Mixed-Mode Oscillations in a piecewise linear system with multiple time scale coupling

• Study of a four dimensional piecewise linear system with three different time scales.
• Proof of existence of an attractive limit cycle with complex Mixed-Mode oscillations.
• Interacting canard phenomena underlying changes and exchanges in oscillations number.

In this work, we analyze a four dimensional slow–fast piecewise linear system with three time scales presenting Mixed-Mode Oscillations. The system possesses an attractive limit cycle along which oscillations of three different amplitudes and frequencies can appear, namely, small oscillations, pulses (medium amplitude) and one surge (largest amplitude). In addition to proving the existence and attractiveness of the limit cycle, we focus our attention on the canard phenomena underlying the changes in the number of small oscillations and pulses. We analyze locally the existence of secondary canards leading to the addition or subtraction of one small oscillation and describe how this change is globally compensated for or not with the addition or subtraction of one pulse.