Generalized boundary equilibria in -dimensional Filippov systems: The transition between persistence and nonsmooth-fold scenarios

We analyze a codimension-two bifurcation in generic n -dimensional Filippov (autonomous, piecewise smooth, discontinuous) systems, when a hyperbolic equilibrium collides with the discontinuity boundary. At generic (codim 1) boundary equilibria (BE), two different scenarios can occur: the ‘persistence’ of the stationary solution, that changes from a standard equilibrium on one side of the discontinuity boundary into a pseudo-equilibrium, and the ‘nonsmooth-fold’ between the standard equilibrium and a coexisting pseudo-equilibrium. Generalized (codim 2) boundary equilibria (GBE) separate persistence and nonsmooth-fold scenarios and are here shown to occur together with a fold bifurcation between pseudo-equilibria (PLP). In a two-parameter plane, the PLP bifurcation curve emanates from the GBE point tangentially to the BE curve. The asymptotic of the PLP curve locally to the GBE point is also derived and tested on a 4-dimensional ecological example.