Multiple-shooting and midpoint-collocation residuals for a phase-fixed periodic orbit

Use the augmented state $u=(x,\theta )\in U\times (0,\mathrm{\infty })$, field $\tilde{f}(u)=(\theta f(x),0)$, and boundary residual $B(u_0,u_N)=(x_N-x_0,\psi (x_0))$ from The phase-bordered periodic-orbit BVP residual. On a mesh $0=s_0<\cdots <s_N=1$, write $u_i=(x_i,{\theta }_i)$.

The endpoint-pair boundary residual and square BVP convention are those of Square nonlinear two-point boundary-value problems and endpoint residuals. The exact multiple-shooting residual has segment rows

$${\mathrm{\Phi }}_{s_{i+1}-s_i}(u_i)-u_{i+1}(0\le i<N)$$

and final row $B(u_0,u_N)$, where ${\mathrm{\Phi }}_h$ is the exact flow of $\tilde{f}$. Its finite-budget candidate procedure is Finite-budget multiple shooting for a nonlinear two-point BVP on this augmented BVP.

For uniform $h=1\mathrm{/}N$, the normalized midpoint-collocation residual has rows

$$\frac{u_{i+1}-u_i}{h}-\tilde{f}\text{\hspace{0.17em}⁣}\left(\frac{u_i+u_{i+1}}{2}\right)(0\le i<N)$$

and final row $B(u_0,u_N)$. This is Normalized implicit-midpoint collocation residual for a two-point BVP for the same augmented BVP; its finite-budget candidate procedure is Finite-budget implicit-midpoint collocation for a nonlinear BVP. Both systems are square: the state dimension is $d+1$, with $d+1$ boundary equations.