Period Homeostasis Near Hopf Bifurcation

Abstract

Homeostasis is a biological phenomenon in which a function of the state of the system remains approximately constant as an input parameter varies. Recent mathematical work has developed a framework for studying homeostasis using methods from singularity theory. In this framework, the requirement that the function of the state remain approximately constant is replaced by the condition that the derivative of the function with respect to the input parameter vanishes at an isolated point. This zero-derivative condition is called infinitesimal homeostasis. Much of the existing theory concerns steady states. In this paper, we develop an analogous theory for oscillatory systems in which the homeostatic quantity is the period of a stable periodic solution. We refer to this behavior as period homeostasis and to the corresponding zero-derivative condition as infinitesimal period homeostasis. Experimental studies in cyanobacteria and cultured human cells suggest that circadian rhythms can disappear through Hopf bifurcation as temperature decreases to a critical value, with the oscillation amplitude tending to zero. Motivated by this evidence, we focus on stable periodic solutions arising from nondegenerate Hopf bifurcation. We show that bifurcation theory and singularity theory can be used to find period homeostasis in parameterized ODE models. Our results connect the onset of oscillations with homeostasis of the period and yield a geometric description of parameter space that locally organizes the qualitative behavior of solutions of the ODE.