variational equation

Japanese: variational equation（英語表記）variationalequation

...If one solution to the normal form nonlinear equation (1) is found, and it is x _j = φ _j ( t ) ( j = 1, ..., n ), to examine other solutions in the vicinity of this solution, one usually performs a variable transformation x _j = φ _j ( t ) + y _j to transform (1) into: The linear equation obtained by taking only the first term of the Taylor expansion on the right-hand side is called the variational equation of (1). The so-called linearization method uses the properties of the solution of (6) to examine the properties of the solution of (1), and most local studies that examine the situation in the vicinity of one solution rely on this linearization method. ...

*Some of the terminology explanations that mention "variational equation" are listed below.

Source | Heibonsha World Encyclopedia 2nd Edition | Information

Japanese:

…正規形の非線形方程式(1)の解が一つ求められたとし，それをx_j＝φ_j(t)(j＝1，……，n)とするとき，この解の近傍にある他の解を調べるには，ふつう変数変換x_j＝φ_j(t)＋y_jを行って(1)を，と変形する。この右辺のテーラー展開の第1項のみをとって得られる線形方程式，を(1)の変分方程式variational equationとよぶ。(6)の解の性質を利用して(1)の解の性質を調べるのがいわゆる線形化の方法で，一つの解の近傍の状況を調べる局所的研究の大部分はこの線形化の方法に依存している。…