Validated forward-Euler tube for the periodic flow and variational equation

Let $f:U\to {\mathbb{R}}^d$ be $C^2$. Set $m=d+d^2+1$ and define the augmented autonomous field

$$H(x,Z,\theta )=(\theta f(x),\theta Df(x)Z,0)$$

on an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^m$. The exact flow and its differentiability on a common compact tube use Initial-data sensitivity of an ODE flow on a common compact tube. Inputs are a rational initial-state box $X_0\subset U$, a rational period interval $\mathrm{\Theta }=[{\theta }_{-},{\theta }_{+}]\subset (0,\mathrm{\infty })$, an integer $N\ge 1$, a compact convex rational tube $K\subset \mathrm{\Omega }$, and a sound range oracle with rational bounds $L,M$ as in Exact range-oracle contract for validated flow and monodromy enclosures. Use the maximum norm and exact rational set arithmetic. Let $h=1\mathrm{/}N$ and

$$Y_0=X_0\times \{I_d\}\times \mathrm{\Theta }\mathrm{.}$$

For $n=0,\dots ,N-1$, first set

$$Q_n=Y_n+[-Mh,Mh{]}^m\mathrm{.}$$

If $Q_n\subseteq ̸K$, return inconclusive. Then request $H^{\mathrm{\#}}(Y_n)$. If it is inconclusive, return inconclusive. Otherwise form

$$Y_{n+1}=Y_n+h{H}^{\mathrm{\#}}(Y_n)+[-LM{h}^2\mathrm{/}2,LM{h}^2\mathrm{/}2{]}^m\mathrm{.}$$

If any required rational set operation is unavailable or the next box is not finite, return inconclusive; otherwise continue. On success return the tube boxes $Q_0,\dots ,Q_{N-1}$ and endpoint box $Y_N$. The first, second, and last coordinate blocks of $Y_N$ enclose the exact endpoint, the exact variational matrix, and the period, respectively. The procedure makes no success claim if the supplied range oracle or $L,M$ bounds are not sound on $K$.