Abstract
Fractional derivatives provide a prominent platform for various chemical and physical system with memory and hereditary properties, while most of the previous differential systems used to describe dynamic phenomena including oscillation quenching are integer order. Here, effects of fractional derivative on the transition process from oscillatory state to stationary state are illustrated for the first time on mean-filed coupled oscillators. It is found the fractional derivative could induce the emergence of a first-order discrete transition with hysteresis between oscillatory and stationary state. However, if the fractional derivative is smaller than the critical value, the transition will be invertible. Besides, the theoretical conditions for the steady state are calculated via Lyapunov indirect method which probe that, the backward transition point is unrelated to mean-field density. Our result is a step forward in enlightening the control mechanism of explosive phenomenon, which is of great importance to highlight the function of fractional-order derivative in the emergence of collective behaviors on coupled nonlinear model.
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