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–k 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 ¼ p0qn ðn=0Þ. In case n–0; 1, the Einstein–Friedmann equation is not integrable and there may occur chaotic behaviours. The cosmological constant (k) 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 logj ¼ 10ð1=/Þ2½1 þ ð/Þ8 ¼ 26:737, which is in nice agreement with the experiment ( 26.730).