Isochron geometry in the presence of saddle objects
In biological, chemical, and physical systems one is often interested in the synchronisation properties of observed oscillating behaviour, represented by periodic orbits or focus equilibria in ordinary-differential-equation models. Here, two dual concepts are crucial: the asymptotic phase with which orbits converge to an oscillation, and its isochrons —each consisting of all points with the same asymptotic phase. From a dynamical systems perspective; isochrons are (𝑛−1)-dimensional manifolds (curves or (hyper-)surfaces) that foliate an 𝑛-dimensional basin of attraction. This thesis addresses the question of what such isochron foliations may look like when they are (one-dimensional) curves. It is important to also consider the backward-time isochrons of repelling oscillations. More specifically, we study how the two foliations by forward-time and backward-time isochrons interact in regions of phase space where they both exist, and how their global properties change during bifurcations. To this end, we present a case study of a specific planar system that features a sequence of relevant bifurcations. We explain how the basins and isochron foliations change throughout the sequence of bifurcations. In particular, we identify structurally stable tangencies between the foliations by forward-time and backward-time isochrons, which are associated with phase sensitivity. In contrast to an earlier reported mechanism involving a pair of tangency orbits, we find isochron foliation tangencies along single orbits. Moreover, the foliation tangencies we report arise from actual bifurcations of the system, as we illustrate in detail. We also study how isochrons accumulate onto a basin boundary, which may involve saddle equilibria and/or extend to infinity. We compute isochrons reliably with a refined boundary value problem set-up, implemented as a toolbox for CoCo in MATLAB; a tutorial with examples is provided. Moreover, we extend our set-up to isochrons that foliate a two-dimensional (un)stable manifold of a saddle periodic-orbit — providing a novel algorithm for computing such manifolds. As is demonstrated with examples of orientable and nonorientable manifolds, a foliation by isochrons can be used to visualise and understand their topological, geometrical, and synchronisation properties.
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