Homoclinic based on parallel multiple shooting

We aim at finding homoclinic orbits for the Cauchy problem

\[\tag{1} \frac{d x}{d t}=f(x)\]

and we write $\phi^t(x_0)$ the associated flow (or semigroup of solutions).

The general method is explained in BifurcationKit.jl.

General method

It amounts to solving a boundary value problem. See ^[DeWitte] for description of the equations on the projectors.

\[\left\{\begin{aligned} & \dot{u}(t)-2 T\cdot F(u(t), p)=0 \\ & F\left(u_0, p\right)=0 \\ & Q^{U^{\perp}, \mathrm{T}}\left(u(0)-u_0\right)=0, \\ & Q^{S^{\perp}, \mathrm{T}}\left(u(1)-u_0\right)=0 \\ & T_{22 U} Y_U-Y_U T_{11 U}+T_{21 U}-Y_U T_{12 U} Y_U=0, \\ & T_{22 S} Y_S-Y_S T_{11 S}+T_{21 S}-Y_S T_{12 S} Y_S=0 \\ & \left\|u(0)-u_0\right\|-\epsilon_0=0 \\ & \left\|u(1)-u_0\right\|-\epsilon_1=0 \\ & \int_0^1 \tilde{u}^*(t)[u(t)-\tilde{u}(t)] d t=0, \\ \end{aligned}\right.\]

Jacobian

The jacobian is computed with automatic differentiation e.g. ForwardDiff.jl

References