Relaxation oscillation in planar discontinuous piecewise smooth fast–slow systems

This paper provides a geometric analysis of relaxation oscillations in the context of planar fast–slow systems with a discontinuous right-hand side. We give conditions that guarantee the existence of a stable crossing limit cycle Γ ϵ when the singular perturbation parameter ϵ is positive and small enough. Moreover, in the singular limit ϵ → 0, the cycle Γ ϵ converges to a crossing closed singular trajectory. We also study the regularization of the crossing relaxation oscillator Γ ϵ and show that a (smooth) relaxation oscillation exists for the regularized vector field, which is a smooth fast–slow vector field with singular perturbation parameter ϵ. Our approach uses tools in geometric singular perturbation theory. We demonstrate the results to a number of examples including a model of an arch bridge with nonlinear viscous damping.