Differential Equations ( IF 0.8 ) Pub Date : 2021-03-19 , DOI: 10.1134/s0012266121020014 V. V. Amel’kin
Abstract
We consider the polynomial Liénard system \(\dot {x}=-y \), \(\dot {y}=x+A(x)-B(x)y \) under the assumption that the real polynomials \(A(x) \) and \(B(x) \) and the derivative \(A^{\prime }(x) \) satisfy the conditions \(A(0)=B(0)=A^{\prime }(0)=0 \). 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 \(x^3A(x)=(\int \nolimits _0^xsB(s)\,ds)^2\).
中文翻译:
Liénard系统的多项式同步中心理论中一个猜想的正解
摘要
我们认为多项式Liénard型系统\(\点{X} = - Y \) ,\(\ {点Y} = X + A(X)-B(X)Y \)的假设下,真正的多项式 \( A(x)\)和\(B(x)\)以及导数\(A ^ {\ prime}(x)\)满足条件\(A(0)= B(0)= A ^ {\质数}(0)= 0 \)。我们证明,当且仅当多项式\(A(x)\)和\(B(x)\)是与奇函数相关的系统,该系统在奇点\(O(0,0)\)上具有一个等时中心。通过身份 \(x ^ 3A(x)=(\ int \ nolimits _0 ^ xsB(s)\,ds)^ 2 \)。