Crises of a Fractional Birhythmic Van der Pol Oscillator Using the Improved Cell Mapping Method

Birhythmicity oscillators have been extensively applied in fields such as biology, physics, and engineering. Studying their global dynamics is crucial for gaining a comprehensive understanding of the intrinsic mechanisms that govern oscillator behavior. This paper focuses on investigating the influence of memory effects on the global dynamics of the fractional birhythmic van der Pol (BVDP) oscillator. To determine the system's global properties, we employ an improved cell mapping method. Specifically, the system's evolution is computed by introducing additional auxiliary variables, creating a space for storing historical information. This method allows us to examine the system's global properties without memory loss. Through a comparison of the global dynamic behaviors of BVDP oscillators with memory to those without memory, we observe that the presence of memory effects results in the emergence of chaotic attractors in the system. This, in turns, results in system instability and heightened sensitivity to initial conditions. Furthermore, our findings suggest that changes in the fractional-order can induce various crises in the oscillator and may have the potential to suppress chaotic oscillations.