Method of slices

When a dynamical system has a homogeneous spatial dimension, patterns can arise at any location in this dimension. The method of slices reduces all states that are identical but for a shift to a unique state in a slice. Patterns that repeat in time but with a shift are relative periodic orbits (RPOs). The key property of the method of slices is that it closes these orbits. 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].