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

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.

The Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian Systems with Polycycles S(2) or S(3) Under Perturbations of Piecewise Smooth Polynomials

2020 ◽  
Vol 30 (15) ◽  
pp. 2050230
Author(s):  
Jiaxin Wang ◽  
Liqin Zhao
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Period Annulus ◽  
Quadratic Hamiltonian

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).

Bifurcation of a Kind of 1D Piecewise Differential Equation and Its Application to Piecewise Planar Polynomial Systems

2019 ◽  
Vol 29 (05) ◽  
pp. 1950072
Author(s):  
Jianfeng Huang ◽  
Yuye Jin
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Planar System ◽  
Order Analysis ◽  
First Order ◽  
Separation Curve ◽  
Straight Line ◽  
Melnikov Functions

This paper deals with a kind of piecewise smooth equation which is linear in the dependent variable. We study the problem of lower bounds for the maximum number of limit cycles of such equations using Melnikov functions. First of all, using the first order Melnikov function, we prove that these differential equations have a sharp upper bound for the number of the limit cycles which bifurcate from the periodic orbits and cross the separation straight line. Furthermore, in some cases the maximum number of these limit cycles is three, up to any order analysis. In the end, we apply this result on a kind of piecewise smooth planar system which has a separation curve with [Formula: see text] up to homomorphism.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

On the Number of Limit Cycles of Discontinuous Liénard Polynomial Differential Systems

2018 ◽  
Vol 28 (14) ◽  
pp. 1850175
Author(s):  
Fangfang Jiang ◽  
Zhicheng Ji ◽  
Yan Wang
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Second Order ◽  
Discontinuous Differential Equations ◽  
Differential Systems ◽  
Lower Upper ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Averaging Theorem

In this paper, we investigate the number of limit cycles for two classes of discontinuous Liénard polynomial perturbed differential systems. By the second-order averaging theorem of discontinuous differential equations, we provide several criteria on the lower upper bounds for the maximum number of limit cycles. The results show that the second-order averaging theorem of discontinuous differential equations can predict more limit cycles than the first-order one.

ASYMPTOTIC EXPANSIONS OF MELNIKOV FUNCTIONS AND LIMIT CYCLE BIFURCATIONS

2012 ◽  
Vol 22 (12) ◽  
pp. 1250296 ◽  
Author(s):  
MAOAN HAN
Keyword(s):  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Asymptotic Expansions ◽  
Melnikov Function ◽  
Heteroclinic Loop ◽  
First Order ◽  
Melnikov Functions ◽  
Limit Cycle Bifurcation ◽  
Nilpotent Center

In the study of the perturbation of Hamiltonian systems, the first order Melnikov functions play an important role. By finding its zeros, we can find limit cycles. By analyzing its analytical property, we can find its zeros. The main purpose of this article is to summarize some methods to find its zeros near a Hamiltonian value corresponding to an elementary center, nilpotent center or a homoclinic or heteroclinic loop with hyperbolic saddles or nilpotent critical points through the asymptotic expansions of the Melnikov function at these values. We present a series of results on the limit cycle bifurcation by using the first coefficients of the asymptotic expansions.

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

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.

Limit Cycle Bifurcations Near a Piecewise Smooth Generalized Homoclinic Loop with a Saddle-Fold Point

2017 ◽  
Vol 27 (05) ◽  
pp. 1750071 ◽  
Author(s):  
Feng Liang ◽  
Dechang Wang
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Homoclinic Loop ◽  
First Order ◽  
Period Annulus ◽  
Straight Line ◽  
Smooth Perturbation

In this paper, we suppose that a planar piecewise Hamiltonian system, with a straight line of separation, has a piecewise generalized homoclinic loop passing through a Saddle-Fold point, and assume that there exists a family of piecewise smooth periodic orbits near the loop. By studying the asymptotic expansion of the first order Melnikov function corresponding to the period annulus, we obtain the formulas of the first six coefficients in the expansion, based on which, we provide a lower bound for the maximal number of limit cycles bifurcated from the period annulus. As applications, two concrete systems are considered. Especially, the first one reveals that a quadratic piecewise Hamiltonian system can have five limit cycles near a generalized homoclinic loop under a quadratic piecewise smooth perturbation. Compared with the smooth case [Horozov & Iliev, 1994; Han et al., 1999], three more limit cycles are found.

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

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.

Bifurcation of Limit Cycles from a Polynomial Degenerate Center

2010 ◽  
Vol 10 (3) ◽  
Author(s):  
Adriana Buică ◽  
Jaume Giné ◽  
Jaume Llibre
Keyword(s):  
Limit Cycles ◽  
Upper Bounds ◽  
Bifurcation Of Limit Cycles ◽  
Period Annulus ◽  
Melnikov Functions

AbstractUsing Melnikov functions at any order, we provide upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the degenerate center ẋ = −y((x

scholarly journals Second Order Melnikov Functions of Piecewise Hamiltonian Systems

2020 ◽  
Vol 30 (01) ◽  
pp. 2050016
Author(s):  
Peixing Yang ◽  
Jean-Pierre Françoise ◽  
Jiang Yu
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Second Order ◽  
Quadratic Polynomial ◽  
First Order ◽  
Melnikov Functions

In this paper, we consider the general perturbations of piecewise Hamiltonian systems. A formula for the second order Melnikov functions is derived when the first order Melnikov functions vanish. As an application, we can improve an upper bound of the number of bifurcated limit cycles of a piecewise Hamiltonian system with quadratic polynomial perturbations.

Sign in / Sign up

Export Citation Format

Share Document