Carryover of a Saddle-Node Bifurcation After Transforming a Parameter into a Variable

Carlos Contreras,Gerda De Vries,Gustavo Carrero

doi:10.1007/s10884-021-10097-z

Abstract

In this article, we introduce and study the carryover of a saddle-node bifurcation, a concept that describes how a saddle-node bifurcation of a dynamical system is carried over into an extended dynamical system obtained by transforming one of the parameters of the original system into a variable. We show that additional transversality and singularity conditions are needed to guarantee the carryover of a saddle-node bifurcation and provide a graphical methodology with a two-parameter bifurcation diagram to verify that such conditions are met. The results are applied to a gene activation model when the parameter describing the signal for activation is transformed into a variable, and to a cell cycle regulatory model when the parameter describing the cell mass is transformed into a variable. In both cases, we show that a saddle-node bifurcation carryover takes place.

Highlights

Consider a dynamical system ż = f (z; μ1, μ2 ), (1)n where z ∈ R and one of the parameters μ1, μ2 ∈ R drives a saddle-node bifurcation as it crosses a critical value
We studied the carryover of a saddle-node bifurcation; we take a system where a saddle-node bifurcation is present, extend it by transforming a parameter into a variable, and find the conditions that guarantee that a saddle-node bifurcation is still present
Our focus was restricted to the case where there are two parameters of interest and the vector field of the new equation does not depend on the variables already present in the original system

Summary

Introduction

N where z ∈ R and one of the parameters μ1 , μ2 ∈ R drives a saddle-node bifurcation as it crosses a critical value. 2. the tangent line to g(μ1 ; μ2 ) = 0 at (μ∗1 , μ∗2 ) is not parallel to the μ1 -axis, respectively This proposition says that in order to find the saddle-node bifurcation points for the extended system, we plot the two-parameter bifurcation diagram of the original system, superimpose the nullclines of the new equation in the extended system, and look for transverse intersections between the saddle-node bifurcation curve and the nullclines. This case is interesting because at (μ1 , μ2 ) = (1, 0), the transversality condition is not satisfied for the original system with respect to μ2 , i.e., Dμ2 f (0; 1, 0) = 0. Since the tangent line of g(μ; λ) is not parallel to the μ-axis at any of the intersection points, we conclude, by Proposition 4, that there is a saddle-node carryover with λ as the bifurcation parameter.

Gene activation

Cell cycle start

Discussion