Relativistic field equations and nonlinear dynamics

We have studied the gravitational field equations on the basis of general relativity and nonlinear dynamics. By integrating the space component of the Einstein–Friedmann equation, we have derived the Higgs-type W-shape potential [U = (1/4)(λ − κp)x2 − (1/8)ζx4], and classified the solutions in the Einstein–Friedmann equation. We have shown that there occurs chaotic behaviours in case the following conditions are satisfied; (i) the expanding ratio , (ii) the curvature ζ = −1, and (iii) the cosmological constant λ < κp. The curvature (ζ) and the cosmological constant (λ) turn out to be important to discuss the chaotic behaviours in the gravitational field. We have also discussed the model and the dimension of the universe with the use of E-infinity theory.