Method of slices

When a dynamical system has a homogeneous spatial dimension, a state can be shifted without altering its physical properties. The method of slices reduces all states that are identical up to a shift, to a unique state in a slice.

Patterns that repeat in time but reappear with a shift are known as relative periodic orbits (RPOs). The key property of the method of slices is that it closes these orbits, so that within the slice RPOs become periodic orbits (POs).

Searching for recurrences, i.e. for POs, in the slice is typically an order of magnitude less work than searching for RPOs in the full space. More importantly, to map out unstable manifolds, and to identify connections between orbits, this is possible with structurally-motivated coordinates only in the sliced space.

The method of slices reduces the continuous shift symmetry. The difficulty in symmetry reduction lies in the uniqueness of the shift for a given state, and in ensuring that as the state changes continuously the shift likewise changes continuously. Work on this method has been funded by the EPSRC [1].

Note that the symmetry-reduced space, i.e. the slice, is not a Poincaré section. Rather than having points that pierce the section, a full trajectory is produced within the slice. Nor is this low dimensional modelling, as the original state can be fully reconstructed.

The method of slices can be applied to any system with a homogeneous dimension. Examples where it have been applied include the Lorenz equations [2] and pipe flow [3].