# 3D Phase Portrait of a System of differential equations

Here I have a system as follows:

$$\frac{dx}{dt}=a(by-x); \frac{dy}{dt}=rx-xz; \frac{dz}{dt}=(xy)^n-bz$$

Here $$x, y$$ and $$z$$ are positive real variables. All the parameters $$a, r$$ and $$b$$ are all positive real numbers and $$n$$ is a natural number.

How can I a make the phase portrait 3D of the system by varying the natural number $$n$$ from say $$1$$ to $$100$$?

3D phase portrait can be built by analogy with 2D using SliceVectorPlot3D[]

p[s1_, a1_, b1_, r1_, n1_] :=
Block[{s = s1, a = a1, b = b1, r = r1, n = n1},
v3D = {a*(b*y - x), r*x - x*z, (x*y)^n - b*z};
SliceVectorPlot3D[v3D/Norm[v3D],
s, {x, -10, 10}, {y, -10, 10}, {z, -10, 10},
PlotTheme -> "Scientific",
VectorColorFunction -> "BlueGreenYellow", VectorScale -> Small,
VectorPoints -> Fine, PlotLabel -> Row[{"n=", n}],
AxesLabel -> Automatic]]

Table[p["XStackedPlanes", 2, 1, 4, n], {n, 1, 100, 33}]

Table[p["YStackedPlanes", 2, 1, 4, n], {n, 1, 100, 33}]

Table[p["ZStackedPlanes", 2, 1, 4, n], {n, 1, 100, 33}] a = 1; b = 1; r = 1;

pf = ParametricNDSolveValue[{x'[t] == a*(b*y[t] - x[t]),
y'[t] == r*x[t] - x[t]*z[t], z'[t] == (x[t]*y[t])^n - b*z[t],
x == x0, y == x0, z == x0}, {x[t], y[t], z[t]}, {t,
20}, {x0, n}];

Manipulate[ParametricPlot3D[pf[x0, n], {t, 0, 20}], {{x0, -1, "x0"}, -2, 2},
{{n, 1, "n"}, 1, 100}]