  $Circuit~ Complexity$, a well known computational technique has recently
become the backbone of the physics community to probe the chaotic behaviour and
random quantum fluctuations of quantum fields. This paper is devoted to the
study of out-of-equilibrium aspects and quantum chaos appearing in the universe
from the paradigm of two well known bouncing cosmological solutions viz.
$Cosine~ hyperbolic$ and $Exponential$ models of scale factors. Besides
$circuit~ complexity$, we use the $Out-of-Time~ Ordered~ correlation~ (OTOC)$
functions for probing the random behaviour of the universe both at early and
the late times. In particular, we use the techniques of well known two-mode
squeezed state formalism in cosmological perturbation theory as a key
ingredient for the purpose of our computation. To give an appropriate
theoretical interpretation that is consistent with the observational
perspective we use the scale factor and the number of e-foldings as a dynamical
variable instead of conformal time for this computation. From this study, we
found that the period of post bounce is the most interesting one. Though it may
not be immediately visible, but an exponential rise can be seen in the
$complexity$ once the post bounce feature is extrapolated to the present time
scales. We also find within the very small acceptable error range a universal
connecting relation between Complexity computed from two different kinds of
cost functionals-$linearly~ weighted$ and $geodesic~ weighted$ with the OTOC.
Furthermore, from the $complexity$ computation obtained from both the
cosmological models and also using the well known MSS bound on quantum Lyapunov
exponent, $\lambda\leq 2\pi/\beta$ for the saturation of chaos, we estimate the
lower bound on the equilibrium temperature of our universe at late time scale.
Finally, we provide a rough estimation of the scrambling time in terms of the
conformal time.
