Out-of-phase mixed-mode oscillations of two strongly coupled identical relaxation oscillators

Abstract

We analyzed bifurcations of periodic regimes generated in the systems of two identical relaxation oscillators under strong coupling through a ''slow'' ͑inhibitory͒ variable. It was numerically shown that complex spatiotemporal behavior is observed near the boundaries of stability of the known antiphase periodic attractor and inhomogeneous steady states. Specifically, the following attractors were found: ͑i͒ a set of cycles of the antiphase type, each of which consists of one full-amplitude excursion and of the different number of smallamplitude high-frequency oscillations ͑the period of antiphase mixed-mode regimes is much greater than that of simple antiphase oscillations͒, ͑ii͒ inhomogeneous regimes of the above described type ͑out-of-phase mixed mode͒ with unequal numbers of small oscillations for different oscillators, ͑iii͒ period doubling cascades of the out-of-phase mixed mode that lead to the appearance of chaotic attractors. We showed that the modes found are not specific for our particular model; however, they are common for several classes of models and sensitive to the stiffness of oscillators. We discuss also conditions for the generation of such regimes. ͓S1063-651X͑96͒06806-7͔

Excitatory and inhibitory coupling in a one-dimensional array of Belousov-Zhabotinsky micro-oscillators: Theory

“…For two coupled BZ oscillators, the IP mode exists for any parameters, but it is stable to large-amplitude perturbations only in a limited region of the parameter space, which we designate as IP. We refer to the other regions as AP/IP or simply AP (where AP oscillations are stable, while IP is unstable to large-amplitude perturbation) and T/IP or simply T. Bistability between IP and AP oscillations (or between IP and T regimes) has been observed in other systems of two coupled oscillators [7,11,27]. We also find bistability between the T and AP regimes.…”

“…Such modes have been found in many other systems of two coupled identical oscillators [11,24,25]. There are two other dynamical modes of two coupled BZ oscillators at rather small r V , when a << b: (iv) one oscillator exhibits large amplitude oscillations (with the amplitude of z close to c 0 ) and the other shows small amplitude oscillations (less than 0.01c 0 ), i.e., almost suppressed oscillations; (v) a chaotic-like mode at still smaller r V , in which both oscillators demonstrate alternating bursting with different numbers of spikes in each burst.…”

“…In this case, the coupling term in eqs. (11) and (13) assumes the familiar form ±C X (x 1 -x 2 )/2, and the meaning of the coupling strength C X is more evident.…”

“…Synchronization, for example, arises from the coupling of oscillators in physics (e.g., Huygens's clocks [7]), chemistry [1], and biology (e.g., in coupled neurons [8]). Such phenomena as quorum sensing [9,10] result primarily from excitatory, i.e., attractive, coupling between oscillators, while multistability and multirhythmicity are due to inhibitory, or repulsive, coupling [11,12].Recently, we developed a new experimental system for studying coupled chemical oscillators with local inhibitory coupling, consisting of an array of small (≅ 100 μm in diameter) identical water droplets separated by a surfactant monolayer from an oil phase and/or from each other. In the latter case, two monolayers of touching droplets can produce a bilayer [13,14].Each droplet contains the reactants of the oscillatory Belousov-Zhabotinsky (BZ) reaction [15,16]: malonic acid (MA), bromate, sulfuric acid, ferroin (catalyst), and a small amount of Ru(bpy) 3 2+ , which serves both as a cocatalyst and to make the BZ reaction photosensitive.Droplets are diffusively coupled through species dissolved in the oil phase, mainly Br 2 , the inhibitor.In that system, employing both one-dimensional (1D) [13] and two-dimensional (2D) [17] geometries, we observed a number of spatiotemporal patterns consisting of both oscillatory and stationary droplets.…”

“…Synchronization, for example, arises from the coupling of oscillators in physics (e.g., Huygens's clocks [7]), chemistry [1], and biology (e.g., in coupled neurons [8]). Such phenomena as quorum sensing [9,10] result primarily from excitatory, i.e., attractive, coupling between oscillators, while multistability and multirhythmicity are due to inhibitory, or repulsive, coupling [11,12].…”