Rotation parameter modulation of dynamics in coupled fractional-order Van der Pol oscillators

Traditional fixed-sign coupling schemes impose rigid directional constraints on interaction pathways, while integer-order differential models inherently lack the capacity to describe memory effects and long-range temporal correlations. To overcome these limitations, here we establish a fractional-order coupled Van der Pol oscillator system with a rotation parameter to investigate how the rotation parameter and fractional-order derivative control the dynamics. The rotation parameter distributes the coupling to the x and y variables via trigonometric functions, precisely regulating the sign and weight of coupling in each direction; this governs bifurcations and qualitatively determines the steady-state mode. Moreover, the memory effect of the fractional-order derivative exhibits quadrant dependence: As the fractional-order derivative decreases, the rotation parameter delays amplitude death in the second and third quadrants but accelerates incoherent oscillation death in the fourth quadrant. Finally, employing phase reduction theory, we derive the phase model and numerically verify the system’s transition path. Our findings break away from traditional frameworks, deepen the understanding of fractional-order coupled systems, elucidate the regulatory mechanisms of rotation parameters, and provide a theoretical foundation for controlling synchronization, chimera states, and death states in real-world systems.