Asymptotic Limit of a Singularly Perturbed Stationary Diffusion Equation: The Case of a Limit Cycle

Details Asymptotic Limit of a Singularly Perturbed Stationary Diffusion Equation: The Case of a Limit Cycle Asymptotic Limit of a Singularly Perturbed Stationary Diffusion Equation: The Case of a Limit Cycle

2010-11-17

arxiv.org/abs/1011.4049v1

Physics

Mathematical Physics

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Scientific paper

A limit cycle for a nonlinear ordinary differential equation has a sustained, stationary oscillation in time; Any non-trivial stationary stochastic process also exhibits stationary oscillations in time, though with randomness and a stationary probability density. A reconciliation of these two views of oscillatory dynamics has been elusive, although it becomes increasingly important in the biochemical modeling of cellular dynamics, where stochatic models based on the chemical master equation and the deterministic model based on the Law of Mass Action are routinely compared. Using a singularly perturbed stationary diffusion equation as a model for the chemical master equation with sufficiently large volume, $\epsilon \leftrightarrow 1/V$, we show that its stationary solution $u(\vx)$ exhibits a clear separation of the exponentially and algebraic small contributions: $u(\vx)=C_{\epsilon}(\vx) e^{-\phi(\vx)/\epsilon}$, in which $\phi(x)\ge 0$ and $=0$ on the entire stable limit cycle. On the limit cycle, $C_0(\vx)$ is inversely proportional to the velocity of the non-uniform periodic oscillation preserving the ergodicity. For an unstable limit cycle, our result suggests the existence of a "boundary loop" in systems of dimension two or higher.

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