  The theory of non-gravitational interaction between a pressure-less dark
matter (DM) and dark energy (DE) is a phenomenologically rich cosmological
domain which has received magnificent attention in the community. In the
present article we have considered some interacting scenarios with some novel
features: the interaction functions do not depend on the external parameters of
the universe, rather, they depend on the intrinsic nature of the dark
components; the assumption of unidirectional flow of energy between DM and DE
has been extended by allowing the possibility of bidirectional energy flow
characterized by some sign shifting interaction functions; and the DE equation
of state has been considered to be either constant or dynamical in nature.
These altogether add new ingredients in this context, and, we performed the
phase space analysis of each interacting scenario in order to understand their
global behaviour. According to the existing records in the literature, this
combined picture has not been reported elsewhere. From the analyses, we
observed that the DE equation of state as well as the coupling parameter(s) of
the interaction models can significantly affect the nature of the critical
points. It has been found that within these proposed sign shifting interacting
scenarios, it is possible to obtain stable late time attractors which may act
as global attractors corresponding to an accelerating expansion of the
universe. The overall outcomes of this study clearly highlight that the sign
shifting interaction functions are quite appealing in the context of
cosmological dynamics and they deserve further attention.
