The predictable world we observe is a simplified, coarse-grained version of the probabilistic
quantum reality. Yet, the quantum world is by definition linear, while the classical world
with nonlinearity can be chaotic. Thus, the real challenge is how to connect our classical
intuitions with the counter-intuitive quantum theories. There the diagnostic tools come in
as help. In recent years, Out-of-Time-Order-Correlator (OTOC) has emerged as diagnostic
tool for quantum mechanical signatures of chaos.
Previous studies have concluded that OTOCs show false positive of chaos in the
neighbourhood of a local maximum in the potential. Though, it is necessary, but it is not
the only condition. In this thesis, by applying a symmetry-breaking perturbations (linear
and nonlinear), we notice that the exponential behaviour of the OTOCs remains remarkably
resilient even in the absence of a local maximum. Therefore, the critical factor lies not in the
presence of a local maximum, but in the dynamic nature of the density of states in the broken
symmetry regions where he slope of the potential is an extrema. Our examination, including
one dimensional potentials with linear perturbation and two dimensional harmonic potential
with nonlinear perturbations, reveal that the universality of this phenomenon.
The latter part of the thesis focusses on the crucial role of curvature of the billiard boundary
on the particle dynamics. In this study, we introduce two geometrically distinct billiards: a
bean- and a peanut-shaped billiard. These systems incorporate both focusing and defocusing
walls with no neutral segments. Our study reveals a strong correlation between classical and
quantum dynamics.