Extreme phase sensitivity in systems with fractal isochrons

Abstract

Sensitivity to initial conditions is usually associated with chaotic dynamics and strange attractors. However, even systems with (quasi)periodic dynamics can exhibit it. In this context we report on the fractal properties of the isochrons of some continuous-time asymptotically periodic systems.We define a global measure of phase sensitivity that we call the phase sensitivity coefficient and show that it is an invariant of the system related to the capacity dimension of the isochrons. Similar results are also obtained with discrete-time systems. As an illustration of the framework, we compute the phase sensitivity coefficient for popular models of bursting neurons, suggesting that some elliptic bursting neurons are characterized by isochrons of high fractal dimensions and exhibit a very sensitive (unreliable) phase response.

An operator-theoretic viewpoint to non-smooth dynamical systems: Koopman analysis of a hybrid pendulum

“…A close examination of the contour plots suggests that merging of trajectories indeed occur for certain initial conditions in (15). Additionally, the phase plots indicate that the kicking of the pendulum introduces a high level of phase sensitivity [5] close to the unstable periodic orbit.…”

“…Clearly, if c is a bijection, the level sets of φ 0 separates the basins of every limit cycle. A method to find eigenfunctions directly from the time-histories of observables involves computing their infinite-time averages (5). These averages are projections of observables (6) onto the eigenspace at λ = 0, and for observables that are integrable on the set A 2 , these averages are well-defined almost everywhere in the kicked region (follows directly from theorem 1).…”

A continuation approach to computing phase resetting curves

“…Figure 7 illustrates that the PTC for system (24) is indeed continuous. Panel (a) reproduces the PTC from Fig.…”

“…System (24) was presented and studied in [20], because experimental data on similar pacemaker cells suggested that the PTC was discontinuous; see already Fig. 6(a).…”

“…Panel (a) reproduces the PTC from Fig. 6(b), but shows the computed values for ϑ new over the wider range [−0.6, 1] to show that a maximum of ϑ new is quickly followed by a minimum of ϑ new (lowest point of dashed Instead of an instantaneous reset, the reset in [20] is obtained by applying a current with amplitude I app for a fixed duration ∆t; the specific case for which a seemingly discontinuous PTC was observed is given by I app = −150 pA and ∆t = 0.02 s. Mathematically, this amounts to replacing the V -equation in system (24) by…”