  In this study, we delve into a novel dynamic system inspired by Montgomery's
pair correlation conjecture in number theory. The dynamic system is intricately
designed to emulate the behavior of the nontrivial zeros of the Riemann zeta
function. Our exploration encompasses bifurcation analysis and Lyapunov
exponents to scrutinize the system's behavior and stability, offering insights
into both small and large initial conditions. Our efforts extend to unveiling
the probability distribution characterizing the dynamics for varying initial
conditions. The dynamic system unfolds intricate behaviors, displaying
sensitivity to initial conditions and revealing complex bifurcation patterns.
Small deviations in the initial conditions unveil significantly different
trajectories, reminiscent of chaotic systems. Lyapunov exponents become our
lens into understanding stability and chaos within the system. A comparative
analysis between the dynamic system's approximate solutions and the actual
nontrivial zeros of the Riemann zeta function enhances our comprehension of
model accuracy and its potential implications for number theory.
  This research illuminates the versatility of dynamic systems as analogs for
studying complex mathematical phenomena. It provides fresh perspectives on the
pair correlation conjecture, establishing connections with nonlinear dynamics
and chaos theory. Notably, we delve into the boundedness of solutions for both
small and large initial conditions, unraveling the distinctive probability
distribution governing the dynamics in each scenario. Furthermore, we introduce
an in-depth analysis of the entropy of our dynamic system for both small and
large initial conditions. The entropy study enhances our understanding of the
predictability and stability of the system, shedding light on its behavior in
different parameter regimes.
