On the non-existence of isochronous tangential centers in Filippov vector fields

Novaes, Douglas; Silva, Leandro

doi:10.1090/proc/16047

On the non-existence of isochronous tangential centers in Filippov vector fields
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by Douglas D. Novaes and Leandro A. Silva PDF

Proc. Amer. Math. Soc. 150 (2022), 5349-5358 Request permission

Abstract:

The isochronicity problem is a classical problem in the qualitative theory of planar vector fields which consists in characterizing whether a center is isochronous or not, that is, if all the trajectories in a neighborhood of the center have the same period. This problem is usually investigated by means of the so-called period function. In this paper, we are interested in exploring the isochronicity problem for tangential centers of planar Filippov vector fields. By computing the period function for planar Filippov vector fields around tangential centers, we show that such centers are never isochronous.

References

A. E. Bell, The horologium oscillatorium of Christian Huygens, Nature 148 (1941), no. 3748, 245-248.
Claudio Buzzi, Claudio Pessoa, and Joan Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst. 33 (2013), no. 9, 3915–3936. MR 3038046, DOI 10.3934/dcds.2013.33.3915
Javier Chavarriga and Marco Sabatini, A survey of isochronous centers, Qual. Theory Dyn. Syst. 1 (1999), no. 1, 1–70. MR 1747197, DOI 10.1007/BF02969404
Carmen Chicone, Ordinary differential equations with applications, 2nd ed., Texts in Applied Mathematics, vol. 34, Springer, New York, 2006. MR 2224508
Carmen Chicone and Marc Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989), no. 2, 433–486. MR 930075, DOI 10.1090/S0002-9947-1989-0930075-2
B. Coll, A. Gasull, and R. Prohens, Center-focus and isochronous center problems for discontinuous differential equations, Discrete Contin. Dynam. Systems 6 (2000), no. 3, 609–624. MR 1757390, DOI 10.3934/dcds.2000.6.609
Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128, DOI 10.1007/978-94-010-2196-8
K. da S. Andrade, O. M. L. Gomide, and D. D. Novaes, Qualitative analysis of polycycles in Filippov systems, arXiv:1905.11950, 2019.
A. F. Filippov, Differential equations with discontinuous righthand sides, Mathematics and its Applications (Soviet Series), vol. 18, Kluwer Academic Publishers Group, Dordrecht, 1988. Translated from the Russian. MR 1028776, DOI 10.1007/978-94-015-7793-9
Francesc Mañosas and Pedro J. Torres, Isochronicity of a class of piecewise continuous oscillators, Proc. Amer. Math. Soc. 133 (2005), no. 10, 3027–3035. MR 2159782, DOI 10.1090/S0002-9939-05-07873-1
Douglas D. Novaes and Gabriel Rondón, Smoothing of nonsmooth differential systems near regular-tangential singularities and boundary limit cycles, Nonlinearity 34 (2021), no. 6, 4202–4263. MR 4281444, DOI 10.1088/1361-6544/ac04be
Douglas D. Novaes and Leandro A. Silva, Lyapunov coefficients for monodromic tangential singularities in Filippov vector fields, J. Differential Equations 300 (2021), 565–596. MR 4302123, DOI 10.1016/j.jde.2021.08.008
I. I. Pleškan, A new method of research on the isochronism of a system of two differential equations, Differencial’nye Uravnenija 5 (1969), 1083–1090 (Russian). MR 0249715
Valery G. Romanovski and Douglas S. Shafer, The center and cyclicity problems: a computational algebra approach, Birkhäuser Boston, Ltd., Boston, MA, 2009. MR 2500203, DOI 10.1007/978-0-8176-4727-8
Valery G. Romanovski, Xingwu Chen, and Zhaoping Hu, Linearizability of linear systems perturbed by fifth degree homogeneous polynomials, J. Phys. A 40 (2007), no. 22, 5905–5919. MR 2345153, DOI 10.1088/1751-8113/40/22/010
Iván Sánchez-Sánchez and Joan Torregrosa, Criticality via first order development of the period constants, Nonlinear Anal. 203 (2021), Paper No. 112164, 21. MR 4164553, DOI 10.1016/j.na.2020.112164