Nonlinear dynamics in the relativistic field equation

We have investigated relativistic equations and chaotic behaviors of the gravitational field with the use of general relativity and nonlinear dynamics. The space component of the Friedmann equation shows chaotic behaviors in case of the inflation (h=G˙/G>0) and open (ζ = −1) universe. In other cases (h ≦ 0, ζ = 0 and ζ = +1), there occurs non-chaotic behaviors. We have shown the following properties of the Friedmann chaos: (1) the sensitive dependence of solutions on the initial values (x0andx˙0) and the parameters (a, b, c and d  ); (2) the self-similarity of solutions in the x–x˙ plane and the x–ρ plane. We carried out the numerical calculations with the use of the microsoft EXCEL. The self-similarity and the hierarchy structure of the universe have been also discussed on the basis of E-infinity theory.