Piecewise differential equations

Specific Domains Numerics
6

Here is a (mildly) interesting example x''+x'+x=\pm p_1 where the sign of p_1 changes when a switching manifold is encountered at x=p_2. To make things more interesting, consider hysteresis in the switching manifold such that p_2\mapsto -p_2 whenever the switching manifold is crossed.

The code is relatively straightforward; the StaticArrays/SVector/MVector can be ignored, they are only for speed.

using OrdinaryDiffEq
using StaticArrays

f(x, p, t) = SVector(x[2], -x[2]-x[1]+p[1])  # x'' + x' + x = ±p₁
h(u, t, integrator) = u[1]-integrator.p[2]  # switching surface x = ±p₂;
g(integrator) = (integrator.p .= -integrator.p)  # impact map (p₁, p₂) = -(p₁, p₂)

prob = ODEProblem(f,  # RHS
                  SVector(0.0, 1.0),  # initial value
                  (0.0, 100.0),  # time interval
                  MVector(1.0, 1.0))  # parameters
cb = ContinuousCallback(h, g)
sol = solve(prob, Vern6(), callback=cb, dtmax=0.1)

Then plot sol[2,:] against sol[1,:] to see the phase plane - a nice non-smooth limit cycle in this case.

Note that if you try to use interpolation of the resulting solution (i.e., sol(t)) you need to be very careful around the points that have a discontinuous derivative as the interpolant goes a little awry. That’s why I’ve used dtmax=0.1 to get a smoother solution output in this case. (I’m probably not using the most appropriate integrator either but it’s the one that I was using in a previous piece of code that I copied-and-pasted :slight_smile:)

If you are interested in the dynamics of piecewise differential equations there are plenty of gotchas to watch out for, particularly if your system is a Filippov system; effects like sliding require a bit more care than shown in the code above. (E.g., I’m not sure if DifferentialEquations.jl can handle a system that changes from an ODE to a DAE part way through solving, such as happens with the sliding vector fields that come from Filippov systems; ever tried it @ChrisRackauckas?)