scholarly journals Rational Limit Cycles on Abel Polynomial Equations

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 885 ◽  
Author(s):  
Claudia Valls
Keyword(s):  
Limit Cycles ◽  
Upper Bounds ◽  
Polynomial Equations ◽  
Abel Polynomial ◽  
Real Polynomials ◽  
Abel Equations

In this paper we deal with Abel equations of the form d y / d x = A 1 ( x ) y + A 2 ( x ) y 2 + A 3 ( x ) y 3 , where A 1 ( x ) , A 2 ( x ) and A 3 ( x ) are real polynomials and A 3 ≢ 0 . We prove that these Abel equations can have at most two rational (non-polynomial) limit cycles when A 1 ≢ 0 and three rational (non-polynomial) limit cycles when A 1 ≡ 0 . Moreover, we show that these upper bounds are sharp. We show that the general Abel equations can always be reduced to this one.

scholarly journals LIMIT CYCLES FOR GENERALIZED ABEL EQUATIONS

2006 ◽  
Vol 16 (12) ◽  
pp. 3737-3745 ◽  
Author(s):  
ARMENGOL GASULL ◽  
ANTONI GUILLAMON
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Periodic Functions ◽  
One Dimensional ◽  
Nonautonomous Differential Equations ◽  
Abel Equations ◽  
The Right

This paper deals with the problem of finding upper bounds on the number of periodic solutions of a class of one-dimensional nonautonomous differential equations: those with the right-hand sides being polynomials of degree n and whose coefficients are real smooth one-periodic functions. The case n = 3 gives the so-called Abel equations which have been thoroughly studied and are well understood. We consider two natural generalizations of Abel equations. Our results extend previous works of Lins Neto and Panov and try to step forward in the understanding of the case n > 3. They can be applied, as well, to control the number of limit cycles of some planar ordinary differential equations.

Characterization of the Existence of Non-trivial Limit Cycles for Generalized Abel Equations

2021 ◽  
Vol 20 (1) ◽  
Author(s):  
M. J. Álvarez ◽  
J. L. Bravo ◽  
M. Fernández ◽  
R. Prohens
Keyword(s):  
Limit Cycles ◽  
Abel Equations

On the rational limit cycles of Abel equations

2018 ◽  
Vol 110 ◽  
pp. 28-32 ◽  
Author(s):  
Changjian Liu ◽  
Chunhui Li ◽  
Xishun Wang ◽  
Junqiao Wu
Keyword(s):  
Limit Cycles ◽  
Abel Equations

scholarly journals Crossing Limit Cycles of Planar Piecewise Linear Hamiltonian Systems without Equilibrium Points

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 755
Author(s):  
Rebiha Benterki ◽  
Jaume LLibre
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Upper Bounds ◽  
Linear Hamiltonian Systems ◽  
Straight Lines

In this paper, we study the existence of limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points. Firstly, we prove that if these systems are separated by a parabola, they can have at most two crossing limit cycles, and if they are separated by a hyperbola or an ellipse, they can have at most three crossing limit cycles. Additionally, we prove that these upper bounds are reached. Secondly, we show that there is an example of two crossing limit cycles when these systems have four zones separated by three straight lines.

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.

Alien limit cycles in Abel equations

2020 ◽  
Vol 482 (1) ◽  
pp. 123525 ◽  
Author(s):  
M.J. Álvarez ◽  
J.L. Bravo ◽  
M. Fernández ◽  
R. Prohens
Keyword(s):  
Limit Cycles ◽  
Abel Equations

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).

scholarly journals On the Number of Limit Cycles in Generalized Abel Equations

2020 ◽  
Vol 19 (4) ◽  
pp. 2343-2370
Author(s):  
Jianfeng Huang ◽  
Joan Torregrosa ◽  
Jordi Villadelprat
Keyword(s):  
Limit Cycles ◽  
Abel Equations
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