Scaling law in saddle-node bifurcations for one-dimensional maps: a complex variable approach |
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Authors: | Jorge Duarte Cristina Januário Nuno Martins Josep Sardanyés |
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Institution: | 1.Department of Mathematics,ISEL—Engineering Superior Institute of Lisbon,Lisbon,Portugal;2.Centro de Análise Matemática, Geometria e Sistemas Dinamicos, Department of Mathematics,Instituto Superior Técnico,Lisbon,Portugal;3.Instituto de Biología Molecular y Celular de Plantas,Consejo Superior de Investigaciones Científicas-UPV,Valencia,Spain |
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Abstract: | The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due
to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations,
where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed
transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium,
found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled
this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, τ, scales according to an inverse square-root power law, τ∼(μ−μ
c
)−1/2, as the bifurcation parameter μ, is driven further away from its critical value, μ
c
. In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional
maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species
ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising
due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement
with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling
laws of one-dimensional discrete dynamical systems with saddle-node bifurcations. |
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Keywords: | |
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