M. A. M. Alwash
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.
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We obtain direct and inverse approximation theorems of $2π$-periodic functions by Taylor--Abel--Poisson operators in the integral metric.
The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation $x\mapsto x^{-1}$, and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution …
The Polynomial Abel differential equations are considered a model problem for the classical Poincaré center--focus problem for planar polynomial systems of ordinary differential equations. Last decades several works pointed out that all the centers of the polynomial Abel differential equations satisfied the composition conditions (also called universal centers). In this work we provide a simple counterexample to this conjecture.
The Abel differential equation $y'=p(x)y^2+q(x)y^3$ with $p,q\in \mathbb R[x]$ is said to have a center on a segment $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(b)=y(a)$. The problem of description of conditions implying that the Abel equation has a center may be interpreted as a simplified version of the classical Center-Focus problem of Poincaré. The Abel equation is said to have a "parametric center" if for each $\varepsilon \in \mathbb R$ the equation $y'=p(x)y^2+\varepsilon q(x)y^3$ has a center. In this paper we show that the Abel eq…
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 one limit cycle for planar differential systems.
We study the structure of the set of harmonic solutions to perturbed nonautonomous, T-periodic, separated variables ODEs on manifolds. The perturbing term is allowed to contain a finite delay and to be T-periodic in time.
For an analytic differential system in $\mathbb R^n$ with a periodic orbit, we will prove that if the system is analytically integrable around the periodic orbit, i.e. it has $n-1$ functionally independent analytic first integrals defined in a neighborhood of the periodic orbit, then the system is analytically equivalent to its Poincaré--Dulac type normal form. This result is an extension for analytic integrable differential systems around a singularity to the ones around a periodic orbit.
This paper is dedicated to present an exact solution for a nonlinear differential equation so-called Abel equation. This equation was known as one of the group of unsolvable differential equations. The present method is applicable for any arbitrary form of nonlinear coefficients of Abel equation. On the other hands, the presented solution in this paper is verified with exact solution for some restricted forms of Abel equation that has been reported by Polyanin and Zaytsev [5]. Also, numerical solution is utilized to verify this method. All verifications have approved that the present method is…
Delay differential equations with periodic coefficients: a numerical insight
本文通过数值方法研究带时滞与非线性负周期反馈的标量微分方程，证明了在周期容许范围内存在与反馈系数同周期的缓慢振荡周期解。
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 rational solutions of the Abel equation $x'=A(t)x^3+B(t)x^2$ where $A,B\in C[t]$. We prove that if $deg(A)$ is even or $deg(B)>(deg(A)-1)/2$ then the equation has at most two rational solutions. For any other case, an upper bound on the number of rational solutions is obtained. Moreover, we prove that if there are more than $(deg(A)+1)/2$ rational solutions then the equation admits a Darboux first integral.
Some inverse problems for semi-infinite periodic generalized Jacobi matrices are considered. In particular, a generalization of the Abel criterion is presented. The approach is based on the fact that the solvability of the Pell-Abel equation is equivalent to the existence of a certainly normalized $J$-unitary $2\times 2$-matrix polynomial (the monodromy matrix).
In this paper we investigate the existence and uniqueness of bounded, periodic and almost periodic solutions for second order differential equations involving reflection of the argument.The relationship between frequency modules of forced term and the solution of the equation is considered.
In this paper, we study the existence and uniqueness of periodic solutions of the differential equation of the form . Here, we obtain some sufficient conditions which guarantee the existence of periodic solutions. This equation is a quite general third- order nonlinear vector differential equation, and one example is given for illustration of the subject.
Symbolic Integration of Differential Forms: From Abel to Zeilberger
本文通过历史与现代方法，探讨微分形式的符号积分，涵盖阿贝尔定理到泽尔伯格抵消法，提供算法框架处理代数与超越积分问题。
Motivated by Lazer-Leach type results, we study the existence of periodic solutions for systems of functional-differential equations at resonance with an arbitrary even-dimensional kernel and linear deviating terms involving a general delay of the form $\int_0^{2π}u(t+s)\,dλ(s)$, where $λ$ is a finite regular signed measure. Our main technique shall be the Coincidence Degree Theorem due to Mawhin.
We establish several delay-independent criteria for the existence and stability of positive periodic solutions of n-dimensional nonautonomous functional differential equation by several fixed point theorems. Examples from positive and negative growth feedback systems demonstrate the results.
We investigate the structure of the set of $T$-periodic solutions to periodically perturbed coupled delay differential equations on differentiable manifolds. By using fixed point index and degree-theoretic methods we prove the existence of branches of $T$-periodic solutions to the considered equations. As main application of our methods, we study a generalized version of the sunflower equation.
Existence of periodic solutions for a scalar differential equation modelling optical conveyor belts
针对模拟光传送带的一维常微分方程，研究了其周期解的存在性，在物理相关特例下证实周期解存在，且一般假设下周期解集合有界。
In this paper we investigate the properties of the set of T-periodic solutions of semi-explicit parametrized Differential-Algebraic Equations with non-autonomous constraints of a particular type. We provide simple, degree theoretic conditions for the existence of branches of T-periodic solutions of the considered equations. Our approach is based on topological arguments about differential equations on implicitly defined manifolds, combined with elementary facts of matrix analysis.
In this paper, we study the existence and uniqueness of pseudo $S$-asymptotically $ω$-periodic mild solutions of class $r$ for fractional integro-differential neutral equations. An example is presented to illustrate the application of the abstract results.
In this paper, we provide a necessary and sufficient condition ensuring the property of exponential dichotomy for periodic linear systems of generalized differential equations. This condition allow us to revisit a recent result of admissibility, obtaining an alternative formulation with a particular simplicity.
In the present paper we establish the necessary and sufficient conditions for two generalized Abel differential equations to be locally equivalent under the action of the pseudogroup of linear transformations of the form $\{x\mapsto f(x),~ y\mapsto g(x)\cdot y + h(x)\}$. These conditions are formulated in terms of differential invariants.
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