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I've spent a bunch of time perusing Stack Exchange to try to find an answer and found nothing; hopefully this isn't a duplicate.

I have a time-dependent vector field $Phi_t(x,y,z)=(xlambda^t,ylambda^-t,z+t)$, $lambda>1$ arbitrary; for my example, I'm restricting to the region $[0,1]times[0,1]timesmathbbR$. What I would like to do is:

• fix some $lambda>1$;


• generate some number of random initial points;


• compute / store the orbits of these points over some length of time;


• plot these orbits in 3D with animation.

I've currently managed to do all of this except animate the flow lines.

Here's my current code:

seeds = RandomReal[-1, 1, 250, 3]; (* 250 random initial points *)
lam = 1.5; (* [Lambda]>1 fixed *)

func[x_, y_, z_, t_] := x lam^t, y lam^(-t), z + t; (* the vector field itself *)
orbit[k_] := Table[func[seeds[[k]], n], n, 0, 9.75, 0.25]; (* function to compute the orbit for a single initial point *)
orbits = orbit[#] & /@ Range[1, Length[seeds], 1]; (* computes orbits for all initial points *)

Graphics3D[
 Red, Arrowheads[-.01, .01], Arrow[BezierCurve[orbits[[#]]]] & /@ Range[1, Length[seeds], 1]
, PlotRange -> -1, 1, -1, 1, -1, 1, Boxed -> False, Axes -> True, AxesEdge -> -1, -1, -1, -1, -1, -1, ViewPoint -> 2.6056479300835718`, 2.1387445365836095`, 0.29388887642263006`, ViewVertical -> 0.3985587476649791`, 0.332086389794556`, 0.8549090912915488`, ImageSize -> 400]

Here's the output:

This is okay, but what I'd really like is something that can either

1. animate one entire flow line a little at a time, then the next flow line a little at a time, etc. (in the same plot); or


2. animate all flow lines simultaneously, a little at a time.

Can anyone help me with this?

Note: By defining an auxiliary function buildorbits[k_,n_]:=orbits[[k, 1 ;; n]];, I can animate single orbits using a very "hackish-feeling" implementation of ListAnimate. For instance:

; is this really my best option, though?

You can use ParametricPlot3D to get smoother orbits:

SeedRandom[1]
seeds = RandomReal[-1, 1, 50, 3];

Animate[ParametricPlot3D[Evaluate[func[seeds[[#]], t] & /@ Range[Length@seeds]], 
 t, 0, tmax, 
 BoxRatios -> 1, 
 PlotStyle -> Arrowheads[Medium],
 ImageSize->400,
 PlotRange -> -60, 60, -1, 1, -10, 10] /. Line -> Arrow,
 tmax, .1, 10]

Not exactly what was asked, but another way to visualize the flow, based on How can I create a fountain effect?:

DynamicModule[x0, y0, z0, last = 0, lam = 1.5, n = 500, colors, 
 replace,
 last = Clock[Infinity];
 x0, y0, z0 = RandomReal[-1, 1, 3, n];
 colors = RandomColor[n];

 Graphics3D[
 GraphicsComplex[
 Dynamic@
 With[t = Clock[Infinity],
 With[dt = (t - last)/2, With[dl = lam^dt,
 last = t;
 x0 = x0*dl; y0 = y0/dl; z0 = z0 + dt; (* integration of velocity *)
 replace = Pick[Range@n, UnitStep[z0 - 1], 1];
 With[pick = UnitStep[z0 - 1],
 x0[[replace]] = RandomReal[-1, 1, Length@replace];
 y0[[replace]] = RandomReal[-1, 1, Length@replace];
 z0[[replace]] = RandomReal[-1, -1 + dt, Length@replace]
 ];
 Transpose@x0, y0, z0
 ]]],
 Point[Range@n, VertexColors -> colors]
 ], PlotRange -> -2, 2, -2, 2, -1, 1, Axes -> True, 
 AxesLabel -> x, y, z
 ],

 Initialization :> (x0, y0, z0 = RandomReal[-1, 1, 3, n])
 ]

The integration is based on the ODE for $Phi$, which is autonomous and linear and can be done by scalings and translation:
$$d over dt,(x,y,z) = (x log lambda, -y log lambda, 1)$$