Einstein–Friedmann equation, nonlinear dynamics and chaotic behaviours

We have studied the Einstein–Friedmann equation [Case 1] on the basis of the bifurcation theory and shown that the chaotic behaviours in the Einstein–Friedmann equation [Case 1] are reduced to the pitchfork bifurcation and the homoclinic bifurcation. We have obtained the following results:(i) “The chaos region diagram” (the p–λ plane) in the Einstein–Friedmann equation [Case 1].(ii) “The chaos inducing chart” of the homoclinic orbital systems in the unforced differential equations.We have discussed the non-integrable conditions in the Einstein–Friedmann equation and proposed the chaotic model: p=p0ρn(n≧0). In case n≠0,1n≠0,1, the Einstein–Friedmann equation is not integrable and there may occur chaotic behaviours. The cosmological constant (λ) turns out to play important roles for the non-integrable condition in the Einstein–Friedmann equation and also for the pitchfork bifurcation and the homoclinic bifurcation in the relativistic field equation. With the use of the E  -infinity theory, we have also discussed the physical quantities in the gravitational field equations, and obtained the formula logκ=-10(1/ϕ)2[1+(ϕ)8]=-26.737logκ=-10(1/ϕ)2[1+(ϕ)8]=-26.737, which is in nice agreement with the experiment (−26.730).