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.