Scaling law in Saddle-node bifurcations for one-dimensional maps: A complex variable approach

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, tau, scales according to an inverse square-root power law, tau similar to (mu-mu (c) )(-1/2), as the bifurcation parameter mu, is driven further away from its critical value, mu (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.

Pozzolanic activity of oil-refining catalyst: evaluation by electron and atomic microscopy

The reuse of waste fluid catalytic cracking (FCC) catalyst as partial surrogate for cement can reduce the environmental impact of both the oil-refinery and cement production industries [1,2]. FCC catalysts can be considered as pozzolanic materials since in the presence of water they tend to chemically react with calcium hydroxide to produce compounds possessing cementitious properties [3,4]. In addition, partial replacement of cement with FCC catalysts can enhance the performance of pastes and mortars, namely by improving their compressive strength [5,6]. In the present work the reaction of waste FCC catalyst with Ca(OH)2 has been investigated after a curing time of 28 days by scanning electron microscopy (SEM) with electron backscattered signal (BSE) combined with X-ray energy dispersive spectroscopy (EDS) carried out with a JEOL JSM 7001F instrument operated at 15 kV coupled to an INCA pentaFetx3 Oxford spectrometer. The polished cross-sections of FCC particles embedded in resin have also been evaluated by atomic force microscopy (AFM) in contact mode (CM) using a NanoSurf EasyScan 2 instrument. The SEM/EDS results revealed that an inward migration of Ca occurred during the reaction. A weaker outward migration of Si and Al was also apparent (Fig. 1). The migration of Ca was not homogeneous and tended to follow high-diffusivity paths within the porous waste FCC catalyst particles. The present study suggests that the porosity of waste FCC catalysts is key for the migration/reaction of Ca from the surrounding matrix, playing an important role in the pozzolanic activity of the system. The topography images and surface roughness parameters obtained by atomic force microscopy can be used to infer the local porosity in waste FCC catalyst particles (Fig. 2).