Phase locking to the n-torus

We discuss phase resetting behaviour when non-small, near instantaneous, perturbations are applied to a dynamical system containing a strong attractor that is equivalent to a winding map on an $n$-dimensional torus. Almost all of the literature to date has focused on the application of phase transition curves derivable for systems with attracting limit cycles, when $n = 1$, and the consequent possibilities for entrainment of the system subject to such periodic largish perturbations. In higher dimensions, the familiar tongues in the amplitude versus period plane, describing solutions entrained with periodic fast perturbations, have subtle structure (since only locally two-dimensional subsets of the $n$-dimensional torus may admit such solutions) whilst the topological class of the $n$-dimensional phase transition mapping is represented by a matrix of winding numbers. In turn this governs the existence of asymptotic solutions in the limits of small perturbations. This paper is an attempt to survey the entrainment behaviour for the full set of alternative (near rigid) phase transition mappings definable on the 2-torus. We also discuss higher dimensional effects and we suggest how this might be important for non-linear neural circuits including delays that routinely exhibit such attractors and may drive one another in cascades of periodic behaviour.