Effects of potential field delay and coupling delay on collective behavior of a fractional-order coupled system in a dichotomous fluctuating potential

Abstract The collective dynamic of a fractional-order globally coupled system with time delays and fluctuating frequency is investigated. The power-law memory of the system is characterized using the Caputo fractional derivative operator. Additionally, time delays in the potential field force and coupling force transmission are both considered. Firstly, based on the delay decoupling formula, combined with statistical mean method and the fractional-order Shapiro–Loginov formula, the “statistic synchronization” among particles is obtained, revealing the statistical equivalence between the mean field behavior of the system and the behavior of individual particles. Due to the existence of the coupling delay, the impact of the coupling force on synchronization exhibits non-monotonic, which is different from the previous monotonic effects. Then, two kinds of theoretical expression of output amplitude gains G and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mi>G</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:math> are derived by time-delay decoupling formula and small delay approximation theorem, respectively. Compared to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mi>G</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:math> , G is an exact theoretical solution, which means that G is not only more accurate in the region of small delay, but also applies to the region of large delay. Finally, the study of the output amplitude gain G and its resonance behavior are explored. Due to the presence of the potential field delay, a new resonance phenomenon termed “periodic resonance” is discovered, which arises from the periodic matching between the potential field delay and the driving frequency. This resonance phenomenon is analyzed qualitatively and quantitatively, uncovering undiscovered characteristics in previous studies.