Small amplitude limit cycles of polynomial differential equations

1983 ◽  
pp. 346-357 ◽  
Author(s):  
N. G. Lloyd
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Polynomial Differential Equations

The number of limit cycles of certain polynomial differential equations

1984 ◽  
Vol 98 (3-4) ◽  
pp. 215-239 ◽  
Author(s):  
T. R. Blows ◽  
N. G. Lloyd
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Equations ◽  
Image Position ◽  
Cubic Systems

SynopsisTwo-dimensional differential systemsare considered, where P and Q are polynomials. The question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.

scholarly journals On the polynomial limit cycles of polynomial differential equations

2011 ◽  
Vol 181 (1) ◽  
pp. 461-475 ◽  
Author(s):  
Jaume Giné ◽  
Maite Grau ◽  
Jaume Llibre
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Polynomial Differential Equations

scholarly journals On the limit cycles of polynomial differential systems with homogeneous nonlinearities

2000 ◽  
Vol 43 (3) ◽  
pp. 529-543 ◽  
Author(s):  
Chengzhi Li ◽  
Weigu Li ◽  
Jaume Llibre ◽  
Zhifen Zhang
Keyword(s):  
Differential Equations ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Homogeneous Polynomials ◽  
Differential Systems ◽  
Polynomial Differential Equations ◽  
Polynomial Differential Systems

AbstractWe consider three classes of polynomial differential equations of the form ẋ = y + establish Pn (x, y), ẏ = x + Qn (x, y), where establish Pn and Qn are homogeneous polynomials of degree n, having a non-Hamiltonian centre at the origin. By using a method different from the classical ones, we study the limit cycles that bifurcate from the periodic orbits of such centres when we perturb them inside the class of all polynomial differential systems of the above form. A more detailed study is made for the particular cases of degree n = 2 and n = 3.

The number of small-amplitude limit cycles of Liénard equations

1984 ◽  
Vol 95 (2) ◽  
pp. 359-366 ◽  
Author(s):  
T. R. Blows ◽  
N. G. Lloyd
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Small Amplitude ◽  
Extensive Literature ◽  
Second Order Differential Equations ◽  
Liénard Equations ◽  
Comprehensive Survey ◽  
Image Position

We consider second order differential equations of Liénard type:Such equations have been very widely studied and arise frequently in applications. There is an extensive literature relating to the existence and uniqueness of periodic solutions: the paper of Staude[6] is a comprehensive survey. Our interest is in the number of periodic solutions of such equations.

scholarly journals Simultaneous Bifurcation of Limit Cycles and Critical Periods

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Regilene D. S. Oliveira ◽  
Iván Sánchez-Sánchez ◽  
Joan Torregrosa
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Critical Periods ◽  
Bifurcation Of Limit Cycles ◽  
Period Function ◽  
Polynomial Differential Equations ◽  
Return Map ◽  
Lie Bracket ◽  
Liénard Systems ◽  
Quadratic Family

AbstractThe present work introduces the problem of simultaneous bifurcation of limit cycles and critical periods for a system of polynomial differential equations in the plane. The simultaneity concept is defined, as well as the idea of bi-weakness in the return map and the period function. Together with the classical methods, we present an approach which uses the Lie bracket to address the simultaneity in some cases. This approach is used to find the bi-weakness of cubic and quartic Liénard systems, the general quadratic family, and the linear plus cubic homogeneous family. We finish with an illustrative example by solving the problem of simultaneous bifurcation of limit cycles and critical periods for the cubic Liénard family.

scholarly journals On the Limit Cycles of the Polynomial Differential Systems with a Linear Node and Homogeneous Nonlinearities

2014 ◽  
Vol 24 (05) ◽  
pp. 1450065 ◽  
Author(s):  
Jaume Llibre ◽  
Jiang Yu ◽  
Xiang Zhang
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Homogeneous Polynomials ◽  
Differential Systems ◽  
Polynomial Differential Equations ◽  
Polynomial Differential Systems ◽  
Existence And Nonexistence

We consider the class of polynomial differential equations ẋ = λx + Pn(x, y), ẏ = μy + Qn(x, y) in ℝ2 where Pn(x, y) and Qn(x, y) are homogeneous polynomials of degree n > 1 and λ ≠ μ, i.e. the class of polynomial differential systems with a linear node with different eigenvalues and homogeneous nonlinearities. For this class of polynomial differential equations, we study the existence and nonexistence of limit cycles surrounding the node localized at the origin of coordinates.

A Cubic System with Eight Small-Amplitude Limit Cycles

1991 ◽  
Vol 47 (2) ◽  
pp. 163-171 ◽  
Author(s):  
E. M. JAMES ◽  
N. G. LLOYD
Keyword(s):  
Limit Cycles ◽  
Small Amplitude ◽  
Cubic System

Small-amplitude limit cycles in polynomial Li�nard systems

1996 ◽  
Vol 3 (2) ◽  
pp. 183-190 ◽  
Author(s):  
Colin J. Christopher ◽  
Noel G. Lloyd
Keyword(s):  
Limit Cycles ◽  
Small Amplitude
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