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.

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 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.

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.

scholarly journals On the Limit Cycles of Linear Differential Systems with Homogeneous Nonlinearities

2015 ◽  
Vol 58 (4) ◽  
pp. 818-823 ◽  
Author(s):  
Jaume Llibre ◽  
Xiang Zhang
Keyword(s):  
Limit Cycles ◽  
Homogeneous Polynomials ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Linear Differential Systems ◽  
Nonexistence And Existence ◽  
Linear Differential

AbstractWe consider the class of polynomial differential systems of the form , where Pn and Qn are homogeneous polynomials of degree n. For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence

scholarly journals The Cubic Polynomial Differential Systems with two Circles as Algebraic Limit Cycles

2018 ◽  
Vol 18 (1) ◽  
pp. 183-193 ◽  
Author(s):  
Jaume Giné ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycles ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Cubic Polynomial ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

AbstractIn this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles.

Sign in / Sign up

Export Citation Format

Share Document