STABILITY OF SYNCHRONIZATION MANIFOLD IN COUPLED FRACTIONAL-ORDER NONLINEAR OSCILLATORS: A MASTER STABILITY FUNCTION APPROACH

The Master Stability Function (MSF) is a crucial tool for understanding the synchronization behavior of coupled nonlinear oscillators. In recent years, the field of fractional calculus and its applications in nonlinear dynamics gained significant attention, leading to an expanded focus on the network dynamics of such systems. This study aims to explore this emerging area by deriving the MSF for coupled fractional-order nonlinear oscillators and investigating their relationship with coupling strength and fractional order. To provide a comprehensive comparison, we utilize the well-known nonlinear oscillators to study the differences and similarities between integer-order and fractional-order MSFs. Our analysis reveals that, similar to integer-order systems, fractional-order coupled nonlinear oscillators exhibit MSFs that can be characterized by the presence of negative values within a finite interval of the normalized coupling parameter. This negative region is crucial as it indicates stable synchronization. Furthermore, we categorize the fractional-order MSFs using the same classifications applied to integer-order MSFs. This classification helps in systematically understanding and comparing the synchronization properties of both types of systems. Our findings are supported by extensive numerical simulations, which demonstrate that the majority of fractional-order coupled oscillators exhibit higher classes of MSF. This higher classification suggests that fractional-order systems have a superior ability to achieve synchronization compared to their integer-order counterparts. In summary, this study underscores the significance of fractional calculus in enhancing our understanding of synchronization in complex systems. The derived MSF for fractional-order nonlinear oscillators provides valuable insights into their dynamic behavior and opens new avenues for research in various scientific and engineering disciplines.