A new fractional-order discrete BVP oscillator model with coexisting chaos and hyperchaos

In this work, the nonlinear conductance of the Bonhoeffer–van der Pol (BVP) model is replaced with an odd function that multiplies sine and cosine, and a novel discrete map with both chaos and hyperchaos is proposed. Fractional calculus is applied to this model to explore its complex dynamics. We focused on the different properties from the previous work, which manifested as the coexistence of various attractors in certain specific parameters, including the coexistence of quasi-periodical, chaotic, periodic, and hyperchaotic attractors. In particular, the rare coexisting phenomenon of hyperchaos and chaos was discovered in this model for the first time. In addition, the attractor can be flexibly controlled to move in the x direction of the phase space. Finally, the discrete model was verified on the DSP platform. The simulation under several sets of parameters shows the theoretical results of the new complex dynamical behaviors of the system. These phenomena indicate that the newly constructed fractional-order discrete model has relatively rich dynamical behaviors.