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differential equations - NDSolve::depdole error


I'm trying to solve a pair of differential equations with NDSolve. They contain a discontinuous function bounce. I've trimmed down my original equations to the following for simplicity:


eqxy = {
  0.1 == bounce[x[t]] - Cos[x[t]] x''[t], 
  y''[t] == -Sin[x[t]] + Cos[x[t]] x''[t]

};

where


bounce[x_] := If[Abs[x] >= 0.045, Abs[x], 0]

The following formatting may be clearer $$ \left( \begin{align} 0.1 &=\text{If}\,[\left| x(t)\right| \geq 0.045,\,\left| x(t)\right|,\,0]-x''(t) \cos(x(t)) \cr y''(t) &=x''(t) \cos (x(t))-\sin (x(t)) \cr \end{align} \right) $$


Now, running


NDSolve[
  Join[eqxy, {x[0] == 0, x'[0] == 1, y[0] == 1, y'[0] == 1}], 
  {x, y}, {t, 0, 1}, Method -> "Automatic"]


yields the error message



NDSolve::depdole: The differential order of a dependent variable in {x'[t], x''[t], y'[t], y''[t]} exceeds the highest order that appears in the differential equations.



although interpolating functions are produced as solutions.


The error seems to be associated with bounce, with its conditional and absolute value functions -- removing bounce eliminates the error message. But how does bounce cause any order excess?


Appreciate any insights. (Hope I've eliminated any typos.)


I'm running Mathematica version 10.3.0.0, Linux x86 (64-bit).



Answer




It's a confirmed bug in event processing.


Try Method -> {Automatic, "DiscontinuityProcessing" -> False} as a workaround.


NDSolve[
  Join[eqxy, {x[0] == 0, x'[0] == 1, y[0] == 1, y'[0] == 1}], 
  {x, y}, {t, 0, 1}, Method -> {Automatic, "DiscontinuityProcessing" -> False}]

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