Bifurcation of Limit Cycles from a Polynomial Degenerate Center

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles [Formula: see text] or [Formula: see text] under the perturbations of piecewise smooth polynomials with degree [Formula: see text]. Roughly speaking, for [Formula: see text], a polycycle [Formula: see text] is cyclically ordered collection of [Formula: see text] saddles together with orbits connecting them in specified order. The discontinuity is on the line [Formula: see text]. If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary [Formula: see text] and [Formula: see text] are respectively [Formula: see text] and [Formula: see text] (taking into account the multiplicity).

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QUADRATIC PERTURBATIONS OF A CLASS OF QUADRATIC REVERSIBLE LOTKA–VOLTERRA SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741350137x ◽

2013 ◽

Vol 23 (08) ◽

pp. 1350137

Author(s):

YI SHAO ◽

A. CHUNXIANG

Keyword(s):

Limit Cycles ◽

Positive Answer ◽

Abelian Integrals ◽

Volterra System ◽

Bifurcation Of Limit Cycles ◽

Number Of Zeros ◽

Period Annulus ◽

Volterra Systems

This paper is concerned with the bifurcation of limit cycles of a class of quadratic reversible Lotka–Volterra system [Formula: see text] with b = -1/3. By using the Chebyshev criterion to study the number of zeros of Abelian integrals, we prove that this system has at most two limit cycles produced from the period annulus around the center under quadratic perturbations, which provide a positive answer for a case of the conjecture proposed by S. Gautier et al.

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Bifurcation of Limit Cycles in a Near-Hamiltonian System with a Cusp of Order Two and a Saddle

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416501807 ◽

2016 ◽

Vol 26 (11) ◽

pp. 1650180 ◽

Cited By ~ 1

Author(s):

Ali Bakhshalizadeh ◽

Hamid R. Z. Zangeneh ◽

Rasool Kazemi

Keyword(s):

Asymptotic Expansion ◽

Hamiltonian System ◽

Limit Cycle ◽

Limit Cycles ◽

Upper Bound ◽

Melnikov Function ◽

Bifurcation Of Limit Cycles ◽

Heteroclinic Loop ◽

First Order ◽

Period Annulus

In this paper, the asymptotic expansion of first-order Melnikov function of a heteroclinic loop connecting a cusp of order two and a hyperbolic saddle for a planar near-Hamiltonian system is given. Next, we consider the limit cycle bifurcations of a hyper-elliptic Liénard system with this kind of heteroclinic loop and study the least upper bound of limit cycles bifurcated from the period annulus inside the heteroclinic loop, from the heteroclinic loop itself and the center. We find that at most three limit cycles can be bifurcated from the period annulus, also we present different distributions of bifurcated limit cycles.

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Limit Cycles in the Equation of Whirling Pendulum With Piecewise Smooth Perturbations

10.21203/rs.3.rs-187696/v1 ◽

2021 ◽

Author(s):

Jihua Yang

Keyword(s):

Limit Cycles ◽

Upper Bounds ◽

Piecewise Smooth ◽

Chebyshev System ◽

Exact Bound ◽

First Order ◽

Pendulum Equation ◽

Melnikov Functions ◽

Switching Line ◽

Special Case

Abstract This paper deals with the problem of limit cycles for the whirling pendulum equation ẋ = y, ẏ = sin x(cos x-r) under piecewise smooth perturbations of polynomials of cos x, sin x and y of degree n with the switching line x = 0. The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained by using the Picard-Fuchs equations which the generating functions of the associated first order Melnikov functions satisfy. Further, the exact bound of a special case is given by using the Chebyshev system.

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On Behavior of the Periodic Orbits of a Hamiltonian System of Bifurcation of Limit Cycles

Mathematical Problems in Engineering ◽

10.1155/2021/1952706 ◽

2021 ◽

Vol 2021 ◽

pp. 1-6

Author(s):

Amor Menaceur ◽

Mohamed Abdalla ◽

Sahar Ahmed Idris ◽

Ibrahim Mekawy

Keyword(s):

Hamiltonian System ◽

Periodic Orbits ◽

Limit Cycles ◽

Polynomial Systems ◽

Upper Bounds ◽

Averaging Theory ◽

Bifurcation Of Limit Cycles ◽

First Order ◽

Nonlinear Anal ◽

Generalized Differential

In light of the previous recent studies by Jaume Llibre et al. that dealt with the finite cycles of generalized differential Kukles polynomial systems using the first- and second-order mean theorem such as (Nonlinear Anal., 74, 1261–1271, 2011) and (J. Dyn. Control Syst., vol. 21, 189–192, 2015), in this work, we provide upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits of Hamiltonian system using the averaging theory of first order.

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ON THE NUMBER OF LIMIT CYCLES IN PERTURBATIONS OF A QUADRATIC REVERSIBLE CENTER

Journal of the Australian Mathematical Society ◽

10.1017/s1446788712000213 ◽

2012 ◽

Vol 92 (3) ◽

pp. 409-423

Author(s):

JUANJUAN WU ◽

LINPING PENG ◽

CUIPING LI

Keyword(s):

Limit Cycles ◽

Abelian Integral ◽

Reversible System ◽

Linear Estimate ◽

Bifurcation Of Limit Cycles ◽

Arbitrary Degree ◽

Number Of Zeros ◽

Period Annulus

AbstractThis paper is concerned with the bifurcation of limit cycles from a quadratic reversible system under polynomial perturbations. It is proved that the cyclicity of the period annulus is two, and also a linear estimate of the number of zeros of the Abelian integral for the system under polynomial perturbations of arbitrary degreenis given.

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Inhomogeneous Fuchs equations and the limit cycles in a class of near-integrable quadratic systems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500027025 ◽

1997 ◽

Vol 127 (6) ◽

pp. 1207-1217 ◽

Cited By ~ 10

Author(s):

Iliya D. Iliev

Keyword(s):

Limit Cycles ◽

Displacement Function ◽

Quadratic System ◽

Degenerate Case ◽

Second Variation ◽

Centre Manifold ◽

Bifurcation Of Limit Cycles ◽

Quadratic Systems ◽

Period Annulus ◽

The Second Variation

We study the bifurcation of limit cycles in general quadratic perturbations of the particular quadratic system which represents one of two codimension-five components in the intersection of two strata in the centre manifold, and Q4. The study of limit cycles for this degenerate case requires us to investigate not the first but the second variation M2 of the displacement function. We prove that up to three limit cycles can emerge from the period annulus surrounding the centre. This implies that the cyclicity of period annuli of nearby systems in and Q4 is at most three as well. Our approach relies upon the possibility of deriving appropriate Picard–Fuchs equations satisfied by the four independent integrals included in M2.

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Limit Cycles Appearing From Piecewise Smooth Perturbations to a Reversible Nonlinear Center

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421502503 ◽

2021 ◽

Vol 31 (16) ◽

Author(s):

Dan Sun ◽

Linping Peng

Keyword(s):

Limit Cycles ◽

Upper Bounds ◽

Piecewise Smooth ◽

Unperturbed System ◽

Complex Method ◽

Discontinuous Systems ◽

Lower And Upper Bounds ◽

Argument Principle ◽

Period Annulus ◽

Limit Cycle Bifurcation

This paper deals with the limit cycle bifurcation from a reversible differential center of degree [Formula: see text] due to small piecewise smooth homogeneous polynomial perturbations. By using the averaging theory for discontinuous systems and the complex method based on the Argument Principle, we obtain lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus around the center of the unperturbed system.

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Bifurcation of Limit Cycles from the Center of a Quintic System via the Averaging Method

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127417500729 ◽

2017 ◽

Vol 27 (05) ◽

pp. 1750072 ◽

Cited By ~ 4

Author(s):

Bo Huang

Keyword(s):

Limit Cycles ◽

Upper Bound ◽

Averaging Method ◽

Unperturbed System ◽

Homogeneous Polynomials ◽

Bifurcation Of Limit Cycles ◽

First Order ◽

Period Annulus

This paper deals with the bifurcation of limit cycles for a quintic system with one center. Using the averaging method, we explain how limit cycles can bifurcate from the periodic annulus around the center of the considered system by adding perturbed terms which are the sum of homogeneous polynomials of degree [Formula: see text] for [Formula: see text]. We show that up to first-order averaging, at most five limit cycles can bifurcate from the period annulus of the unperturbed system for [Formula: see text], at most [Formula: see text] limit cycles can bifurcate from the periodic annulus of the unperturbed system for any [Formula: see text], and the upper bound is sharp for [Formula: see text] and for [Formula: see text].

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CYCLICITY OF PERIOD ANNULUS OF A QUADRATIC REVERSIBLE SYSTEM WITH DEGENERATE CRITICAL POINT

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741350106x ◽

2013 ◽

Vol 23 (06) ◽

pp. 1350106 ◽

Cited By ~ 1

Author(s):

HAIHUA LIANG ◽

JIANFENG HUANG

Keyword(s):

Critical Point ◽

Limit Cycles ◽

Upper Bound ◽

Outer Boundary ◽

Reversible System ◽

Bifurcation Of Limit Cycles ◽

Period Annulus ◽

Degenerate Critical Point

This paper is concerned with the bifurcation of limit cycles from the period annulus of a quadratic reversible system. The outer boundary of the period annulus contains a degenerate critical point. The exact upper bound of the number of limit cycles is given. Our result shows that the conjecture on the cyclicity of (r4) system is correct.

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