The variational equation along an ODE solution

Let $D\subseteq \mathbb{R}\times {\mathbb{R}}^n$ be open, let $F:D\to {\mathbb{R}}^n$ be $C^1$ in the state variable, let $x:J\to {\mathbb{R}}^n$ be a solution of

$x^{\mathrm{\prime }}(t)=F(t,x(t))$

on an interval $J$, and let $t_0\in J$. The variational equation along $x$ is the linear matrix ODE

$A^{\mathrm{\prime }}(t)=D_{x}F(t,x(t))A(t),A(t_0)=I_n,$

where $D_{x}F(t,x(t))$ is the Jacobian matrix of partial derivatives of the state variables, read via The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case from the $C^1$ regularity recorded in $C^k$ maps and multi-index derivative notation in Euclidean space. Its solutions are matrix-valued curves $A:J\to M_n(\mathbb{R})$.

For an autonomous equation $x^{\mathrm{\prime }}=V(x)$ with $V$ of class $C^1$, this becomes

$A^{\mathrm{\prime }}(t)=DV(x(t))A(t),A(t_0)=I_n\mathrm{.}$

It is the linearized equation governing first-order variation of nearby solutions with respect to their initial data.