Transition from Amplitude to Oscillation Death via Turing Bifurcation Aneta Koseska, 1,2, * Evgenii Volkov, 3 and Ju ¨rgen Kurths 1,4 1 Institute of Physics, Humboldt-University, 10099 Berlin, Germany 2 Department of Systemic Cell Biology, Max Planck Institute of Molecular Physiology, 44227 Dortmund, Germany 3 Department of Theoretical Physics, Lebedev Physical Institute, 119991 Moscow, Russia 4 Potsdam Institute of Climate Impact Research, 14473 Potsdam, Germany (Received 25 March 2013; revised manuscript received 16 May 2013; published 10 July 2013) Coupled oscillators are shown to experience two structurally different oscillation quenching types: amplitude death (AD) and oscillation death (OD). We demonstrate that both AD and OD can occur in one system and find that the transition between them underlies a classical, Turing-type bifurcation, providing a clear classification of these significantly different dynamical regimes. The implications of obtaining a homogeneous (AD) or inhomogeneous (OD) steady state, as well as their significance for physical and biological applications and control studies, are also pointed out. DOI: 10.1103/PhysRevLett.111.024103 PACS numbers: 05.45.a, 05.45.Xt, 89.75.Fb Since the work of Van der Pol and Van der Mark [1], the studies of coupled nonchaotic oscillators have provided a rich source of ideas and insights regarding the role of different coupling types, as well as the dependence on the oscillator structure in the generation of new dynamical regimes [2–4]. It has been shown that even ensembles consisting of identical oscillators may generate a variety of rhythms that differ in their period and phase relations based on the coupling organization [5–8]. Apart from such rhythmogenic activity, coupling can even suppress oscil- lations in a network by different mechanisms. Here, we distinguish between two main manifestations of oscillation quenching, amplitude and oscillation death phenomena, which are structurally different. Generally, the amplitude death (AD) refers to a situation where oscillations are suppressed when individual oscilla- tors are coupled and return to the steady state of the system instead. Thus, the amplitude death results in a homoge- neous steady state (HSS), since all of the oscillators popu- late the same state. Three main mechanisms can lead to this phenomenon: (a) a sufficiently large variance of the fre- quency distribution [2,9], (b) existence of time delay in the coupling [10–13], and (c) coupling of identical oscillators through dissimilar (or conjugate) variables [14,15] (for a recent review on AD, see Ref. [16]). On the other hand, the second manifestation of oscillation quenching—the oscil- lation death (OD) phenomenon—has a significantly differ- ent background of occurrence compared to AD. Namely, OD is a result of breaking the system’s symmetry through a pitchfork bifurcation of the unstable steady state, whereby the homogeneous steady state splits, giving rise to two additional branches. In the limiting case of two coupled oscillators, one follows the upper, whereas the second oscillator follows the lower branch. Thus, OD is mani- fested as a stabilized inhomogeneous steady state (IHSS), displaying further the possibility for the occurrence of additional limit cycle(s) in the same phase space area. The idea of the broken symmetry steady state pioneered by Turing [17] for stationary media received its mathe- matical formulation by Prigogine and Lefever [18] for two identical oscillating elements—Brusselators, coupled in a diffusionlike manner. Furthermore, it has been shown theo- retically that OD is model independent, persisting for large parametric regions in several models of diffusively coupled chemical [19] or biological oscillators [7,20–24]. Experimentally, the extinction of oscillations in chemical reactors coupled by mutual mass exchange was initially reported by Dolnik and Marek [25]. Later on, Crowley and Epstein demonstrated for two coupled, slightly nonident- ical chemical oscillators that the basis for the OD is a specific, vector-type coupling, namely, coupling via a slow recovery variable [26]. Recently, OD has been ex- perimentally observed in chemical nanooscillators (micro- fluidic Belousov-Zhabotinsky-octane droplets), diffusively coupled via signaling species (Br 2 , in this case) [27]. However, in certain systems, i.e., in neurobiology, a mani- festation of both oscillation quenching types is present: OD constitutes a well-known phenomenon in neurons, the winner-take-all situation [28], whereas AD mainly serves to suppress neuronal oscillations [29]. Because of their significantly different representations, as inhomogeneous (OD) and homogeneous (AD) steady states, both oscillation quenching types allow the genera- tion of two structurally different dynamical regimes with different meaning. This is important not only from a view- point of dynamical control but also from an application aspect: it has been shown that OD can be interpreted as a background mechanism of cellular differentiation [30,31], whereas AD is mainly used as a stabilization control in physical or chemical systems [32,33]. Thus, OD is espe- cially significant for biology, since in contrast to AD, it can provide presence of heterogeneity in a stable homogeneous medium. This possibility is further widened by the fact that OD is a source of a stable inhomogeneous limit cycle PRL 111, 024103 (2013) PHYSICAL REVIEW LETTERS week ending 12 JULY 2013 0031-9007= 13=111(2)=024103(5) 024103-1 Ó 2013 American Physical Society