  We employ a typical genetic circuit model to explore how noise can influence
the dynamic structure. With the increase of a key interactive parameter, the
model will deterministically go through two bifurcations and three dynamic
structure regions. We find that a new quasi-periodic component, which is not
allowed by deterministic dynamics, will be generated by noise inducing in the
first two regions, and this quasi-period will be more and more stable along
with the noise increasing. Especially, in the second region, the quasi-period
will compete with a stable limit cycle and perform a new transient rhythm.
Furthermore, we ascertain the entropy production rate (EPR) and the heat
dissipation rate (HDR), and discover a minimal value with elucidating
theoretically. In the end, we unveil the mechanism of the quasi-periods
forming, and show a practical biological instance. We expect that this work is
helpful to solve some biological or ecological problems, such as genetic origin
of periodical cicadas and population dynamics with fluctuation.
