相关论文: Maximum number of limit cycles for Abel equation h…

In this paper, we investigate the maximum number of limit cycles of the reduced Abel equation $\dot{x}=A(t)x^{3}+B(t)x^{2}$ on an interval $[0,T]$. The Smale-Pugh problem asks whether this maximum number is bounded in terms of a given class…

This paper studies the number of limit cycles, known as the Smale-Pugh problem, for the generalized Abel equation \begin{align*} \frac{dx}{d\theta}=A(\theta)x^p+B(\theta)x^q, \end{align*} where $A$ and $B$ are are piecewise trigonometrical…

We study the number of rational limit cycles of the Abel equation $x'=A(t)x^3+B(t)x^2$, where $A(t)$ and $B(t)$ are real trigonometric polynomials. We show that this number is at most the degree of $A(t)$ plus one.

We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +C(t)x. Specifically, we find bounds on the number of its rational solutions when A(t), B(t) and C(t) are polynomials with real or complex coefficients; and on the number of…

A criterion is obtained for the semi-stability of the isolated singular positive closed solutions, i.e., singular positive limit cycles, of the Abel equation $x'=A(t)x^3+B(t)x^2$, where $A,B$ are smooth functions with two zeros in the…

In this paper, the existence and number of non-contractible limit cycles of the Josephson equation $\beta \frac{d^{2}\Phi}{dt^{2}}+(1+\gamma \cos \Phi)\frac{d\Phi}{dt}+\sin \Phi=\alpha$ are studied, where $\phi\in \mathbb S^{1}$ and…

In this paper we consider the limit cycles of the planar system $$\frac{d}{dt}(x,y)=\mathbf X_n+\mathbf X_m, $$ where $\mathbf X_n$ and $\mathbf X_m$ are quasi-homogeneous vector fields of degree $n$ and $m$ respectively. We prove that…

New criteria are established for upper bounds on the number of limit cycles of periodic Abel differential equations having two periodic invariant curves, one of them bounded. The criteria are applied to obtain upper bounds of either zero or…

The study of the dynamics of a continuous observable and non-controllable three-dimensional symmetric piecewise linear system with three zones can be reduced to the study of the existence of limit cycles for the piecewise differential…

In this paper, we study the maximum number of limit cycles for the piecewise smooth system of differential equations $\dot{x}=y, \ \dot{y}=-x-\varepsilon \cdot (f(x)\cdot y +{\rm sgn}(y)\cdot g(x))$. Using the averaging method, we were able…

A solution of the Abel equation $\dot{x}=A(t)x^3+B(t)x^2$ such that $x(0)=x(1)$ is called a periodic orbit of the equation. Our main result proves that if there exist two real numbers $a$ and $b$ such that the function $aA(t)+bB(t)$ is not…

Let $x'=S(t,x)$ be a differential equation in the cylinder, linear piecewise in $x$ and with trigonometric coefficients in $t$. In this paper, we provide an upper bound on the number of limit cycles in terms of the number of regions of the…

We analyze the dynamics of a 4-parameter family of planar ordinary differential equations, given by a polynomial of degree 5 that is equivariant under a symmetry of order 6. We obtain the number of limit cycles as a function of the…

We find an upper bound for the number of limit cycles, bifurcating from the 8-loop of the Duffing oscillator $x"= x-x^{3}$ under the special cubic perturbation $$ x"= x-x^{3}+\lambda_{1}y+\lambda_{2}x^{2}+\lambda_{3}xy+\lambda_{4}x^{2}y .…

We analyze the dynamics of a class of $\mathbb{Z}_{2n}$-equivariant differential equations on the plane, depending on 4 real parameters. This study is the generalisation to $\mathbb{Z}_{2n}$ of previous works with $\mathbb{Z}_4$ and…

This paper is devoted to study the limit cycle problem of a cubic reversible system with an isochronous center, when it is perturbed inside a class of polynomials. An upper bound of the number of limit cycles is obtained using the Abelian…

The present paper is devoted to the study of the maximum number of limit cycles bifurcated from the periodic orbits of the quadratic isochronous center $\dot{x}=-y+\frac{16}{3}x^{2}-\frac{4}{3}y^{2},\dot{y}=x+\frac{8}{3}xy$ by the averaging…

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…

We study limit cycles in piecewise complex systems with switching manifold $\mathbb{S}^1$. Using M\"obius transformations we establish an equivalence between circular and straight-line discontinuities that preserves periods, stability, and…

In this paper, using geometric properties of the field rotation parameters, we present a solution of Smale's Thirteenth Problem on the maximum number of limit cycles for Li\'{e}nard's polynomial system. We also generalize the obtained…