Periodicity Hub and Nested Spirals in the Phase Diagram of a Simple Resistive Circuit

Abstract

We report the discovery of a remarkable ''periodicity hub'' inside the chaotic phase of an electronic circuit containing two diodes as a nonlinear resistance. The hub is a focal point from where an infinite hierarchy of nested spirals emanates. By suitably tuning two reactances simultaneously, both current and voltage may have their periodicity increased continuously without bound and without ever crossing the surrounding chaotic phase. Familiar period-adding current and voltage cascades are shown to be just restricted one-parameter slices of an exceptionally intricate and very regular onionlike parameter surface centered at the focal hub which organizes all the dynamics.

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

“…2(b) and 2(c). These enlargements serve to show where the zig-zag networks are located and to illustrate a nice by-product of the diode circuit: its control space displays several large groups of nested spiral networks and hubs of the type mentioned in the Introduction, not just single spirals as known so far [8,9]. The discovery of a circuit with several large spirals is important because such a circuit can help bypass the major obstruction in the experimental and theoretical study of spirals: the fact that known spirals are strongly distorted and compressed due to the scales used.…”

“…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.…”

“…As an example of a wide-ranging regularity in parameter space we mention the infinite alternation of spirals of chaos and of periodicity which emanate from certain periodicity hubs, namely, from exceptional focal points ubiquitous in the control space of some systems [8]. In this alternation, each spiral corresponds to a certain periodic phase characterized individually by an infinite cascade of periodic waveforms specific to it which evolve and get continuously more complex when parameters are suitably changed along the spiral towards its focus.…”

“…In this alternation, each spiral corresponds to a certain periodic phase characterized individually by an infinite cascade of periodic waveforms specific to it which evolve and get continuously more complex when parameters are suitably changed along the spiral towards its focus. Their period seems to accumulate to a specific value near the focal point while the number of peaks (local maxima) in each period seems to grow without bound, diverging as one approaches the focal point more and more [8,9]. Knowledge of such infinite networks of periodic phases has a direct practical application.…”

Deterministic Chaos Theory: Basic Concepts

“…13 (b) shows better the same periodic window. Such windows are also called shrimps [37] and have been observed in several dynamical systems [38,39].…”