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

*To*: mathgroup at smc.vnet.net*Subject*: [mg101802] Re: False divergence of the NDSolve solution: how to avoid*From*: Roland Franzius <roland.franzius at uos.de>*Date*: Fri, 17 Jul 2009 05:05:44 -0400 (EDT)*References*: <h3n5uh$2gp$1@smc.vnet.net>

Alexei Boulbitch schrieb: > 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. y is a dummy. You have to consider eigenvalues of the matrix for |x,y,z|->oo. At |(x,z)| -> oo easily you will find by elimination of y x'' == - x' z^2 z' == 18 z - x'^2 z - z^3 If, by chance of rounding errors z < 0 z will run away to -oo. > > 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}]; s=NDSolve[{eq1, eq2, eq3, x[0] == 0.669, y[0] == 0.881, z[0] == 0.988}, {x, y, z}, {t, 0, 40}]; NDSolve::mxst: Maximum number of 10000 steps reached at the point t == \ 27.928120676960916`. >> > 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]}] The Plot explodes at t=27 exponentially > 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? Its always extremely difficult to control solutions of osciallating nonlinear systems for a long time:-( At least in this case use Abs[z[t]] on the right to cross the line immediately if an error of z<0 is occurs or choose a t-step adaptive method if z is approaching 0. -- Roland Franzius

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

**Re: eigenvector centrality**

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

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