       Re: phase-space versus controlling parameter surface

• To: mathgroup at smc.vnet.net
• Subject: [mg91733] Re: phase-space versus controlling parameter surface
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Sun, 7 Sep 2008 05:32:28 -0400 (EDT)
• References: <g9t6r5\$jd9\$1@smc.vnet.net>

```Hi,

your input is a bit corrupted but, and a bundel of lines
is not a surface

ode = {x'[t] == v[t],
v'[t] == -x[t] 3 - k v[t] + B Cos[t]};
init = {x == 1, v == 0};

DuffingLine[kk_?NumericQ, Bb_?NumericQ, {t_, t0_, t1_}] :=
Module[{sol},
sol = NDSolve[Join[ode, init] /. {k -> kk, B -> Bb},
{x[t], v[t]},  {t, t0, t1}];
ParametricPlot3D[{t, x[t], v[t]} /. sol, {t, t0, t1}][]
]

alllines =
Table[DuffingLine[0.1, 1, {t, 0, 4 Pi}], {k, 0.0001, 1, 0.01}, {b,
0.0, 2, 0.05}];

alllines =
Table[DuffingLine[0.1, 1, {t, 0, 4 Pi}], {k, 0.0001, 1, 0.01}, {b,
0.0, 2, 0.05}];

Graphics3D[alllines]

BTW MathGL3d line illumination will give you the illusion of a surface
with

Get["MathGL3d`"]

MVShow3D[Graphics3D[alllines /. _Hue :> Sequence[]],
MVNewScene -> True]

Regards
Jens

Luca Petrone wrote:
> Dear All,
>
> I am interested in plotting a surface of the phase-space versus a controlling parameter, e.g. the B in a Duffing's equation
>
> x'[t] == v[t]v'[t] == - x[t]^3 - k v[t] + B Cos[t]
>
> that is=2C in the space {x[t]=2C v[t]=2C B} for a known k and B varying from Bmin to BmaxI tryed something like :
>
> ParametricPlot3D[ Evaluate[{x[t]=2C v[t]=2C B} /.    NDSolve[{v'[t] == - x[t]^3 - k v[t] + B Cos[t]=2C            x'[t] == v[t]=2C x ===
>  1=2C v == 0}=2C {x=2C v}=2C {t=2C 0=2C 2000}=2C     MaxSteps -> Infinity]  ]=2C {t=2C 1950=2C 1950 + 4 Pi}=2C {B=2C 0.2=2C 0.6}]
>
> but without success.Is there any way to get it ?
> Thank you very much for your help.
>
> Yours=2C
> Luca P.Milano - Italy
> _________________________________________________________________
>

```

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