Description:
<jats:p>We investigate stationary and nonstationary probability densities for a weakly forced nonlinear physical or chemical system that displays self-oscillations in the absence of forcing. The period and amplitude of forcing are taken as adjustable constraints. We consider a homogeneous reaction system described by a master equation. Our method of solution is based on the Wentzel–Kramers–Brillouin (WKB) expansion of the probability density with the system size as the expansion parameter. The first term in this expansion is the stochastic potential (eikonal). In the absence of forcing, the probability density is logarithmically flat on the limit cycle. With periodic forcing, the phenomenon of phase locking can occur whereby a stable cycle, which is close to the unforced cycle, adopts a constant relative phase to the forcing. A saddle cycle also exists and has a different constant relative phase. For such phase-locked solutions, the distribution over the relative phases is peaked on the stable cycle and exhibits a logarithmically flat region (a plateau) that originates on the saddle cycle. This plateau is due to a nonzero relative phase slippage: large fluctuations from the stable cycle over the saddle cycle are overwhelmingly more probable in a certain relative phase direction, which depends upon the location of the parameters within an entrainment region. This distribution of relative phases is logarithmically equivalent to that of a Brownian particle in a periodic potential with a constant external force in the strong damping and weak noise limits. For parameter values outside of an entrainment region (for which a quasiperiodic solution exists), the distribution in relative phase is logarithmically flat. For this regime, we investigate the evolution of an initially localized density and show that the width grows proportionally with the square root of time. The proportionality factor depends upon both the position (phase) on the cross section of the peak of the density and the distance in parameter space from the boundary of the entrainment region. For parameter values that approach the boundary of an entrainment region, this proportionality factor tends to infinity. We also determine an expression for the first order correction to the stochastic potential for both entrained and quasiperiodic solutions. A thermodynamic interpretation of these results is made possible by the equality of the stochastic potential with an excess work function.</jats:p>