Solving complex valued differential equations and dealing with branch cuts

Question

I'm gonna solve dx/dt=sqrt(1+(i*x)^3) for the initial condition x=-2-2i, and plot the real versus imaginary parts of x. I'm expecting to get a closed contour. However, what I get is only an incomplete portion of the actual plot. Can anyone help me to fix this problem in Matlab or Mathematica? Thanks in advance

Answer 1

Take a second and imagine someone who hasn't been looking over your shoulder watching your screen and reading the original problem description. How much could they figure out just and only given your written description above?

Now, is this close to the solution that you might be seeing?

In[1]:= f=x[t]/.NDSolve[{x'[t]==Sqrt[1+(I x[t])^3],x[0]== -2-2I}, x[t], {t,0,1.2}][[1]]; Plot[{Re[f], Im[f]}, {t, 0, 1.2}]

NDSolve::mxst: Maximum number of 10000 steps reached at the point t == 1.08993270437`. >>

Out[2]= ...PlotSnipped...

where the solution appears to blow up at 1.09? And why would it blow up there? Is that were your plot becomes incomplete?

Have I guessed the wrong range? Have I guessed the wrong initial condition? Don't know.

That doesn't try to do a ContourPlot yet, that is still just trying to see what the solution might look like.