Abel Differential Equations Research Articles

On the equivalence problem of generalized Abel ODEs under the action of the linear transformations pseudogroup

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 ↦ f(x), y ↦ g(x) · y + h(x)}. These conditions are formulated in terms of differential invariants.

The center problem and composition condition for Abel differential equations

The classical Poincaré center-focus problem for planar polynomial systems of ordinary differential equations can be transformed, in certain particular cases, to the center problem of a trigonometric Abel differential equation. Several research papers focused on the study of the center problem for trigonometric Abel differential equations. Polynomial Abel differential equations are also considered in the literature as a model problem. In this work we make a survey of the most important results in this context and we provide the state of the art of several related conjectures. We give two new results on these conjectures.

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 m˜k=∫abPkq 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⩽2+deg⁡q+3(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 polynomial Abel equation y′=py2+εqy3 for p,q polynomials and ε≪1. In particular, our result implies that for p satisfying a well-known generic condition, the number of periodic solutions near the zero solution does not exceed 5+deg⁡q+3deg2⁡p. This is the first such bound depending solely on the degrees of the Abel equation.

The Integrating Factors for Riccati and Abel Differential Equations

We can recast the Riccati and Abel differential equationsinto new forms in terms of introduced integrating factors.Therefore, the Lie-type systems endowing with transformation Lie-groups$SL(2,{\mathbb R})$ can be obtained.The solution of second-order linearhomogeneous differential equation is an integrating factorof the corresponding Riccati differential equation.The numerical schemes which are developed to fulfil the Lie-group property have better accuracy and stability than other schemes.We demonstrate that upon applying the group-preserving scheme (GPS) to the logistic differential equation, it is not only qualitatively correct for all values of time stepsize $h$, and is also the most accurate one among all numerical schemes compared in this paper.The group-preserving schemes derived for the Riccati differential equation, second-order linear homogeneous and non-homogeneous differential equations, the Abel differential equation and higher-order nonlinear differential equations all have accuracy better than $O(h^2)$.

Universal curves in the center problem for Abel differential equations

We study the center problem for the class ${\mathcal{E}}_{{\rm\Gamma}}$ of Abel differential equations $dv/dt=a_{1}v^{2}+a_{2}v^{3}$, $a_{1},a_{2}\in L^{\infty }([0,T])$, such that images of Lipschitz paths $\tilde{A}:=(\int _{0}^{\cdot }a_{1}(s)\,ds,\int _{0}^{\cdot }a_{2}(s)\,ds):[0,T]\rightarrow \mathbb{R}^{2}$ belong to a fixed compact rectifiable curve ${\rm\Gamma}$. Such a curve is said to be universal if whenever an equation in ${\mathcal{E}}_{{\rm\Gamma}}$ has center on $[0,T]$, this center must be universal, i.e. all iterated integrals in coefficients $a_{1},a_{2}$ of this equation must vanish. We investigate some basic properties of universal curves. Our main results include an algebraic description of a universal curve in terms of a certain homomorphism of its fundamental group into the group of locally convergent invertible power series with the product being the composition of series, explicit examples of universal curves, and approximation of Lipschitz triangulable curves by universal ones.

Algebraic geometry of the center-focus problem for Abel differential equations

The Abel differential equation $y^{\prime }=p(x)y^{3}+q(x)y^{2}$ with polynomial coefficients $p,q$ is said to have a center on $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(a)=y(b)$. The problem of giving conditions on $(p,q,a,b)$ implying a center for the Abel equation is analogous to the classical Poincaré center-focus problem for plane vector fields. Center conditions are provided by an infinite system of ‘center equations’. During the last two decades, important new information on these equations has been obtained via a detailed analysis of two related structures: composition algebra and moment equations (first-order approximation of the center ones). Recently, one of the basic open questions in this direction—the ‘polynomial moments problem’—has been completely settled in Pakovich and Muzychuk [Solution of the polynomial moment problem. Proc. Lond. Math. Soc. (3)99(3) (2009), 633–657] and Pakovich [Generalized ‘second Ritt theorem’ and explicit solution of the polynomial moment problem. Compositio Math.149 (2013), 705–728]. In this paper, we present a progress in the following two main directions: first, we translate the results of Pakovich and Muzychuk [Solution of the polynomial moment problem. Proc. Lond. Math. Soc. (3)99(3) (2009), 633–657] and Pakovich [Generalized ‘second Ritt theorem’ and explicit solution of the polynomial moment problem. Compositio Math.149 (2013), 705–728] into the language of algebraic geometry of the center equations. Applying these new tools, we show that the center conditions can be described in terms of composition algebra, up to a ‘small’ correction. In particular, we significantly extend the results of Briskin, Roytvarf and Yomdin [Center conditions at infinity for Abel differential equations. Ann. of Math. (2)172(1) (2010), 437–483]. Second, applying these tools in combination with explicit computations, we start in this paper the study of the ‘second Melnikov coefficients’ (second-order approximation of the center equations), showing that in many cases vanishing of the moments and of these coefficients is sufficient in order to completely characterize centers.

Parametric Center-Focus Problem for Abel Equation

The Abel differential equation $$y'=p(x)y^3 + q(x) y^2$$ with meromorphic coefficients $$p,q$$ is said to have a center on $$[a,b]$$ if all its solutions, with the initial value $$y(a)$$ small enough, satisfy the condition $$y(a)=y(b)$$ . The problem of giving conditions on $$(p,q,a,b)$$ implying a center for the Abel equation is analogous to the classical Poincaré Center-Focus problem for plane vector fields. Following Briskin et al. (Ergodic Theory Dyn Syst 19(5):1201–1220, 1999; Isr J Math 118:61–82, 2000); Cima et al. (Qual Theory Dyn Syst 11(1):19–37, 2012; J Math Anal Appl 398(2):477–486, 2013) we say that Abel equation has a “parametric center” if for each $$\epsilon \in \mathbb C$$ the equation $$y'=p(x)y^3 + \epsilon q(x) y^2$$ has a center. In the present paper we use recent results of Briskin et al. (Algebraic Geometry of the Center-Focus problem for Abel differential equations, arXiv:1211.1296 , 2012); Pakovich (Comp Math 149:705–728, 2013) to show show that for a polynomial Abel equation parametric center implies strong “composition” restriction on $$p$$ and $$q$$ . In particular, we show that for $$\deg p,q \le 10$$ parametric center is equivalent to the so-called “Composition Condition” (CC) (Alwash and Lloyd in Proc R Soc Edinburgh 105A:129–152, 1987; Briskin et al. Ergodic Theory Dyn Syst 19(5):1201–1220, 1999) on $$p,q$$ . Second, we study trigonometric Abel equation, and provide a series of examples, generalizing a recent remarkable example given in Cima et al. (Qual Theory Dyn Syst 11(1):19–37, 2012), where certain moments of $$p,q$$ vanish while (CC) is violated.

Weak and strong composition conditions for the Abel differential equation

We establish an equivalence between two forms of the composition condition for the Abel differential equation with trigonometric coefficients.

Periodic orbits from second order perturbation via rational trigonometric integrals

The second order Poincaré-Pontryagin-Melnikov perturbation theory is used in this paper to study the number of bifurcated periodic orbits from certain centers. This approach also allows us to give the shape and the period up to the first order. We address these problems for some classes of Abel differential equations and quadratic isochronous vector fields in the plane. We prove that two is the maximum number of hyperbolic periodic orbits bifurcating from the isochronous quadratic centers with a birational linearization under quadratic perturbations of second order. In particular the configurations (2,0) and (1,1) are realizable when two centers are perturbed simultaneously. The required computations show that all the considered families share the same iterated rational trigonometric integrals.

Homotopy analysis method for solving Abel differential equation of fractional order

Abstract In this study, the homotopy analysis method is used for solving the Abel differential equation with fractional order within the Caputo sense. Stabilityand convergence of the proposed approach is investigated. The numerical results demonstrate that the homotopy analysis method is accurate and readily implemented.

A Chiellini Type Integrability Condition for the Generalized First Kind Abel Differential Equation

The Chiellini integrability condition of the first order first kind Abel equation $dy/dx=f(x)y^2+g(x)y^3$ is extended to the case of the general Abel equation of the form $dy/dx=a(x)+b(x)y+f(x)y^{\alpha -1}+g(x)y^{\alpha}$, where $\alpha \in \Re$, and $\alpha > 1$. In the case $\alpha =2$ the generalized Abel equations reduces to a Riccati type equation, for which a Chiellini type integrability condition is obtained.

An explicit bound of the number of vanishing double moments forcing composition

We give two new characterizations of pairs of polynomials or trigonometric polynomials that form a composition pair. One of them proves that the cancellation of a given number of double moments implies that they form a composition pair. This number only depends on the maximum degree of both polynomials. This is the first time that composition is characterized in terms of the cancellation of an explicit number of double moments. Our results allow to recognize the composition centers for polynomial and trigonometric Abel differential equations.

On centers for Generalized Abel Differential Equation

A new condition is given for generalized Abel differential equation to have a center. We apply the results to some polynomial differential systems in the plane to find necessary and sufficient center conditions.

Generalized Weierstrass Integrability of the Abel Differential Equations

We study the Abel differential equations that admit either a generalized Weierstrass first integral or a generalized Weierstrass inverse integrating factor.

Analysis to the solutions of Abel's differential equation of the first kind under the transformation y=u(x)z(x)+v(x)

Analysis to the solutions of Abel's differential equation of the first kind under the transformation y=u(x)z(x)+v(x)

On the Derivation of a Closed-Form Expression for the Solutions of a Subclass of Generalized Abel Differential Equations

We investigate the properties of a general class of differential equations described bydy(t)/dt=fk+1(t)y(t)k+1+fk(t)y(t)k+⋯+f2(t)y(t)2+f1(t)y(t)+f0(t),withk>1a positive integer andfi(t), 0≤i≤k+1, withfi(t), real functions oft. Fork=2, these equations reduce to the class ofAbel differential equations of the first kind,for which a standard solution procedure is available. However, fork>2no general solution methodology exists, to the best of our knowledge, that can lead to their solution. We develop a general solution methodology that for odd values ofkconnects the closed form solution of the differential equations with the existence of closed-form expressions for the roots of the polynomial that appears on the right-hand side of the differential equation. Moreover, the closed-form expression (when it exists) for the polynomial roots enables the expression of the solution of the differential equation in closed form, based on the class of Hyper-Lambert functions. However, for certain even values ofk, we prove that such closed form does not exist in general, and consequently there is no closed-form expression for the solution of the differential equation through this methodology.

Algebraic geometry of Abel differential equation

A solution $$y(x)$$ of an Abel differential equation $$(1) \ y^{\prime }=p(x)y^2 + q(x) y^3$$ is called “closed” on $$[a,b]$$ if $$y(a)=y(b)$$ . The equation $$(1)$$ is said to have a center on $$[a,b]$$ if all its solutions (with the initial value $$y(a)$$ small enough) are closed. The problems of counting closed solutions (Smale–Pugh problem) is strongly related to the classical Hilbert 16th problem of bounding the number of limit cycles of a plane polynomial vector field. In turn, the problem of giving conditions on $$(p,q,a,b)$$ for $$(1)$$ to have a center on $$[a,b]$$ is analogous to the classical Poincare center–focus problem for plane vector fields. It is well known that both in the classical and in the Abel equation cases the center conditions are given by an infinite system of polynomial equations in the parameter space. The complexity of this system presents one of the main difficulties in the center–focus problem, as well as in the bounding of closed trajectories. In recent years two important structures have been related to the center equations for $$(1)$$ : composition algebra and moment vanishing. In the present paper we give an overview of these results (sometimes providing also new ones), stressing their algebraic–geometric interpretation and consequences. The second part of the paper is devoted to a rather detailed study of a specific example of the Abel equation which possesses algebraic solutions. We identify complex closed solutions, stressing their ramification properties. In particular, we analyze the continuation paths along which these solutions become closed.

A simple solution of some composition conjectures for Abel equations

Trigonometric Abel differential equations appear in the study of the number of limit cycles and the center-focus problem for certain families of planar polynomial systems. The composition centers are a class of centers for trigonometric Abel equations which have been widely studied during last years. We characterize this type of centers as the ones given by couples of trigonometric polynomials for which all the generalized moments vanish. They also coincide with the strongly and the highly persistent centers. Our result gives a simple and self-contained proof of the so called Composition Conjecture for trigonometric Abel differential equations. We also prove a similar version of this result for Abel equations with polynomial coefficients.

SOLUTION OF HIGHER-ORDER ABEL EQUATIONS BY DIFFERENTIAL TRANSFORM METHOD

In this paper, the second-, third- and fourth-order Abel equations are solved. The differential transform method (DTM) is used to compute approximate solutions of the nonlinear ordinary Abel differential equations. The results are compared with the results obtained by the classical Runge–Kutta (RK4) method. Figures are presented to show the reliability and simplicity of the method.

An identity derived from the solution of a class of differential equations for the evolution of a key agreement protocol

In this paper we prove an identity involving the Lambert W function which is derived through the solution of an instance of the Abel differential equation of the first kind. For x ∈ ℜ, the Lambert W function is defined as the principle branch of the function that satisfies the following equation Lambert W (x) e LambertW(x) = x. The Abel Dif dy(t) ferential Equation of the First Kind is defined as . The identity is proved through a connection of the solution of a special case of this type of differential equations with the solutions of a third degree polynomial equation.