In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part. Let $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ and $q(x) = b(x,x)$. Let $x_0 \in \mathbf{R}^n$ and $f_{x_0}$ the solution of $y' = \varphi(y),\ y(0) = x_0$. For any function $y$ defined on $\mathbf{R}_+$, define: $$\varepsilon ( y ) : \mathbf { R } _ { + } \rightarrow \mathbf { R } ^ { n }, \quad t \mapsto \varphi ( y ( t ) ) - a ( y ( t ) )$$
Verify the equality $$\forall t \in \mathbf { R } _ { + } , \quad q \left( f _ { x _ { 0 } } \right) ^ { \prime } ( t ) = - \left\| f _ { x _ { 0 } } ( t ) \right\| ^ { 2 } + 2 b \left( f _ { x _ { 0 } } ( t ) , \varepsilon \left( f _ { x _ { 0 } } ( t ) \right) \right)$$