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). A special case is that of relative equilibria (travelling waves), which are steady in a moving frame. These are reduced to equilibria within the slice.

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, and achieves this with the aid of a template or reference state. The difficulty in symmetry reduction lies in the uniqueness of the shift for a given state, and in ensuring that when the state changes continuously in time, the shift likewise changes continuously. Work on the method of slices has been funded by the EPSRC [1].

Other approaches that involve shifts include, e.g., the selection of a moving Galilean frame. As this does not reduce states that differ by a shift to a unique state, however, it is not a symmetry reduction method. The moving frame will not reduce all relative equilibria to equilibria, for example.

Note that a full trajectory is produced within the slice, which is convenient for analysing stable and unstable manifolds, and connections between states. The slice is not the same as Poincaré section, where one has only points that pierce the section. Nor is the method of slices low dimensional modelling either, 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].