Can OrdinaryDiffEq.jl solve Fillippov system?

[xiaoming]

@ChrisRackauckas

Those aren’t the same equation. In the first case you have -A*sign(x)* .... In the second case, you have -A *s*..., so presumably s = sign(x) where s = p[1]. So does p[1] = sign(x)? In the second case you have the condition u[2]-integrator.p[1], so when u[2] = integrator.p[1] you switch the parameters. When you start, p[1] = 1, so when u[2] = 1 you swap p[1] = -1. Now next time around, the sign change would occur when u[2] = -1. In your original model, the point where the sign changed did not move after the first crossing, so these are not the same model.

Thank you for pointing out why my second code was wrong! Yes, the solution was oscillating around u[2]=1, since the vector fields in the two sides of u[2]=1 are squeezing towards u[2]=1. Just like this picture:

In non-smooth nonlinear dynamics, this phenomenon is called the sliding mode. My question is whether the existing solvers can handle this sliding mode more accurately. For example, detect when the solution enters the sliding mode or when it exits, and the solution can just slide along the switch surface, without oscillating.