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Equilibrium locus of the flow on circular networks of cells

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  • 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.

    Mathematics Subject Classification: Primary: 57R99, 37C25; Secondary: 34H05.

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  • Figure 1.  The homotopy path generated with the following parameters ${\mathit{\boldsymbol{\lambda}}}_1 = (1, 1, 1)$, ${\mathit{\boldsymbol{\lambda}}}_2 = ( 1.39328599, 8.30098374, 3.98355604)$ and $s = 1$. The first equilibrium point $E_0 = (1/3, 1/3, 1/3)$ is represented by a green ball, while the final equilibrium point $E_1 = ( 0.53112814, 0.1203633, 0.34850856)$ is rendered as a red triangle. Of course all over the path the constraint $e_1 + e_2 + e_3 = s$ holds.

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