Algebraic geometry of Abel differential equation
Consider a system of differential equations \[ \dot{x}= -y + F(x,y), \qquad \dot{y}=x+G(x,y), \tag{\(*\)} \] where \(F\) and \(G\) are analytic functions without constant and linear terms. This system has a center at the origin if all the solutions around the origin are periodic. The classical center-focus problem consists in finding necessary and sufficient conditions on \(F\) and \(G\) to have a center at the origin. Center conditions are given by an infinite system of polynomial equations in the coefficients of \(F\) and \(G\) and the ideal generated by these equations determines the center variety. Hence, the algebraic geometry enters to this problem in a natural way. If \(F\) and \(G\) are homogeneous polynomials then system \((*)\) can be reduced by a Cherkas transformation to the Abel equation \[ \frac{d \rho}{ d \theta} = p(\theta) \rho^2 + q(\theta) \rho^3, \tag{\(**\)} \] where \(\theta \in [0, 2 \pi]\) and \(p\) and \(q\) are trigonometric polynomials. Under this tranformation periodic solutions of system \((*)\) become periodic solutions of \((**)\). Hence, a natural extended setting for the center-focus problem is to consider a general (not necessary polynomial) Abel equation \[ y'= p(x) y^2 + q(x) y^3, \] with \(x \in [a,b]\) and repalacing the periodicity of the solutions by the condition \(y(a)=y(b)\). In this case the center conditions are expressed in a rather explicit form as a linear combinations of iterated integrals of \(p\) and \(q\). Recently two important structures have been related to the center equations for equation \((**)\): composition algebra and moment vanishing. In the present paper the authors give an overview of the relations between the center set, composition set and moment set from the results obtained in previous works, 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. The authors identify complex periodic solutions, stressing their ramification properties. In particular, the authors analyze the continuation paths along which these solutions become periodic.
scientific article; zbMATH DE number 6345557
scientific article