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False divergence of the NDSolve solution: how to avoid

• To: mathgroup at smc.vnet.net
• Subject: [mg101771] False divergence of the NDSolve solution: how to avoid
• From: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>
• Date: Thu, 16 Jul 2009 08:21:45 -0400 (EDT)

Dear Community,

I am simulating a system of ODE using v6. Here are the equations:

eq1 = x'[t] == y[t];
eq2 = y'[t] == 1/x[t] - 1.4 - (4.5 + y[t])*(1 + z[t]^2);
eq3 = z'[t] == 18*z[t] - 0.75*(4.5 + y[t])^2*z[t] - z[t]^3;

It is simulated at x>0. This system at x>0 seems to be globally stable. 
To understand it observe that at large x, y, and z one finds
y' ~ - y*z^2 and z' ~ - z^3. In other words, there is a kind of a 
non-linear "returning force" for y and z, while x follows the dynamics 
of y.

However, when solving it  on Mathematica I sometimes find trajectories 
that counterintuitively  diverge.
Check this for example:

NDSolve[{eq1, eq2, eq3, x[0] == 0.669, y[0] == 0.881,
   z[0] == 0.988}, {x, y, z}, {t, 0, 40}];
       
Plot[{Evaluate[x[t] /. s], Evaluate[y[t] /. s],
  Evaluate[z[t] /. s]}, {t, 0, 45}, PlotRange -> All,
 PlotStyle -> {Red, Green, Blue},
 AxesLabel -> {Style["t", 16], Style["x, y, z", 16]}]

My guess is that this is due to some peculiarity in the numeric method 
used, and the method should be probably changed, or its parameters 
specified. I am however, not experienced in numeric approaches for 
solving ODEs.

Now comes the question:
Can you give me a hint, of
(i)  what may be the reason of such a behavior?
and
(ii) What should I do to avoid such a false divergence?

Thank you, Alexei

-- 
Alexei Boulbitch, Dr., habil.
Senior Scientist

IEE S.A.
ZAE Weiergewan
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Luxembourg

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• Follow-Ups:
  • Re: False divergence of the NDSolve solution: how to avoid
    • From: Daniel Lichtblau <danl@wolfram.com>