epl draft Diverse routes to oscillation death in a coupled oscillator system

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 multiparameter-sensitive 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 [3–6]. 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 [4,10–12]. 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, 13–16] (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 saddle-node 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 [27] 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 alternates between periodic, quasiperiodic and even chaotic behavior, reflecting the complex temporal variability observed in real biological systems [22,23]. Our model is a set of five coupled oscillators that successfully reproduces many phenomena seen in the cardiovascular system (CVS), e.g. modulation [20] and synchronization [24]. Each oscillator has its own characteristic frequency and amplitude [25] (see Table 1) and emulates a particular physiological function. Heart and respiration are obvious physiological processes; the myogenic oscillation is related to the intrinsic self-regulatory activity of the smooth muscle tissue in the walls of the blood vessels; the neurogenic oscillation is associated with the neural control by the central nervous system; and the nitric oxide related endothelial oscillation is associated with metabolic activity mediated by the endothelial tissue that lines the whole CVS [26]. The basic unit, ẋi = −xiqi − yiωi + Pi(X,Y) ẏi = −yiqi + xiωi + Qi(X,Y), (1) is the Poincaré oscillator [27] with: i = 1, . . . ,m, (m = 5);