The phase-bordered periodic-orbit BVP residual

Let $f:U\to {\mathbb{R}}^d$ be an autonomous $C^2$ vector field on an open set $U$, with the autonomous ODE convention of Autonomous ordinary differential equations and the $C^2$ map convention of $C^k$ Euclidean maps and diffeomorphisms. Let $\psi (x)=c^{\mathrm{\top }}(x-x_{\mathrm{r}\mathrm{e}\mathrm{f}})$ be an affine phase function with $c\mathrm{\ne }0$. On the scaled interval $s\in [0,1]$, introduce the augmented state $u=(x,\theta )\in U\times (0,\mathrm{\infty })$ and vector field

$$\tilde{f}(x,\theta )=(\theta f(x),0)\mathrm{.}$$

For endpoint states $u_0=(x_0,{\theta }_0)$ and $u_1=(x_1,{\theta }_1)$, define the boundary map

$$B(u_0,u_1)=(x_1-x_0,\psi (x_0))\in {\mathbb{R}}^{d+1}\mathrm{.}$$

This is a square nonlinear BVP in the sense of Square nonlinear two-point boundary-value problems and endpoint residuals: its state dimension and number of boundary equations are both $d+1$. The period coordinate is constant along solutions because its derivative is zero. The common-tube differentiability of the exact flow is given by Initial-data sensitivity of an ODE flow on a common compact tube, and monodromy/Floquet terminology follows Local periodic-orbit, Poincare-return, and Floquet data.

Whenever the exact autonomous flow ${\varphi }_t$ exists on the required interval, the associated phase-bordered shooting residual is

$$G(x,\theta )=({\varphi }_{\theta }(x)-x,\psi (x))\in {\mathbb{R}}^{d+1},\theta >0.$$

It is defined only for initial data whose flow exists through time $\theta$. Phase transversality at a nonstationary periodic point $x_{\ast }$ means $c^{\mathrm{\top }}f(x_{\ast })\mathrm{\ne }0$. Neither existence of a zero nor invertibility of its Jacobian is part of this definition.