Equilibrium Locus of The Flow on Circular Networks of Cells

TL;DR

This paper provides a geometric analysis of the equilibrium flow locus in circular cell networks, revealing its manifold structure, and introduces a homotopy method for efficient equilibrium computation across parameter variations.

Contribution

It characterizes the equilibrium locus as a smooth manifold with corners and develops a homotopy approach for computing equilibrium points efficiently.

Findings

Equilibrium locus forms a smooth manifold with corners.

Homotopy method enables efficient equilibrium computation.

System can be driven to any reachable equilibrium asymptotically.

Abstract

We perform a geometric study of the equilibrium locus of the flow that models the diffusion process over a circular network of cells. We prove that when considering the set of all possible values of the parameters, the equilibrium locus is a smooth manifold with corners, while for a given value of the parameters, it is an embedded smooth and connected curve. For different values of the parameters, the curves are all isomorphic. Moreover, we show how to build a homotopy between different curves obtained for different values of the parameter set. This procedure allows the efficient computation of the equilibrium point for each value of some first integral of the system. This point would have been otherwise difficult to be computed for higher dimensions. We illustrate this construction by some numerical experiments. Eventually, we show that when considering the parameters as inputs, one can easily bring the system asymptotically to any equilibrium point in the reachable set, which we also easily characterize.