  A Fubini instanton is a bounce solution which describes the decay of a vacuum
state located at the top of the tachyonic potential {\it via} the tunneling
without a barrier. We investigate various types of Fubini instantons of a
self-gravitating scalar field under a tachyonic quartic potential. With gravity
taken into account, we show there exist various types of unexpected solutions
including oscillating bounce solutions. We present numerically oscillating
Fubini bounce solutions in anti-de Sitter and de Sitter spaces. We construct
the parametric phase diagrams of the solutions, which is the extension of our
previous work. Of particular significance is that there always exist solutions
in all parameter spaces in anti-de Sitter space. The regions are divided
depending on the number of oscillations. On the other hand, de Sitter space
allows solutions with codimension-one in parameter spaces. We numerically
evaluate semiclassical exponents which give the finite tunneling probabilities.
