Bursting oscillations in coupling Mathieu-van der Pol oscillator under parametric excitation

The bursting oscillation is a fast-slow oscillation. A complex four-dimensional coupled Mathieu-van der Pol oscillator is analyzed to investigate the bursting oscillations. The multiscale phenomena appear in four-dimensional coupled Mathieu-van der Pol oscillator due to the significant difference between the natural frequencies of the system and frequencies of the parametric excitation. Analyzing the bifurcation diagrams of four-dimensional coupled Mathieu-van der Pol oscillator using the bifurcation theory and fast-slow analysis, we identify four different bursting oscillation modes. Additionally, an intriguing phenomenon is observed as the parameters of the fast and slow systems change in the orbits. This phenomenon is called as the slow channel effect. The trajectory traverses the bifurcation point without the immediate bifurcation behaviors but gradually converges to a stable limit cycle after an apparent delay. Four-dimensional coupled Mathieu-van der Pol oscillator has a composite Hopf bifurcation which means that a Hopf bifurcation respectively corresponds to the large and small limit cycles. This research provides the insights into the mechanism of the bursting oscillations.