Diverse routes to oscillation death in a coupled-oscillator system

Abstract

We study oscillation death (OD) in a well-known coupled-oscillator system that has been used to model cardiovascular phenomena. We derive exact analytic conditions that allow the prediction of OD through the two known bifurcation routes, in the same model, and for different numbers of coupled oscillators. Our exact analytic results enable us to generalize OD as a multiparametersensitive phenomenon. It can be induced, not only by changes in couplings, but also by changes in the oscillator frequencies or amplitudes. We observe synchronization transitions as a function of coupling and confirm the robustness of the phenomena in the presence of noise. Numerical and analogue simulations are in good agreement with the theory.Coupled oscillator systems exhibit a variety of phenomena relevant to physics, biology, and other branches of science and technology. Here, we study oscillation death (OD) [1], a form of synchronization [2] in which the oscillators interact in such a way as to quench each other's oscillations [3][4][5][6]. This intriguing phenomenon was noted in the 19th century by Rayleigh [2], who found that adjacent organ pipes of the same pitch can reduce each other to silence. Since then, OD has been studied in diverse applications including oceanography [7], chemical engineering [8], solid-state lasers [9] and a variety of other experimental systems [4,[10][11][12]. OD is known to occur via two distinct bifurcation mechanisms: (i) Hopf bifurcation, where the coupling induces stability at the origin of the phase space, thus collapsing the orbits to zero, which can happen only if the oscillators are sufficiently different [5,[13][14][15][16] (or for identical oscillators if there are delays [17,18] in the coupling); or (ii) for non-identical oscillators, saddle-node bifurcation [4] in which new fixed points appear on/near the coupled limit cycles, annihilating the periodic orbits. Recently, Karnatak et al. [19] were able to produce OD in two identical coupled oscillators through the saddlenode route, using dissimilar non-delayed coupling.In this Letter, we show that a coupled-oscillator system, which has been used extensively in modeling coupled rhythmic processes in mathematics, physics and biology [26] can undergo OD via both bifurcation routes. We obtain exact analytic conditions for OD, and compare the theory with numerical simulations and analogue electronic experiments. We thus generalize OD as a phenomenon that occurs, not only through coupling-increased dissipativity [2], but also when a measure of dispersion among the parameters of the coupled system is exceeded [21]. We also show that, near the onset of death, the coupled system PACS 82.40.Bj -Oscillations, chaos, and bifurcations PACS 05.45.Xt -Synchronization; coupled oscillators PACS 05.45.-a -Nonlinear dynamics and chaos

Diverse routes of transition from amplitude to oscillation death in coupled oscillators under additional repulsive links

“…The dotted lines are the coexist-ing unstable steady states indicating the origin of SNB at ǫ = 2.414. The HSS coexists (zero black line) with the IHSS here in contrast to a previous report [24] where an unstable origin bifurcates into IHSS via SNB but the origin remains unstable. No transition from HSS to IHSS was observed although a negative mean-field coupling or a repulsive link was used there, possibly due to a symmetric use of the repulsive coupling.…”

Chronotaxic systems with separable amplitude and phase dynamics

“…For example, the model [36] of the cardiovascular system as five coupled self-sustained autonomous limit cycle oscillators reproduces the main characteristic features of the observed cardio-respiratory interactions. However, it was shown [37,38] that an explanation of the variability of cardiac and respiratory frequencies within the model [36] requires consideration of the effect of noise [37] or for the system to be near the onset of oscillation death [38]. In contrast to this, frequencies of oscillations in the systems [31][32][33][34][35], despite being externally perturbed, appear to have stable patterns in time that are resistant to external perturbations.…”

Coupling functions: Universal insights into dynamical interaction mechanisms

“…Another important group of physical phenomena attributable to interactions are those associated with oscillation and amplitude deaths (Bar-Eli, 1985;Mirollo and Strogatz, 1990;Prasad, 2005;Suárez-Vargas et al, 2009;Koseska, Volkov, and Kurths, 2013a;Zakharova et al, 2013;Schneider et al, 2015). Oscillation death is defined as a complete cessation of oscillation caused by the interactions, when an inhomogeneous steady state is reached.…”

“…When the distribution g of the natural frequencies is an even, unimodal, and nonincreasing function, and the coupling is weak, the incoherent state is neutral (Strogatz and Mirollo, 1991), but the order parameter r vanishes (at a polynomial rate) if g is smooth (Fernandez, Gérard-Varet, and Giacomin, 2014). Moreover, on increasing the coupling, the incoherent solution r ¼ 0 bifurcates for…”