The saltation matrix gives the first-order variation across a transverse hybrid reset

Consider a transition in Smooth hybrid ODE modes, guards, resets, and deterministic event priority with state dimension $d\ge 1$, from mode $m^{-}$ to mode $m^{+}$ at $(\tau ,x^{-})$. Let $g(t,x)=0$ be its $C^1$ guard, let $R(t,x)$ be its $C^1$ reset, and write $f^{-}=f_{m^{-}}(\tau ,x^{-})$ and $f^{+}=f_{m^{+}}(\tau ,R(\tau ,x^{-}))$. For each incoming and outgoing mode, assume its IVPs initialized at time $\tau$ from states near $x^{-}$ and $R(\tau ,x^{-})$, respectively, have unique solutions on a common finite interval and remain in a bounded convex open tube whose compact closure lies inside the open ODE domain. Use the initial state as the parameter and the identity as its $C^1$ initial map in Parameter-dependent ODE flows have the variational tangent on a common tube. For event-time variations to the left of $\tau$, impose the same tube hypothesis on the time-reversed incoming field. Thus the relevant mode flows are $C^1$ in initial state, and the incoming flow is jointly $C^1$ in time and initial state near $(\tau ,x^{-})$. Assume also that $\kappa :={\mathrm{\partial }}_{t}g(\tau ,x^{-})+D_{x}g(\tau ,x^{-})f^{-}\mathrm{\ne }0,$ and the nominal event crosses in its declared orientation. For every sufficiently small incoming perturbation $\delta x^{-}$ at the common pre-event time $\tau$, initialize the perturbed incoming mode at $x^{-}+\delta x^{-}$, and let ${\tau }_{\delta }$ be its local guard root near $\tau$; assume this root is the selected transition. Put $t_0=\max \{\tau ,{\tau }_{\delta }\}$, and define $\delta x^{+}$ as the right-limit difference ${\lim }_{t\downarrow t_0}(x_{\delta }^{+}(t)-x^{+}(t))$, with both outgoing trajectories evaluated at the same clock time $t>t_0$. Then $\delta x^{+}=S\delta x^{-}+o(\mathrm{\parallel }\delta x^{-}{\mathrm{\parallel }}_2),S=D_{x}R+\frac{(f^{+}-D_{x}R{f}^{-}-{\mathrm{\partial }}_{t}R)D_{x}g}{{\mathrm{\partial }}_{t}g+D_{x}g{f}^{-}},$ where all derivatives and vector fields are evaluated at the nominal event and the matrix product in the numerator is a column times a row. No claim is made when the crossing derivative vanishes or a nearby trajectory follows a different transition.

The batch-96 supplier Parameter-dependent ODE flows have the variational tangent on a common tube is present on library/numerical-optimization-inverse-problems-and-uncertainty-quantification/discrete-continuous-and-pde-adjoint-sensitivities.md, with its companion page at the same canonical category. The earlier missing-page observation checked the different dynamics-category path. The exact common-tube interface and identity-initial-map/time-reversal specialization above are retained; this placement reconciliation grants no independent proof acceptance.