Measuring the transient time of amplitude death in coupled oscillators

Amplitude death (AD) denotes the cessation of oscillations that has been discovered recently under many typical coupling strategies and scenarios. However, the time needed for the coupled systems to achieve AD when the coupling works, also known as transient time, is still unclear. Here, by considering the microscopic processes from oscillations to AD under distinct coupling schemes, we show that the system parameters have a big impact on the transient time, and there always is the minimum (the shortest time) to reach AD state. In particular, we find the curves of transient time with shape V in the parameter space when the stable AD region is finite, which indicates that the moderate variable parameter facilitates the shorter transient time. For the infinite AD domain, the change of transient time show a trend with shape L, that is, the transient time can maintain minimum when the variable parameter surpasses a certain threshold. Moreover, through analyzing the eigenvalues of coupled system, we observe that the real part of the largest eigenvalue can better fit the variation of transient time. Therefore, the real part of the largest eigenvalue is used as an index for the search of shortest transient time to reach AD state without checking all early-time dynamics of the coupled systems. Our results provide a new insight to treat the transient behaviors of coupled dynamical systems, and give a general method to find the more efficient strategy for inducing AD.