Jump bifurcations in some degenerate planar piecewise linear differential systems with three zones

• Borning of limit cycles from a continuum of homoclinic and heteroclinic connections.
• Bifurcation characterized by the birth of a limit cycle from a continuum of equilibria.
• The Morris–Lecar model for the activity of a single neuron activity is studied.

We consider continuous piecewise-linear differential systems with three zones where the central one is degenerate, that is, the determinant of its linear part vanishes. By moving one parameter which is associated to the equilibrium position, we detect some new bifurcations exhibiting jump transitions both in the equilibrium location and in the appearance of limit cycles. In particular, we introduce the scabbard bifurcation, characterized by the birth of a limit cycle from a continuum of equilibrium points.Some of the studied bifurcations are detected, after an appropriate choice of parameters, in a piecewise linear Morris–Lecar model for the activity of a single neuron activity, which is usually considered as a reduction of the celebrated Hodgkin–Huxley equations.