Periodic orbits from second order perturbation via rational trigonometric integrals

Prohens, R.; Torregrosa, Joan

doi:10.1016/j.physd.2014.05.002

Cited by 6 publications

(7 citation statements)

References 27 publications

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“…We also introduce new special integral functions as well as some of their properties and relations. The proofs follow closely the results from [32].…”

Section: Appendix: Explicit Computations Of the Integralssupporting

confidence: 67%

New lower bound for the Hilbert number in piecewise quadratic differential systems

Cruz

Novaes

Torregrosa

2019

Journal of Differential Equations

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We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by H p (n) the extension of the Hilbert number to degree n piecewise polynomial differential systems, then H p (2) ≥ 16. As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, all the limit cycles appear in one nest bifurcating from the period annulus of some isochronous quadratic centers.2010 Mathematics Subject Classification. Primary: 37G15, 37C27. Secundary: 37G10, 34C07. Key words and phrases. Non-smooth differential system, limit cycles in piecewise quadratic differential systems, first and second order perturbations of isochronous quadratic systems, Hilbert number for piecewise quadratic differential systems.

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“…We also introduce new special integral functions as well as some of their properties and relations. The proofs follow closely the results from [32].…”

Section: Appendix: Explicit Computations Of the Integralssupporting

confidence: 67%

New lower bound for the Hilbert number in piecewise quadratic differential systems

Cruz

Novaes

Torregrosa

2019

Journal of Differential Equations

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“…The main application in [15] is the computation of the Lyapunov constants for piecewise systems and their use in the center-focus problem. Other applications of this method can be found in [30,31]. This procedure is useful not only to discuss the weak 16th Hilbert's problem, but also to study related problems such that the persistence of centers under small perturbations, and the study of the period function for centers, see [5].…”

Section: Introductionmentioning

confidence: 99%

Limit cycles in planar piecewise linear differential systems with nonregular separation line

Cardin

Torregrosa

2016

Physica D: Nonlinear Phenomena

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Abstract. In this paper we deal with lanar piecewise linear differential systems defined in two zones. We consider the case when the two linear zones are angular sectors of angles α and 2π − α, respectively, for α ∈ (0, π). We study the problem of determining lower bounds for the number of isolated periodic orbits in such systems using Melnikov functions. These limit cycles appear studying higher order piecewise linear perturbations of a linear center. It is proved that the maximum number of limit cycles that can appear up to a sixth order perturbation is five. Moreover, for these values of α, we prove the existence of systems with four limit cycles up to fifth order and, for α = π/2, we provide an explicit example with five up to sixth order. In general, the nonregular separation line increases the number of periodic orbits in comparison with the case where the two zones are separated by a straight line.

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“…More concretely, in the latter this theory has been employing to obtain the upper bound for the number of zeros of the Melnikov functions, see e.g. [4,7,9,13,15,16,19,20,25,26,30,31,[40][41][42]44] and the references therein. Actually these studies provide lower bounds for the so called weaken Hilbert 16th problem, see e.g.…”

Section: Introduction and Statements Of Main Resultsmentioning

confidence: 99%

“…with ξ [0,π] = 1 and ν = 1 had been applied by Huang et al in [26]. Prohens and Torregrosa [41] in 2014 used the explicit expressions of the families in (8) with ν = 1 and α ∈ Z + to investigate periodic orbits via the second order perturbations of some quadratic isochronous centers. These explicit expressions in some particular forms has also been utilized in [9,29] for obtaining the lower bounds on the Hilbert number of some planar piecewise smooth differential systems.…”

Section: Introduction and Statements Of Main Resultsmentioning

confidence: 99%

A new Chebyshev criterion and its application to planar differential systems

Huang¹,

Liang²,

Zhang³

2020

Preprint

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This paper establishes a new Chebyshev criterion for some family of integrals. By virtue of this criterion we obtain several new Chebyshev families. With the help of these new families we can answer the conjecture posed by Gasull et al in 2015. Their applications to other two planar differential systems also show that our approach is simpler and in a unified way to handle many kinds of planar differential systems for estimating the number of limit cycles bifurcating from period annulus.

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Periodic orbits from second order perturbation via rational trigonometric integrals

Cited by 6 publications

References 27 publications

New lower bound for the Hilbert number in piecewise quadratic differential systems

New lower bound for the Hilbert number in piecewise quadratic differential systems

Limit cycles in planar piecewise linear differential systems with nonregular separation line

A new Chebyshev criterion and its application to planar differential systems

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