       Re: False divergence of the NDSolve solution: how to avoid

• To: mathgroup at smc.vnet.net
• Subject: [mg101819] Re: False divergence of the NDSolve solution: how to avoid
• From: sean <sean_incali at yahoo.com>
• Date: Sat, 18 Jul 2009 04:50:29 -0400 (EDT)
• References: <h3n5uh\$2gp\$1@smc.vnet.net>

```I think this arises from NDSolve not having the solutions beyond the
point where it reaches the max number of steps.  You can increase the
number of steps NDSolve takes to a higher number of you can just set
it to infinity.(obviously is not a good idea if your system is
weird...)

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;

DynamicModule[{},
Manipulate[
s=NDSolve[{eq1,eq2,eq3,x==0.669,y==0.881,z==0.988}, =
{x,y,z},
{t,0,tf}, MaxSteps->Infinity];
Plot[{Evaluate[x[t]/.s], Evaluate[y[t]/.s], Evaluate[z[t]/.s]}, {t,
0,tf}, PlotRange-> All, PlotStyle->{Red,Green,Blue}, AxesLabel->{Style
["t",16], Style["x, y, z",16]}], {tf, 10, 100}]
]

On Jul 16, 5:20 am, Alexei Boulbitch <Alexei.Boulbi... at iee.lu> wrote:
> 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 == 0.669, y == 0.881,
>    z == 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
> 11, rue Edmond Reuter
> L-5326 Contern
> Luxembourg
>
> Phone: +352 2454 2566
> Fax:   +352 2454 3566
>
> Website:www.iee.lu
>
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