24 crossing limit cycles in only one nest for piecewise cubic systems

Gouveia, Luiz F. S.; Torregrosa, Joan

doi:10.1016/j.aml.2019.106189

Cited by 20 publications

(16 citation statements)

References 21 publications

(31 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hilbert's sixteenth problem has also been considered in the context of planar piecewise polynomial vector fields (see, for instance, [12,18] and the references therein). Providing upper bounds for the maximum number of limit cycles in this context has also been shown to be very challenging, even in the linear case.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set

Novaes

2022

Preprint

View full text Add to dashboard Cite

The second part of the Hilbert's sixteenth problem consists in determining the upper bound H(n) for the number of limit cycles that planar polynomial vector fields of degree n can have. For n ≥ 2, it is still unknown whether H(n) is finite or not. The main achievements obtained so far establish lower bounds for H(n). Regarding asymptotic behavior, the best result says that H(n) grows as fast as n 2 log(n). Better lower bounds for small values of n are known in the research literature. In the recent paper "Some open problems in low dimensional dynamical systems" by A. Gasull, Problem 18 proposes another Hilbert's sixteenth type problem, namely improving the lower bounds for L(n), n ∈ N, which is defined as the maximum number of limit cycles that planar piecewise linear differential systems with two zones separated by a branch of an algebraic curve of degree n can have. So far, L(n) ≥ [n/2], n ∈ N, is the best known general lower bound. Again, better lower bounds for small values of n are known in the research literature. Here, by using a recently developed second order Melnikov method for nonsmooth systems with nonlinear discontinuity manifold, it is shown that L(n) grows as fast as n 2 . This will be achieved by providing lower bounds for L(n), which improves every previous estimates for n ≥ 4.

show abstract

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…The pseudo-Hopf bifurcation was considered by Filippov in his book [13] (see item b of page 241) and has been explored in [9]. When working with cyclicity problems, this bifurcation provides a usefull mechanism to improve the number of limit cycles (see, for instance, [12,18,33]).…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set

Novaes

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Application: Bifurcation of Limit Cycles. It is well known that Lyapunov coefficients can be used to study the appearance of small amplitude limit cycles in smooth and non-smooth vector fields around weak focuses (see, for instance, [11] for smooth vector fields and [3,7,8] for non-smooth vector fields). In this section, we apply the classical ideas to study the appearance of limit cycles around monodromic tangential singularities.…”

Section: Definitionmentioning

confidence: 99%

Lyapunov coefficients for monodromic tangential singularities in Filippov vector fields

Novaes,

Silva

2020

Preprint

View full text Add to dashboard Cite

In planar analytic vector fields, a monodromic singularity can be distinguished between a focus or a center by means of the Lyapunov coefficients, which are given in terms of the power series coefficients of the first-return map around the singularity. In this paper, we are interested in an analogous problem for monodromic tangential singularities of piecewise analytic vector fields Z = (Z + , Z − ). More specifically, for positive integers k + and k − , we consider a (2k + , 2k − )-monodromic tangential singularity, which is defined as an invisible contact of multiplicity 2k + and 2k − between the discontinuity manifold and, respectively, the vectors fields Z + and Z − for which a first-return map is well defined around the monodromic tangential singularity. We first prove that such a first-return map is analytic in a neighborhood of the monodromic tangential singularity. This allow us to define the Lyapunov coefficients. Then, as a consequence of a general property for pair of involutions, we obtain that the index of the first non-vanishing Lyapunov coefficient is always even. In addition, a recursive formula for computing all the Lyapunov coefficients is obtained. Such a formula is implemented in a Mathematica algorithm in the appendix. We also provide results regarding limit cycles bifurcating from monodromic tangential singularities. Several examples are analysed.

show abstract

“…A large number of works on this subject for smooth differential systems have emerged, such as in [2,5,7,14,17,18] and references therein. In recent years, much attention has been paid to this problem for piecewise smooth systems [1,4,6,8,10,11,12,22,23]. It is well known that there are two main methods, Melnikov function method and the averaging method, which are used to investigate the number of periodic solutions for smooth or piecewise smooth planar differential equations, see [2,5,6,7,10,11,12,13,14,15,16,17,18,21,22,23] and references therein.We denote by H p (n) the number of limit cycles of piecewise smooth polynomial differential systems of degree n. In [8], Huan and Yang provided an example and numerical simulations to illustrate that H p (1) ≥ 3.…”

mentioning

confidence: 99%

“…Recently, da Cruz, Novaes and Torregrosa in [1] showed a best lower bound for the piecewise quadratic differential systems, H p (2) ≥ 16. In recent work [4], Gouveia and Torregrosa provided that H p (3) ≥ 24, which is the best lower bound for the number of limit cycles for piecewise cubic polynomial vector fields.…”

mentioning

confidence: 99%

Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters

Cai¹,

Han²

2021

Communications on Pure &Amp; Applied Analysis

View full text Add to dashboard Cite

show abstract

24 crossing limit cycles in only one nest for piecewise cubic systems

Cited by 20 publications

References 21 publications

On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set

On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set

Lyapunov coefficients for monodromic tangential singularities in Filippov vector fields

Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters

Contact Info

Product

Resources

About