Possible evolution of a bouncing universe in cosmological models with
  non-minimally coupled scalar fields
  We explore dynamics of cosmological models with bounce solutions evolving on
a spatially flat Friedmann-Lemaitre-Robertson-Walker background. We consider
cosmological models that contain the Hilbert-Einstein curvature term, the
induced gravity term with a negative coupled constant, and even polynomial
potentials of the scalar field. Bounce solutions with non-monotonic Hubble
parameters have been obtained and analyzed. The case when the scalar field has
the conformal coupling and the Higgs-like potential with an opposite sign is
studied in detail. In this model the evolution of the Hubble parameter of the
bounce solution essentially depends on the sign of the cosmological constant.
