Transition from Amplitude to Oscillation Death via Turing Bifurcation

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.

Figure 1

Role of coupling in the evolution from AD to OD. Parameters: ${\omega }_2=2$ and (a) $\Delta =1$, (b) $\Delta =2$, (c) $\Delta =3.385$, and (d) $\Delta =4$. Additionally, the accompanying limit cycles are also shown. Thin solid lines denote a stable HSS, thick solid lines a stable IHSS, dash-dotted lines an unstable steady state; dotted lines denote unstable limit cycles, whereas colored (online) solid lines denote stable limit cycle solutions.

Figure 2

Two parameter ($\epsilon$ vs $\Delta$) bifurcation diagram denoting the transition between (un)stable homogeneous and inhomogeneous steady states for the system of two coupled Stuart-Landau oscillators [Eqs. (2)].

Figure 3

Conjecture lines denoting stable limit cycles. The parameters are the same as in Fig. 1. Solid lines correspond to time traces of the first oscillator ($x_1$), whereas dashed lines correspond to the second oscillator ($x_2$).