Upper bounds of limit cycles in Abel differential equations with invariant curves

Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation

Given two polynomials $P,q$ we consider the following question: "how large can the index of the first non-zero moment $\tilde{m}_k=\int_a^b P^k q$ be, assuming the sequence is not identically zero?". The answer $K$ to this question is known as the moment Bautin index, and we provide the first general upper bound: $K\leqslant 2+\mathrm{deg} q+3(\mathrm{deg} P-1)^2$. The proof is based on qualitative analysis of linear ODEs, applied to Cauchy-type integrals of certain algebraic functions. The moment Bautin index plays an important role in the study of bifurcations of periodic solution in the pol…

On the number of zeros of Abelian integrals for discontinuous quadratic differential systems

Applying the Picard-Fuchs equation to the discontinuous differential system, we obtain the upper bounds of the number of zeros for Abelian integrals of four kinds of quadratic differential systems when it is perturbed inside all discontinuous polynomials with degree $n$. Furthermore, by using the {\it Chebyshev criterion}, we obtain the sharp upper bounds on each period annulus for $n=2$.

具有线性三角函数系数的Abel方程极限环的最大个数

Maximum number of limit cycles for Abel equation having coefficients with linear trigonometric functions

本文针对系数为线性三角函数的Abel方程，推广已有思想提出新准则，完全解决了Smale-Pugh问题的最简单情形，证明极限环最大个数为三。

One Upper Estimate on the Number of Limit Cycles of Even Degree Liénard Equations in the Focus Case

We give an explicit upper bound for a number of limit cycles of the Liénard equation $\dot{x}=y-F(x)$, $\dot{y}=-x$ of even degree in the case its unique singular point $(0,0)$ is a focus.

Upper bounds for the number of limit cycles of some planar polynomial differential systems

We give an effective method for controlling the maximum number of limit cycles of some planar polynomial systems. It is based on a suitable choice of a Dulac function and the application of the well-known Bendixson-Dulac Criterion for multiple connected regions. The key point is a new approach to control the sign of the functions involved in the criterion. The method is applied to several examples.

Periodic Solutions of Abel Differential Equations

For a class of polynomial non-autonomous differential equations of degree n, we use phase plane analysis to show that each equation in this class has n periodic solutions. The result implies that certain rigid two-dimensional systems have at most one limit cycle which appears through multiple Hopf bifurcation.

Upper bounds for the number of orbital topological types of planar polynomial vector fields "modulo limit cycles"

The paper deals with planar polynomial vector fields. We aim to estimate the number of orbital topological equivalence classes for the fields of degree n. An evident obstacle for this is the second part of Hilbert's 16th problem. To circumvent this obstacle we introduce the notion of equivalence modulo limit cycles. This paper is the continuation of the author's paper in [Mosc. Math. J. 1 (2001), no. 4] where the lower bound of the form 2^{cn^2} has been obtained. Here we obtain the upper bound of the same form. We also associate an equipped planar graph to every planar polynomial vector field…

Josephson 方程的极限环个数

The number of limit cycles of Josephson equation

本文通过变量变换将 Josephson 方程的不可缩极限环转化为 Abel 方程的极限环，证明其至多有两个且上界尖锐，并结合已有结果得到可缩与不可缩极限环总数至多为两个。