Einstein–Friedmann equation, nonlinear dynamics and chaotic behaviours

Abstract

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 = p 0 ρ n ( n ≧ 0 ) . In case n ≠ 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 log κ = - 10 ( 1 / ϕ ) 2 [ 1 + ( ϕ ) 8 ] = - 26.737 , which is in nice agreement with the experiment (−26.730).