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This work is the extension of author’s research, where the modified theory of induced gravity (MTIG) is proposed. In the framework of the MTIG, the mechanism of phase transitions and the description of multiphase behavior of the cosmological scenario are proposed. The theory describes two systems (stages): Einstein (ES) and “restructuring” (RS). This process resembles the phenomenon of a phase transition, where different phases (Einstein’s gravitational systems, but with different constants) pass into each other. The hypothesis that such transitions are random and lead to stochastic behavior of cosmological parameters is considered. In our model, effective gravitational and cosmological “constants” arise, which are defined by the “mean square” of the scalar fields. These parameters can be compared with observations related to the phenomenon of dark energy. The aim of the work is to solve equations of MTIG for the case of a quadratic potential and compare them with observational cosmology data. The interaction of fundamental scalar fields and matter in the form of an ideal fluid is introduced and investigated. For the case of Friedmann-Robertson-Walker space-time, numerical solutions of nonlinear MTIG equations are obtained using the qualitative theory of dynamical systems and mathematical computer programs. For the case of a linear potential, examples joining of solutions, the ES and RS stages, of the evolution of the cosmological model are given. It is shown that the values of such parameters as “Hubble parameter” and gravitational and cosmological “constants” in the RS stage contain solutions oscillating near monotonically developing averages or have stochastic behavior due to random transitions to different stages (RS or ES). Such a stochastic behavior might be at the origin of the tension between CMB measurements of the value of the Hubble parameter today and its local measurements.

Oscillating Cosmological Solutions in the Modified Theory of Induced Gravity