Certified residual, Jacobian, trajectory, and monodromy enclosures for periodic shooting

Let $f(x,\mu )$ be $C^2$ on an open set in ${\mathbb{R}}^d\times {\mathbb{R}}^p$, let $q=(x,\mu )$, and set $g(q)=(f(x,\mu ),0_p)$. Write $Pq=x$. For each fixed $\mu$, let ${\mathrm{\Phi }}_{\theta }(q)$ be the time-$\theta$ flow of $q^{\mathrm{\prime }}=g(q)$, so its parameter coordinates remain constant. For an affine phase $\psi (x)=c^{\mathrm{\top }}(x-x_{\mathrm{r}\mathrm{e}\mathrm{f}})$, define

$$G(q,\theta )=\left[\begin{array}{c}P{\mathrm{\Phi }}_{\theta }(q)-Pq\\ \psi (Pq)\end{array}\right]\mathrm{.}$$

For $p=0$ and $c\mathrm{\ne }0$ this is the shooting residual in The phase-bordered periodic-orbit BVP residual. Its Jacobian is

$$DG(q,\theta )=\left[\begin{array}{cc}P\mathcal{M}-P& f(P{\mathrm{\Phi }}_{\theta }(q),\mu )\\ c^{\mathrm{\top }}P& 0\end{array}\right],\mathcal{M}=D_q{\mathrm{\Phi }}_{\theta }(q)\mathrm{.}$$

For a rational box $Q=X\times \mathrm{\Lambda }\times \mathrm{\Theta }$ with $\mathrm{\Theta }\subset (0,\mathrm{\infty })$, a validated shooting jet enclosure consists of: (i) a box containing $P{\mathrm{\Phi }}_{\theta }(q)$ for every $q,\theta$ in the input; (ii) a box containing the residual $G(Q)$; (iii) an interval matrix $J(Q)$ containing every $DG(q,\theta )$; and (iv) trajectory and monodromy boxes containing every scaled trajectory $P{\mathrm{\Phi }}_{\theta s}(q)$, $0\le s\le 1$, and the physical-state block of $\mathcal{M}$. The input and range-oracle boxes may have arbitrary rational endpoints, as in Exact range-oracle contract for validated flow and monodromy enclosures. The returned interval enclosures have finite dyadic endpoints under Real boxes and interval Jacobian enclosures. Before applying dyadic interval arithmetic, round every rational operand outward to finite dyadic endpoints using Directed dyadic rounding encloses with explicit underflow and overflow bounds, then use Finite real interval operations enclose their exact values; exact rational set operations in the Euler provider remain exact. If any conversion overflows or is unavailable within the supplied cap, return inconclusive. The exact derivative formula follows from Derivative of the nonlinear shooting residual, and its flow sensitivity is on the common tube of Initial-data sensitivity of an ODE flow on a common compact tube. A finite success record is admissible only when obtained by the range-oracle procedure of Validated forward-Euler tube for the periodic flow and variational equation, whose exact flow and monodromy enclosure is guaranteed by The validated Euler tube encloses the exact periodic flow and monodromy matrix, with all queries and interval operations successful and outward under Exact range-oracle contract for validated flow and monodromy enclosures. For this call instantiate that procedure and its soundness theorem with state dimension $n=d+p$, initial state box $X\times \mathrm{\Lambda }$, field $g(q)=(f(x,\mu ),0_p)$, and period interval $\mathrm{\Theta }$. Its augmented field is $H_g(q,Z,\theta )=(\theta g(q),\theta Dg(q)Z,0)$ with $Z\in {\mathbb{R}}^{n\times n}$; the supplied range provider is for this $H_g$ on its actual compact convex rational tube $K_g\subset {\mathbb{R}}^{n+n^2+1}$. Initialize $Y_0=(X\times \mathrm{\Lambda })\times \{I_n\}\times \mathrm{\Theta }$ and require every Euler tube query and inclusion check in this same $K_g$ to succeed. Retain the full $n\times n$ endpoint variation enclosure for $\mathcal{M}=D_q{\mathrm{\Phi }}_{\theta }(q)$ when forming $P\mathcal{M}-P$: its first $d$ rows include both the state and parameter sensitivity columns, while its lower parameter rows are $(0,I_p)$ because the parameters are constant. The physical monodromy is the first $d$ rows and first $d$ columns of $\mathcal{M}$.

To enclose the period column, let $X_T$ be the first $d$ coordinate projection of the successful endpoint enclosure $Y_N$. The exact augmented endpoint lies in $X_T\times \mathrm{\Lambda }$: its parameter equals its initial value in $\mathrm{\Lambda }$, even if a returned interval coordinate is wider. Form the rational query box $Q_T=(X_T\times \mathrm{\Lambda })\times \{I_n\}\times \mathrm{\Theta }$ and check $Q_T\subseteq K_g$ before requesting $H_g^{\mathrm{\#}}(Q_T)$ from that same supplied provider. If the inclusion or query fails, return inconclusive. The first $d$ entries of its vector-field block enclose every $\theta f(P{\mathrm{\Phi }}_{\theta }(q),\mu )$; divide these entries outward by the positive interval $\mathrm{\Theta }$ to enclose the entire period column. Thus the query includes every admitted parameter and the matching augmented identity. For $p=0$, $n=d$ and the parameter factor is the singleton empty tuple, giving the fixed-parameter construction. These residual/Jacobian boxes are in the named coordinate order $(q,\theta )=(x,\mu ,\theta )$; a section callback whose variables are ordered $(x,\theta ,\mu )$ applies that fixed column permutation before appending its affine section row. Any unavailable query or inclusion returns inconclusive. A finite Newton, multiple-shooting, or collocation residual alone is not a validated jet enclosure.