Gain of Entrainment in Nonlinear Cascades

Abstract

We consider the gain of entrainment (GOE)--the difference between the average steady-state output under a periodic input and the steady-state output under a constant input with the same mean--for an $n$-stage feedforward cascade of stable first-order filters interleaved with static nonlinearities. The main result is an exact decomposition of GOE as a weighted sum of local Jensen gaps, where each gap quantifies the mean shift generated by a nonlinearity, and each weight is a product of downstream incremental gains divided by linear time constants. We provide a Bregman-divergence interpretation of the decomposition, and a second-order small-amplitude of GOE separating local curvature, fluctuation energy, and differential gains. We demonstrate the theoretical results using a Michaelis-Menten cascade showing that any nonconstant periodic feeding strictly reduces the average terminal product relative to constant feeding with the same mean.

Disclosure

“eering studies, for introducing him to cascade systems in her Industrial Automation course. Her inspiring lectures sparked an early interest in these systems and ultimately helped motivate the ideas developed in this work. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process The authors used ChatGpt for editing and proofreading. After using ChatGpt, the authors reviewed and edited the content as needed and take full responsibility for the con”

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2026-08-23T05:17:12.257536Z

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