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 the periodic orbit shooting section of the BifurcationKit.jl documentation.

General method

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

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

Usage

A typical workflow is to

1. compute a branch of periodic orbits with BifurcationKit using a Shooting discretization,
2. build the homoclinic problem from the last point of the branch with generate_hom_problem,
3. continue the homoclinic orbit with continuation:

# sh is a Shooting discretization, brpo a branch of periodic orbits
𝐇𝐨𝐦, xhom, pars, _ = generate_hom_problem(sh, brpo.sol[end].x, BK.setparam(brpo, brpo.sol[end].p), BK.getlens(brpo))
br_hom = continuation(𝐇𝐨𝐦, xhom, lens, PALC(), ContinuationPar(); kwargs...)

See the tutorials for fully worked examples.

Jacobian

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

References