Ito process for 2D system

Question

I have the following 2D dynamical system that I solve with NDSolve:

q[v_,x_]:=-v+c Tanh[3*v]-Sign[x];
sol = NDSolve[{x'[t] == v[t], 
   v'[t] == q[x[t],v[t]], x[0] == 0.01, 
   v[0] == 0.01}, {x[t], v[t]}, {t, 0, 100}];

The plot of the solution:

Plot[{Evaluate[x[t] /. sol], Evaluate[v[t] /. sol]}, {t, 0, 100}]

Now I wish to add uncorrelated Noise to the second equation, this is what I have so far:

A[v_]:=1/Cosh[3*v];    
proc = ItoProcess[{\[DifferentialD]x[t] == -v[t]\[DifferentialD]t, \[DifferentialD]v[
          t] == -q[x[t], v[t]] \[DifferentialD]t + 
         A[v[t]] \[DifferentialD]w[t]}, {x[t], v[t]}, {{x, v}, {0.01, 0.01}}, 
      t, w \[Distributed] WienerProcess[]]

and for the plot:

RandomFunction[proc, {0., 100., 0.01}]

I get the message:

"The specification "description" is not a random process recognizedby the system"

what is the source of the error and what is the proper way to write the 2d SDE to obtain both trajectories of x[t] and v[t]?

Answer 1

I am not an expert in dynamical systems, but in this way I have no error:

A[v_] := 1/Cosh[3*v];
q[v_, x_] := -v + c Tanh[3*v] - Sign[x];
c = 10^-4;
sol = NDSolveValue[{x'[t] == v[t], v'[t] == q[x[t], v[t]], 
    x[0] == 0.01, v[0] == 0.01}, {x[t], v[t]}, {t, 0, 100}];

Plot[{sol[[1]], sol[[2]]}, {t, 0, 100}]

As you can see, I substituted your NDsolve with NDSolveValue and changed the Plot consistently.

proc = ItoProcess[{\[DifferentialD]x[
       t] == -v[t] \[DifferentialD]t, \[DifferentialD]v[
       t] == -q[x[t], v[t]] \[DifferentialD]t + 
      A[v[t]] \[DifferentialD]w[t]}, {x[t], 
    v[t]}, {{x, v}, {0.01, 0.01}}, t, 
   w \[Distributed] WienerProcess[]];
RandomFunction[proc, {0., 100., 0.01}];
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