Stern-Brocot trees in spiking and bursting of sigmoidal maps

Abstract

We study the global organization of oscillations in sigmoidal maps, a class of models which reproduces complex locking behaviors commonly observed in lasers, neurons, and other systems which display spiking, bursting, and chaotic sequences of spiking and bursting. We find periodic oscillations to emerge organized regularly according to the elusive Stern-Brocot tree, a symmetric and more general tree which contains the better-known asymmetric Farey tree as a sub-tree. The Stern-Brocot tree provides a natural and encompassing organization to classify nonlinear oscillations. The mathematical algorithm for generating both trees is exactly the same, differing only in the initial conditions. Such degeneracy suggests that the wrong tree might have been attributed to locking phenomena reported in some of the earlier works.

Antiperiodic oscillations

“…The first 80 × 10 5 integration steps were discarded as transient. The chaotic/periodic/antiperiodic nature of solutions was determined and recorded in so-called isospike diagrams16: after the transient we integrated for an additional 80 × 10 5 time-steps and recorded extrema (maxima and minima) of a given variable of interest, up to 800 extrema, counting the number of peaks and checking whether for repetitions. Such high-resolution computations are numerically very demanding and, therefore, were performed on a SGI Altix cluster of 1536 high-performance processors running during a period of several weeks to compute many stability diagrams, three of them presented in Fig.…”

Nested arithmetic progressions of oscillatory phases in Olsen's enzyme reaction model

“…Although observing just quite limited sequences of oscillations, they interpreted them as belonging to a Farey tree, apparently unaware that the arithmetic sum rule p 1 /q 1 p 2 /q 2 ¼ (p 1 þ p 2 )/(q 1 þ q 2 ), which they used as evidence to claim observing the asymmetric Farey tree, is a sum rule that can be used equally well to identify a distinct and much more general organization: the symmetric Stern-Brocot tree. [28][29][30]59 In other words, although frequently used in older literature, fits of the aforementioned arithmetic sum rule to noisy and small sets of data are by no means enough for a conclusive and unambiguous identification of the nature of the underlying organizational tree. Thus, it would be nice to reconsider the experiments and check if they could eventually support the Stern-Brocot scenario.…”

“…1(c) are examples of isospike diagrams, [28][29][30][31][32][33] namely, diagrams recording the number of peaks per period of a given component of the oscillations. Such diagrams are the main tool used here to characterize the unfolding of oscillations in control parameter space.…”

Zig-zag networks of self-excited periodic oscillations in a tunnel diode and a fiber-ring laser

“…For many decades, the great interest has been to investigate mainly the structure of the phase space of flows, with particular emphasis on the possible transitions from order to chaos and a plethora of instabilities associated with these transitions [1][2][3]. More recently, extensive numerical simulations have revealed unexpected regularities in a complementary setting, namely, in the control parameter space of systems as diverse as electronic circuits, laser systems, and modulational interactions in a plasma, in chemical and biophysical oscillators, and in many other paradigmatic flows covering a large spectrum of practical applications [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Such regularities emerged while attempting to classify systematically all collective oscillations supported by the aforementioned applications.…”