Stern–Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer–van der Pol and the FitzHugh–Nagumo models of excitable systems

Abstract

We report a systematic two-parameter study of the organization of mixed-mode oscillations and periodadding sequences observed in an extended Bonhoeffer-van der Pol and in a FitzHugh-Nagumo oscillator. For both systems, we construct isospike diagrams and show that the number of spikes of their periodic oscillations are organized in a remarkable hierarchical way, forming a Stern-Brocot tree. The SternBrocot tree is more general than the Farey tree. We conjecture the Stern-Brocot tree to also underlie the hierarchical structure of periodic oscillations of other systems supporting mixed-mode oscillations.

Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows

“…Since homoclinic loci have an infinite number of subsidiaries, we see nothing preventing the existence of an infinite number of such new independent hubs, as anticipated by numerical simulations [6]. We emphasize that the existence of the extra folding points F j , with j > 1 is a novel and general feature that depends on specific details of the attractor of the system being considered: it does not follow from the local unfolding of a higher codimension bifurcation point in any obvious way (although there may still be a relationship with mixed-mode oscillation theory [37], see for exam-ple [38]). Correspondingly, the homoclinic orbits at the various hubs have a rather different structure (Fig.…”

“…They are therefore a very powerful way to begin the analysis of nonlinear systems and can also be applied to laboratory experiments which, of course, only detect stable structures. A complementary tool that may be useful in analyzing dynamical systems is the direct study of the oscillations as parameters are tuned [37].…”

Circuits with Oscillatory Hierarchical Farey Sequences and Fractal Properties

“…In fact, all the parameters α, β, a, b and K in (1) may be considered as the bifurcation parameters. We shall analyze the local maximum values of LAOs and SAOs and show that, by slowly changing any of the above parameters, we can obtain the local maximum values of x i (t), i = 1, 2, 3, that can be described by the Farey arithmetic [17] and related to the Stern-Brocot tree [8,24]. In our analysis we use the letters L and s in L s to describe the number of local maximum values of the steady-state LAOs and SAOs, respectively, and not the number of LAOs and SAOs.…”

Stability mosaics in a forced Brusselator

“…Figure 1 illustrates the typical situation studied in this paper. For a given window in the frequency versus the parameter A control space, it shows two distinct stability diagrams: a typical Lyapunov stability diagram is shown in Figure 1a, while an isospike diagram [17,[22][23][24] is shown in Figure 1b. This latter diagram represents in colors parameter regions characterized by periodic oscillations having the same number of spikes per period, as recorded in the colorbar under the diagram.…”