Generalizing the transition from amplitude to oscillation death in coupled oscillators

Amplitude death (AD) and oscillation death (OD) are two structurally different oscillation quenching types in coupled nonlinear oscillators. The transition from AD to OD has been recently realized due to the interplay between heterogeneity and coupling strength [A. Koseska et al., Phys. Rev. Lett. 111, 024103 (2013)]. We identify here the transition from AD to OD in nonlinear oscillators with couplings of distinct natures. It is demonstrated that the presence of time delay in the coupling cannot induce such a transition in identical oscillators, but it can indeed facilitate its occurrence with a low degree of heterogeneity. Moreover, it is further shown that the AD to OD transition is reliably observed in identical oscillators with dynamic and conjugate couplings. The coexistence of AD and OD and rich stable OD configurations after the transition are revealed, which are of great significance for potential applications in physics, biology, and control studies.

Figure 1

Comparison of the evolution from AD to OD in the coupled system (1). Thin dashed lines denote unstable steady states, whereas solid red (light gray) and black (dark gray) lines denote stable IHSSs (OD) and stable HSSs (AD), respectively. See text for further details.

Figure 2

The AD and OD regimes for the coupled system (1) in ($K,\Delta$) parameter space for (a) $\tau =0$, (b) $\tau =0.1$, (c) $\tau =0.2$, (d) $\tau =0.3$, and $w_1=2$. The black (dark gray) and red (light gray) regions denote the stable homogeneous (AD) and inhomogeneous (OD) steady states, respectively. The dashed blue line represents the critical coupling $K_c$ for the birth of the IHSS if $K>K_c$.

Figure 3

The AD and OD regimes for the coupled system (1) with $w_1=10$ in the parameter space of ($K,\Delta$) for (a) $\tau =0$, (b) $\tau =0.005$, (c) $\tau =0.01$, and (d) $\tau =0.16$.

Figure 4

The AD and OD regions for the coupled system (1) in the parameter space of $(\tau ,K)$ for (a) $\Delta =1$, (b) $\Delta =1.4$, (c) $\Delta =1.5$, and (d) $\Delta =1.6$ with $w_1=10$.

Figure 5

Transition from AD to OD in two coupled identical Stuart-Landau oscillators for $w=5$ with (a) and (b) dynamic and conjugate couplings, respectively, and (c) and (d) three Stuart-Landau oscillators with dynamic coupling for the $\alpha \alpha \beta$ and $\alpha \beta \gamma$ OD patterns after the transition.