New approach to study the van der Pol equation for large damping

Abstract

We present a new approach to establish the existence of a unique limit cycle for the van der Pol equation in case of large damping. It is connected with the bifurcation of a stable hyperbolic limit cycle from a closed curve composed of two heteroclinic orbits and of two segments of a straight line forming continua of equilibria. The proof is based on a linear time scaling (instead of the nonlinear Liénard transformation in previous approaches), on a Dulac–Cherkas function and on the property of rotating vector fields.

Keywords

• relaxation oscillations
• dulac–cherkas function
• rotated vector field