Positive Solution of One Conjecture in the Theory of Polynomial Isochronous Centers of Lienard Systems

摘要

We consider the polynomial Lienard system (x) over dot = -y, (y) over dot = x + A(x) - B(x)y under the assumption that the real polynomials A(x) and B(x) and the derivative A'(x) satisfy the conditions A(0) = B(0) = A'(0) = 0. We prove that this system has an isochronous center at the singular point O(0,0) if and only if the polynomials A(x) and B(x) are odd functions related by the identity x(3)A(x) = (integral(x)(0) sB(s) ds)(2).