Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory

Liu, Shanshan; Han, Maoan

doi:10.3934/dcdss.2020133

Cited by 3 publications

(1 citation statement)

References 17 publications

(30 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, this theorem does not tell any information about how to determine an upper bound for the maximum number of 2π-periodic solutions of system (2.3) by averaged functions. In order to give an upper bound Han and his cooperator developed the averaging method, see Theorem 1.1 of [13] or Lemmas 2.1-2.3 of [21] where the relationship between the maximum number of zeros of the first two order averaged functions and the maximum number of 2π-periodic solutions of system (2.3) were established. Regarding the higher order averaged functions, an almost same proof to the one of [13, Theorem 1.1] yields the following result for system (2.3).…”

Section: And This Upper Bound Is Reached Formentioning

confidence: 99%

Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0

Li¹,

Llibre²

2021

CPAA

View full text Add to dashboard Cite

In this paper we study the maximum number of limit cycles bifurcating from the periodic orbits of the center <inline-formula><tex-math id="M1">\begin{document}$ \dot x = -y((x^2+y^2)/2)^m, \dot y = x((x^2+y^2)/2)^m $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ m\ge0 $\end{document}</tex-math></inline-formula> under discontinuous piecewise polynomial (resp. polynomial Hamiltonian) perturbations of degree <inline-formula><tex-math id="M3">\begin{document}$ n $\end{document}</tex-math></inline-formula> with the discontinuity set <inline-formula><tex-math id="M4">\begin{document}$ \{(x, y)\in\mathbb{R}^2: xy = 0\} $\end{document}</tex-math></inline-formula>. Using the averaging theory up to any order <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula>, we give upper bounds for the maximum number of limit cycles in the function of <inline-formula><tex-math id="M6">\begin{document}$ m, n, N $\end{document}</tex-math></inline-formula>. More importantly, employing the higher order averaging method we provide new lower bounds of the maximum number of limit cycles for several types of piecewise polynomial systems, which improve the results of the previous works. Besides, we explore the effect of 4-star-symmetry on the maximum number of limit cycles bifurcating from the unperturbed periodic orbits. Our result implies that 4-star-symmetry almost halves the maximum number.

show abstract

Section: And This Upper Bound Is Reached Formentioning

confidence: 99%

Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0

Li¹,

Llibre²

2021

CPAA

View full text Add to dashboard Cite

show abstract

The number of limit cycles for some polynomial systems with multiple parameters

Cai

Han

2022

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Melnikov functions and limit cycle bifurcations for a class of piecewise Hamiltonian systems

Hou,

Han

2024

MATH

View full text Add to dashboard Cite

<abstract>This study evaluated the number of limit cycles for a class of piecewise Hamiltonian systems with two zones separated by two semi-straight lines. First, we obtained explicit expressions of higher Melnikov functions. Then we applied these expressions to find the upper bounds of the number of limit cycles bifurcated from a period annulus of a piecewise polynomial Hamiltonian system.</abstract>

show abstract

Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory

Cited by 3 publications

References 17 publications

Limit cycles of piecewise polynomial differential systems with the discontinuity line <i>xy</i> = 0

Limit cycles of piecewise polynomial differential systems with the discontinuity line <i>xy</i> = 0

The number of limit cycles for some polynomial systems with multiple parameters

Melnikov functions and limit cycle bifurcations for a class of piecewise Hamiltonian systems

Contact Info

Product

Resources

About