Description:
<jats:p>We investigate fluctuational properties near a limit cycle for a homogeneous chemical reaction system using a master equation approach. Our method of solution is based on the WKB expansion of the probability density in the inverse of the system size. The first two terms of this series give the leading asymptotic behavior. The eikonal equation for the leading order term has the structure of a Hamilton–Jacobi equation. Its solutions are determined by the associated characteristic equations, which also give fluctuational trajectories. In the vicinity of the limit cycle, the characteristic equations are the variational equations for the associated Hamiltonian system, and its solutions may be expressed as linear combinations of Floquet eigenfunctions. These eigenfunctions fall into three sets according to whether the real part of the characteristic exponent is less than, equal to, or greater than zero. Eigenfunctions corresponding to characteristic exponents with the real part less than zero span the stable subspace; they describe exponentially fast relaxation to the limit cycle in the deterministic system. Eigenfunctions corresponding to characteristic exponents with the real part greater than zero span the unstable subspace; they describe most probable fluctuational trajectories away from the limit cycle. The remaining two eigenfunctions are associated with a double zero characteristic exponent and span the center subspace. One eigenfunction is due to the translational invariance of the periodic orbit and the other (generalized eigenfunction) to the one-parameter family of periodic orbits in Hamiltonian systems. The generalized eigenfunction describes diffusion along the limit cycle of a probability distribution front for which the gradient is perpendicular to the isochrons of the limit cycle. We develop an explicit formula for the time evolution of an initially localized density based on all these eigenfunctions. We show that relaxation of the density is exponentially fast in directions transverse to the limit cycle and slow (linear in time) along the limit cycle. In addition, we give a simple formula for the probability diffusion coefficient that characterizes dephasing along the orbit. A formula for the stationary distribution is ob- tained from the nonstationary density by removing the center and stable subspace. For this dens- ity, we give a new derivation of an identity: The marginal probability density along the limit cycle equals a constant times the inverse of the speed on the cycle, which is the invariant density along the limit cycle of the deterministic system.</jats:p>