Theoretical results regarding two-dimensional ordinary-differential equations (ODEs) with second-degree polynomial right-hand sides are summarized, with a focus on multistability, limit cyclesand limit  cycle bifurcations  . 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 ( obtained by generating noisy time-series using the Gillespie algorithm  ) 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.
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.
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a focus on multistability, limit cyclesand
an emphasis on
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cycles, limit
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and multistability
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obtained by generating
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using
generating by
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, and the underlying probability distributions obtained by solving the chemical master equation (CME)