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

Test Models for Statistical Inference: Two-Dimensional Reaction Systems Displaying Limit Cycle Bifurcations and Bistability