Limit Cycles in Planar Piecewise Linear Hamiltonian Systems with Three Zones Without Equilibrium Points

Fonseca, Alexander Fernandes da; Llibre, Jaume; Mello, Luis Fernando

doi:10.1142/s0218127420501576

Cited by 21 publications

(9 citation statements)

References 12 publications

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“…Thus recently in [3,4,14,15] the authors studied the extension of the 16th Hilbert problem for discontinuous piecewise dierential systems formed by linear centers which have either two or more zones and they are separated by either conics, or reducible cubics, or irreducible cubics. In [2,5,6,9] the authors considered discontinuous piecewise linear Hamiltonian systems without equilibrium points where such systems are separated by either two parallel straight lines, or conics, or reducible cubics, or irreducible cubics, in each of these classes of piecewise dierential systems the authors determined the maximum number of crossing limit cycles that these piecewise linear systems can exhibit.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Limit Cycles of Planar Discontinuous Piecewise Linear Hamiltonian Systems Without Equilibria Separated by Nonregular Curves

Jimenez

Llibre

Valls

2022

Int. J. Bifurcation Chaos

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The problem of determining the existence, maximum number and positions of the limit cycles of the planar discontinuous piecewise linear differential systems is an important problem in the qualitative theory of differential systems. In this paper, we study two families of piecewise linear Hamiltonian systems without equilibria in [Formula: see text] separated by a nonregular curve. We provide the maximum number of crossing limit cycles that each family can have and show when this maximum is reached. In this way we are solving for each family the extended 16th Hilbert problem.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Limit Cycles of Planar Discontinuous Piecewise Linear Hamiltonian Systems Without Equilibria Separated by Nonregular Curves

Jimenez

Llibre

Valls

2022

Int. J. Bifurcation Chaos

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“…Recently, some researchers have been trying to estimate the number of limit cycles in discontinuous piecewise Hamiltonian differential systems with three zones. In this direction, we have papers with one limit cycle, see [8,18] and more than two limit cycles, see [26,27,28]. In this work, we contribute along these lines.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Limit Cycles Bifurcating from a Periodic Annulus in Discontinuous Planar Piecewise Linear Hamiltonian differential System with Three Zones

Pessoa,

Ribeiro

2021

Preprint

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In this paper, we study the number of limit cycles that can bifurcating from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines. We prove that if the central subsystem, i.e. the system defined between the two parallel lines, has a real center and the others subsystems have centers or saddles, then we have at least three limit cycles that appear after perturbations of periodic annulus. For this, we study the number of zeros of a Melnikov function for piecewise Hamiltonian system and present a normal form for this system in order to simplify the computations.

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“…In [14] authors proved that if Σ is a straight line then discontinuous piecewise linear differential centers have no crossing limit cycle. There are many other papers devoted to study the existence and the number of limit cycles of these systems when the curve of separation is a straight line, see for instance [2,8,9,10,11,12,15,18].…”

Section: Crossing Limit Cyclesmentioning

confidence: 99%

Limit cycles of discontinuous piecewise linear differential systems formed by centers or Hamiltonian without equilibria separated by irreducible cubics

Damene

Benterki

2021

Moroccan Journal of Pure and Applied Analysis

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The main goal of this paper is to provide the maximum number of crossing limit cycles of two different families of discontinuous piecewise linear differential systems. More precisely we prove that the systems formed by two regions, where, in one region we define a linear center and in the second region we define a Hamiltonian system without equilibria can exhibit three crossing limit cycles having two or four intersection points with the cubic of separation. After we prove that the systems formed by three regions, where, in two noadjacent regions we define a Hamiltonian system without equilibria, and in the third region we define a center, can exhibit six crossing limit cycles having four and two simultaneously intersection points with the cubic of separation.

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Limit Cycles in Planar Piecewise Linear Hamiltonian Systems with Three Zones Without Equilibrium Points

Abstract: We study the existence of limit cycles in planar piecewise linear Hamiltonian systems with three zones without equilibrium points. In this scenario, we have shown that such systems have at most one crossing limit cycle.

Cited by 21 publications

References 12 publications

Limit Cycles of Planar Discontinuous Piecewise Linear Hamiltonian Systems Without Equilibria Separated by Nonregular Curves

Limit Cycles of Planar Discontinuous Piecewise Linear Hamiltonian Systems Without Equilibria Separated by Nonregular Curves

Limit Cycles Bifurcating from a Periodic Annulus in Discontinuous Planar Piecewise Linear Hamiltonian differential System with Three Zones

Limit cycles of discontinuous piecewise linear differential systems formed by centers or Hamiltonian without equilibria separated by irreducible cubics

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