conatus.expert.find_rpo_in_slice

conatus.expert.find_rpo_in_slice(slc, integrator, section, x0, T0, *, tol, fd_step, max_iterations, steady_tol)

Periodic orbit of the reduced flow through the Poincaré section (hyperplane or any g): unknowns (x̂, T), equations reduce(Φ_T(x̂)) − x̂ = 0, slice condition ⟨x̂, t’⟩ = 0 and g(x̂) = 0, solved by Gauss–Newton (least squares: the reduced residual has one component that duplicates the slice condition) with a forward-difference Jacobian of declared step fd_step; converged when the relative residual is ≤ tol. integrator.flow must accept any T (ODE integrators). θ is the unwrapped net phase along the converged orbit.

A relative equilibrium is a fixed point of the reduced flow, hence a “periodic orbit” of any period: a solution with in-slice speed ‖v̂(x̂)‖/‖v(x̂)‖ < steady_tol (declared) is refused (ConvergenceError), never returned as an orbit (found by the example, where the closest recurrence of a trajectory leaving the relative equilibrium converged onto it).

Parameters:

• T0 (float)

• tol (float)

• fd_step (float)

• max_iterations (int)

• steady_tol (float)

Return type:

SliceRPO