  Theoretical results regarding two-dimensional ordinary-differential equations
(ODEs) with second-degree polynomial right-hand sides are summarized, with an
emphasis on limit cycles, limit cycle bifurcations and multistability. The
results are then used for construction of two reaction systems, which are at
the deterministic level described by two-dimensional third-degree kinetic ODEs.
The first system displays a homoclinic bifurcation, and a coexistence of a
stable critical point and a stable limit cycle in the phase plane. The second
system displays a multiple limit cycle bifurcation, and a coexistence of two
stable limit cycles. The deterministic solutions (obtained by solving the
kinetic ODEs) and stochastic solutions (noisy time-series generating by the
Gillespie algorithm, and the underlying probability distributions obtained by
solving the chemical master equation (CME)) of the constructed systems are
compared, and the observed differences highlighted. The constructed systems are
proposed as test problems for statistical methods, which are designed to detect
and classify properties of given noisy time-series arising from biological
applications.
