Analysis of nested repelling mixed-mode oscillations in a slow–fast hard-type van der Pol oscillator subject to weak periodic perturbations

In a previous work (Sekikawa and Inaba, 2025), it was discovered that an unstable canard can be generated by a slow–fast hard-type autonomous van der Pol (vdP) oscillator with fifth-order nonlinearity. In this Letter, we analyze repellers in such a hard-type vdP oscillator when a weak periodic perturbation is applied. We confirm that the repelling oscillations generate successive unnested and nested unstable mixed-mode oscillation-incrementing bifurcations (MMOIBs). According to the numerical results presented here, successive unnested and nested MMOIBs emerge as many times as the system is tracked. The time-series waveforms of the repelling mixed-mode oscillations (MMOs) observed here closely resemble those observed in the case of attracting MMOs, and the bifurcation structures of nested repelling MMOs are also similar to those of the nested attracting MMOs. Nested repelling MMOs represent an entirely novel phenomenon, and we believe that this work could open the path for new research in nonlinear systems.