Scaling law in saddle-node bifurcations for one-dimensional maps: a complex variable approach

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 onedimensional discrete dynamical systems with saddlenode bifurcations.

Dynamical mechanism behind ghosts unveiled in a map complexification

“…The same scaling exponent is found in the supercritical Pitchfork bifurcation in flows [28], and in the Pitchfork and the period-doubling bifurcation in maps [29]. For the s-n, transients scale as τ ∼ |µ − µ c | −1/2 for flows and maps [2,14,30]. Remarkably, this scaling law was found experimentally in an electronic circuit modelling Duffing's oscillator [5].…”

Discontinuity-induced bifurcation cascades in flows and maps with application to models of the yeast cell cycle

“…for all i 0. The argument that led us to conclude the existence of the sequence µ + n in the case of the grazing bifurcation of the two-cycle periodic orbit carries over essentially verbatim to the present case, with the di↵erence that (following [13] and Duarte et al [20], but see also 4.2), µ + n ⇠ n 2 for su ciently large n. With µ = k 0 2,fold k 0 2 , the border collision bifurcation sequence corresponds to the sequence of Type I grazing bifurcations of the corresponding periodic trajectories in Sec. 2.4.…”

“…(103-104) for µ = 0.0034 corresponding to a fixed point of the composition ( R L ) n R with n = 40. For small µ, the number of iterates near the ghost of the nonhyperbolic period-two orbit grows as 1/ p µ[13,20].…”

Bifurcation Gaps in Asymmetric and High-Dimensional Hypercycles

“…The biological implications of these gaps could be relevant within the framework of so-called delayed transitions [Sardanyés & Solé, 2005;Sardanyés & Solé, 2006]. It is known that after a s-n bifurcation a saddle remnant (also named ghost) appears in the phase space [Strogatz, 2000;Fontich & Sardanyés, 2008;,Duarte et al, 2012;Strogatz & Westervelt, 1989;Sardanyés & Solé, 2005]. In this phenomenon, the flow takes a long-lasting excursion just after the s-n bifurcation before going to the only asymptotically globally stable attractor, which involves extinction.…”