Divergence of numerical solution of discontinuous nonllinear ordinary ODE

Questions
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1

Hi, I am facing troubles to solve the following 2nd order ordinary differential equation

{x'[t] == v[t],  v'[t] == 0.4 Sign[v[t]] UnitBox[v[t], x[t]] - 0.1 v[t] - x[t]}

Apparently, this is a nonlinear ODE, moreover, it contains a discontinuity (step) in both variables. I solve it numerically using

sol = NDSolve[{x'[t] == v[t], v'[t] == 0.4 Sign[v[t]] UnitBox[v[t], x[t]] - 0.1 v[t] - x[t], x[0] == 1., v[0] == 1.}, 
{v[t], x[t]}, {t, 0, 30}]

Plotting the solution using

ParametricPlot[Evaluate[{x[t], v[t]} /. sol], {t, 0, 30}]

I get

phase_portraits_for_oscillators.png
phase_portraits_for_oscillators.png360×332 19.3 KB

This suggests that there is a limit cycle. Indeed, this is confirmed by plotting the phase portrait by

f2 = StreamPlot[{v, 0.4` Sign[v] UnitBox[v, x] - 0.1` v - x}, {x, -2, 2}, {v, -2, 2}, StreamStyle -> LightGray];

and plotting it together with the previous solution

Show[f1, f2]

phase_portraits_for_oscillators_2.png
phase_portraits_for_oscillators_2.png360×332 38.1 KB

But now, if I set the initial conditions to inside the orbit, the solution fails

sol = NDSolve[{x'[t] == v[t], v'[t] == 0.4 Sign[v[t]] UnitBox[v[t], x[t]] - 0.1 v[t] - x[t], x[0] == 0.1., v[0] == 0.1.}, 
    {v[t], x[t]}, {t, 0, 30}]

f3 = ParametricPlot[Evaluate[{x[t], v[t]} /. sol], {t, 0, 4.1}, PlotStyle -> Red]

Show[f1, f2, f3]`

phase_portraits_for_oscillators_4.png
phase_portraits_for_oscillators_4.png360×332 40.5 KB

What can be done about it? I tried changing the solver and it did not help. I tried to insert the WhenEvent together with CrossDiscontinuity option. No way.

2

I think it helps to highlight the unitbox:

Show[StreamPlot[{v, 
   0.4` Sign[v] UnitBox[v, x] - 0.1` v - x}, {x, -0.6, 0.6}, {v, -0.6,
    0.6}, StreamStyle -> LightGray, 
  Prolog -> {LightBlue, Rectangle[{-0.5, -0.5}, {0.5, 0.5}]}], f3]

box.jpg
box.jpg360×353 17.1 KB

As you can see, the blue box is the source of the trouble. The curves are smooth until they hit the box. The part where it goes wrong is a very difficult part of the box. If you look at the stream, it is directing the curve back into the edge of the box.

box2.jpg
box2.jpg360×348 16.2 KB

It’s getting pushed back into the discontinuity basically over and over. This part probably needs some very special treatment.