7(2), 788–794 (2017)
Llibre, J., Mereu, A.C., Novaes, D.D.: Averaging theory for discontinuous piecewise differential systems. J. Differ. Equ. 258, 4007–4032 (2015)
Llibre, J., Novaes, D.D., Teixeira, M.A.: On the birth of limit cycles for non-smooth dynamical systems. Bull. Sci. math. 139(3), 229–244 (2015)
Liang, F., Han, M., Romanovski, V.G.: Bifurcation of limit cycles by perturbating a piecewise linear Hamiltonian system with a homoclinic loop. Nonlinear Anal-Real. 75, 4355–4374 (2012)
Xiong,Yanqin, Jianqiang, H.: A class of reversible quadratic systems with piecewise polynomial perturbations. Appl. Math. Comput. 362, 124527 (2019)
Wang, Y., Han, M.: Limit cycles bifurcations by perturbing a class of integrable systems with polycycle. J. Math. Anal. Appl. 418, 357–386 (2014)
Wang, Y., Han, M.: Dana Constantinesn, On the limit cycles of perturbed discontinuous planar systems with 4 switching lines. Chaos Soliton Fract. 83, 158–177 (2016)
Yang, J., Zhao, L.: Bounding the number of limit cycles of discontinuous differential systems by using Picard-Fuchs equations. J. Differ. Equ. 264, 5734–5757 (2018)
Cen, X., Liu, C., Yang, L., Zhang, M.: Limit cycles by perturbing quadratic isochronous centers inside piecewise polynomial differential systems. J. Differ. Equ. 265, 6083–6126 (2018)
Llibre, J., Mereu, A.C.: Limit cycles for discontinuous quadratic differential systems with two zones. J. Math. Anal. Appl. 413, 763–775 (2014)
Itikawa, J., Llibre, J., Mereu, C.: Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones. Discrete Cont. Dyn. B. 22, 3259–3272 (2017)
Li, S., Liu, C.: A linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system. J. Math. Anal. Appl. 428, 1354–1367 (2015)
Coll, B., Gasull, A., Prohens, R.: Bifurcation of limit cycles from two families of centers. Dyn. Cont. Dis. Ser. A. 12, 275–287 (2005)
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This study was funded by National Natural Science Foundation of China (Nos. 11931016 and 11771296), Hunan Province Natural Science Foundation (No. 2018JJ3866), School Youth Foundation of Central South University of Forestry and Technology (No. QJ2017012B), Post Doctor Start-up Foundation of Zhejiang Normal University (No. ZC304019016).
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Xiaoyan Chen declares that she has no conflict of interest. Maoan Han declares that he has no conflict of interest.
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Chen, X., Han, M. A Linear Estimate of the Number of Limit Cycles for A Piecewise Smooth Near-Hamiltonian System. Qual. Theory Dyn. Syst. 19, 61 (2020). https://doi.org/10.1007/s12346-020-00398-x
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DOI: https://doi.org/10.1007/s12346-020-00398-x