Stochastic Mathieu equation: Is this a numerical instability?

Question

So I am a beginner with stochastic differential equations and came across Mathematica's capabilities for solving them. I am solving the stochastic Mathieu equation with a harmonic forcing term that has a stochastic fluctuation imposed.

The linear set of ODEs to solve this are (ready for Ito form):

$ dx(t)=y(t) dt,\\ dy(t)=\lambda (t) dt \cos (\nu t) \sin (x(t))-\epsilon \,\textbf{dw(t)} \sin (x(t))$

Where $\lambda$ is time-varying as in the code below. $\nu$ possesses some physical characteristic I cannot place at the moment but takes real values. $\epsilon$ is a small number $<<1$. $w$ is the Wiener process distribution with zero drift and volatility (Ariaratnam et al.) (google drive link to paper. Only introduction is relevant for now.).

tmax = 100;
\[Lambda][t_] := 
 Piecewise[{{.95, t <= tmax/4}, {1.1, tmax/4 < t <= tmax/2}, {1.25, 
    t > tmax/
      2}}]; \[Nu] = 105.; \[Epsilon] = 0.001; x0 = 0.1; y0 = 0.1; \
drift = 0; volatility = 1.;

sme = ItoProcess[
   {
    \[DifferentialD]x[t] == y[t] \[DifferentialD]t,
    \[DifferentialD]y[
       t] == \[Lambda][t] Cos[\[Nu] t] Sin[
        x[t]] \[DifferentialD]t - \[Epsilon] Sin[
        x[t]] \[DifferentialD]w[t]
    }, x[t],
   {{x, y}, {x0, y0}},
   {t, 0}, w \[Distributed] WienerProcess[drift, volatility]
   ];
path = RandomFunction[sme, {0., tmax, 1}, 1]

My main issue is with the path = RandomFunction[sme, {0., tmax, 1}, 1] line. Here, it is my understanding that the {0., tmax, 1} is a time-stepping from 0 to tmax in steps of 1.

With a "time-step" of 1, I get the following dynamics which are reasonable, I would think:

ListLinePlot[path, PlotRange -> All, GridLines -> None, 
 ImageSize -> 250 {1, 1}, Frame -> True, FrameStyle -> Black, 
 FrameLabel -> {"time", "position"}, 
 PlotStyle -> {Thickness[0.005], Black}, 
 BaseStyle -> {FontSize -> 14, Bold}, Axes -> False]

However, if I change the time-step to, say, 0.1, I see an instability (or is it?) with some sort of monotonically increasing position.

Am I going wrong somewhere? Any input or assistance, I am grateful for.

Edit: The position plot is for x[t].

Answer 1

I think your dt=1 solution is the wrong one. The Cos[ν t] term is very rapidly varying with your parameter value of ν=105.:

Plot[Cos[ν t], {t, 0, tmax}, PlotPoints -> 200]

The apparent periodic-like solution you get with dt=1. might be due to an aliasing effect of sampling the high frequency cosine with period 1. To get the right solution, I'd pick a time step much less than the period of the forcing.

Going out on a limb here, I'd guess that we could replace that rapidly varying Cos[ν t] term by its average of zero.

sme = ItoProcess[{\[DifferentialD]x[t] == y[t] \[DifferentialD]t,
  \[DifferentialD]y[t] == -\[Epsilon] Sin[x[t]] \[DifferentialD]w[t]},
  x[t], {{x, y}, {x0, y0}}, {t, 0}, 
  w \[Distributed] WienerProcess[drift, volatility]];
path = RandomFunction[sme, {0., tmax, 0.01}, 1]

Further speculation: the time average of -ϵ Sin[x[t]] \[DifferentialD]w[t] is zero, so the net effect of noise is zero. We might get some insights from the resulting ODE:

sol = NDSolve[{x'[t] == y[t], y'[t] == 0, x[0] == x0, y[0] == y0}, {x, y}, {t, 0, tmax}];
Plot[x[t] /. sol, {t, 0, tmax}]

I think for more interesting dynamics, you'll want to lower ν.