We study the precise asymptotic behavior of a non-trivial solution that converges to zero, as time tends to infinity, of dissipative systems of nonlinear ordinary differential equations. The nonlinear term of the equations may not possess a Taylor series expansion about the origin. This absence technically cripples previous proofs in establishing an asymptotic expansion, as an infinite series, for such a decaying solution. In the current paper, we overcome this limitation and obtain an infinite series asymptotic expansion, as time goes to infinity. This series expansion provides large time approximations for the solution with the errors decaying exponentially at any given rates. The main idea is to shift the center of the Taylor expansions for the nonlinear term to a non-zero point. Such a point turns out to come from the non-trivial asymptotic behavior of the solution, which we prove by a new and simple method. Our result applies to different classes of non-linear equations that have not been dealt with previously.

我们研究非线性常微分方程的耗散系统的非平凡解随着时间趋于无穷大收敛到零的精确渐近行为。方程的非线性项可能不具有关于原点的泰勒级数展开。这种缺失在技术上削弱了先前建立渐近展开式的证明，作为无限级数，为这样一个衰减的解决方案。在当前的论文中，我们克服了这个限制并随着时间趋于无穷大而获得了无限级数渐近展开。这个级数展开为解提供了大的时间近似值，误差以任何给定的速率呈指数衰减。主要思想是将非线性项的泰勒展开式的中心移动到非零点。事实证明，这一点来自解决方案的非平凡渐近行为，我们通过一种新的简单方法证明了这一点。我们的结果适用于以前没有处理过的不同类别的非线性方程。