# Dynamical System

I'm Trying to solve dynamical system in friedmann universe. But, I have problems with initial conditions to $$\Omega_{\Lambda}$$ and $$\Omega_{k}$$. The code that i'm implementing is

ode1[x0_, y0_] := NDSolve[{x'[n]] == 2 (1 + (3 \[Gamma] - 2)/2 (1 - y[n]) - (3 \[Gamma])/2 x[n]) x[n],  y'[n] ==  2 ((3 \[Gamma] - 2)/2 (1 - y[n]) - (3 \[Gamma])/ 2 x[n]) y[n], x0] == x0, y[0] == y0}, {x[n], y[n]}, {n, -0.7, 0.54}, MaxSteps -> Infinity]

Where $$\gamma=1$$, $$n=\ln a$$, $$x=\Omega_{\Lambda}$$ and .$$y=\Omega_{k}$$ For example, I try

sol[1] = ode1[-0.5, 1.5]; sol[2] = ode1[1, 0]; sol[3] = ode1[0, 1]; sol[4] = ode1[0.8, 0.2]; sol[5] = ode1[-0.3, 1.3]; sol[6] = ode1[1.3,-0.3]; sol[7] = ode1[1.2, -0.2]; sol[8] = ode1[0.2, 0.8];

ParametricPlot[Evaluate[Table[{x[n], y[n]} /. sol[i], {i,8}]], {n, -0.7, 0.54}, PlotStyle -> {Thick}, PlotRange -> {{-0.5, 1.5}, {-0.5, 1}}, PlotPoints -> 100, AxesLabel -> {"x", "y"}, Axes -> True]

The plot shows anything.

Use code

ode1[x0_, y0_] :=
NDSolve[{x'[n] ==
2 (1 + (3 \[Gamma] - 2)/2 (1 - y[n]) - (3 \[Gamma])/2 x[n]) x[n],
y'[n] ==
2 ((3 \[Gamma] - 2)/2 (1 - y[n]) - (3 \[Gamma])/2 x[n]) y[n],
x[0] == x0, y[0] == y0} /. \[Gamma] -> 1, {x, y}, {n, -0.7, 0.54},
MaxSteps -> Infinity]
sol[1] = ode1[-0.5, 1.5]; sol[2] = ode1[1, 0]; sol[3] = ode1[0, 1];
sol[4] = ode1[0.8, 0.2]; sol[5] = ode1[-0.3, 1.3];
sol[6] = ode1[1.3, -0.3]; sol[7] = ode1[1.2, -0.2];
sol[8] = ode1[0.2, 0.8];

ParametricPlot[
Evaluate[Table[{x[n], y[n]} /. sol[i], {i, 8}]], {n, -0.7, 0.54},
PlotStyle -> {Thick}, PlotRange -> All, PlotPoints -> 100,
AxesLabel -> {"x", "y"}, Axes -> True]

Phase space of the system

reg = ImplicitRegion[
1 - x - y < 0 && -.5 <= x <= 1.5 && -1 <= y <= 1, {x, y}]; reg1 =
ImplicitRegion[
1 - x - y > 0 && -.5 <= x <= 1.5 && -1 <= y <= 1, {x, y}];

sp = StreamPlot[{2 (1 + (3 \[Gamma] - 2)/2 (1 - y[n]) - (3 \[Gamma])/
2 x[n]) x[n],
2 ((3 \[Gamma] - 2)/2 (1 - y[n]) - (3 \[Gamma])/2 x[n]) y[
n]} /. \[Gamma] -> 1, {x[n], y[n]} \[Element] reg1,
FrameLabel -> {Subscript[\[CapitalOmega], \[CapitalLambda]],
Subscript[\[CapitalOmega], K]}, StreamPoints -> Fine]

rp = Show[RegionPlot[reg, BoundaryStyle -> None],
Graphics[{Black, Circle[{0, 0}, 2],
Inset[\[CapitalOmega] < 0, {1, 0.5}]}]]
Show[sp, rp]

sp1 = StreamPlot[{2 (1 + (3 \[Gamma] - 2)/2 (1 - y[n]) - (3 \[Gamma])/
2 x[n]) x[n], 1 - x[n] - y[n]} /. \[Gamma] -> 1, {x[n], -.5,
1.5}, {y[n], 0., 2},
FrameLabel -> {Subscript[\[CapitalOmega], \[CapitalLambda]], \
\[CapitalOmega]}, StreamPoints -> Fine]

• Thanks for your answer, the problem is that I want to reproduce the figure 2 [link] (arxiv.org/abs/physics/0108066v1) in page 13. Oct 24, 2019 at 20:18
• @Fisjog Show how to do it? Oct 24, 2019 at 21:11
• thanks a lot! @alex-trounev Oct 24, 2019 at 22:34