Study of a homoclinic canard explosion from a degenerate center

Canard explosion is an appealing event occurring in singularly perturbed systems. In this phenomenon, upon variation of a parameter within an exponentially small range, the amplitude of a small limit cycle increases abruptly. In this letter we analyze the canard explosion in a limit cycle related to a degenerate center (with zero Jacobian matrix). We provide a second-order approximation of the critical value of the parameter for which the canard explosion occurs. Numerical results are compared with the analytical predictions and excellent agreements are found. As in this problem the canard explosion ends in a homoclinic connection, a very good approximation for the homoclinic curve in the parameter plane is also obtained.

Automated summary

This article studies the phenomenon of canard explosion in singularly perturbed systems. By analyzing the limit cycle related to a degenerate center, an approximation of the critical parameter for canard explosion is provided and compared with numerical results, showing excellent agreement. Additionally, a good approximation of the homoclinic curve in the parameter plane is obtained.