adjusted [*adaptively*]{} in each iteration step.

Provided that ${\mathsf{F}}'(u)$ is invertible on a suitable subset of $\Xi\subset X$, we define the [*Newton transform*]{} by $$u\mapsto{\mathsf{N}_{{\mathsf{F}}}}(u):=-{\mathsf{F}}'(u)^{-1}{\mathsf{F}}(u).$$ Then, rearranging terms in , we notice that $$\frac{u_{n+1}-u_{n}}{\Delta t_n}={\mathsf{N}_{{\mathsf{F}}}}(u_n), \qquad n\ge 0,$$ i.e., can be seen as the discretization of the [*Davydenko-type system*]{}, $$\label{eq:davy}
\begin{split}
\dot{u}(t)&={\mathsf{N}_{{\mathsf{F}}}}(u(t)), \quad t\geq 0,\qquad
 u(0)=u_0,
 \end{split}$$ by the forward Euler scheme, with step size $\Delta t_n>0$.

For $t\in[0,\infty)$, the solution $u(t)$ of , if it exists, defines a trajectory in $X$ that starts at $u_0$, and that will potentially converge to a zero of ${\mathsf{F}}$ as $t\to\infty$. Indeed, this can be seen (formally) from the integral form of , that is, $${\mathsf{F}}(u(t))={\mathsf{F}}(u_0)e^{-t},\qquad t\ge 0,$$ which implies that ${\mathsf{F}}(u(t))\to 0$ as $t\to\infty$.

Now taking the view of dynamical systems, our goal is to compute an upper bound for the value of the step sizes $\Delta t_n>0$ from , $n\ge 0$, so that the discrete forward Euler solution $\{u_n\}_{n\ge 0}$ from  stays reasonably close to the continuous solution of . Specifically, for a prescribed tolerance $\tau>0$, a Taylor expansion analysis (see [@AmreinWihler:15 Section 2] for details) reveals that $$u(t)=u_0+t{\mathsf{N}_{{\mathsf{F}}}}(u_0)+\frac{t^2}{2h}\eta_h+\mathcal{O}(t^3)+\mathcal{O}(t^2h\|{\mathsf{N}_{{\mathsf{F}}}}(u_0)\|_X^2),$$ where, for any sufficiently small $h>0$, we let $\eta_h={\mathsf{N}_{{\mathsf{F}}}}(u_0+h{\mathsf{N}_{{\mathsf{F}}}}(u_0))-{\mathsf{N}_{{\mathsf{F}}}}(u_0)$. Hence, after the 