Measuring and Controlling the Chaotic Motion of Profits

The study of economic systems has generated deep interest in exploring the complexity of chaotic motions in economy. Due to important developments in nonlinear dynamics, the last two decades have witnessed strong revival of interest in nonlinear endogenous business chaotic models. The inability to predict the behavior of dynamical systems in the presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we study a specific economic model from the literature. More precisely, a system of three ordinary differential equations gather the variables of profits, reinvestments and financial flow of borrowings in the structure of a firm. Firstly, using results of symbolic dynamics, we characterize the topological entropy and the parameter space ordering of kneading sequences, associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. Finally, we show that complicated behavior arising from the chaotic firm model can be controlled without changing its original properties and the dynamics can be turned into the desired attracting time periodic motion (a stable steady state or into a regular cycle). The orbit stabilization is illustrated by the application of a feedback control technique initially developed by Romeiras et al. [1992]. This work provides another illustration of how our understanding of economic models can be enhanced by the theoretical and numerical investigation of nonlinear dynamical systems modeled by ordinary differential equations.

Strong and weak Allee effects and chaotic dynamics in Richards' growths

In this paper we define and investigate generalized Richards' growth models with strong and weak Allee effects and no Allee effect. We prove the transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, depending on the implicit conditions, which involve the several parameters considered in the models. New classes of functions describing the existence or not of Allee effect are introduced, a new dynamical approach to Richards' populational growth equation is established. These families of generalized Richards' functions are proportional to the right hand side of the generalized Richards' growth models proposed. Subclasses of strong and weak Allee functions and functions with no Allee effect are characterized. The study of their bifurcation structure is presented in detail, this analysis is done based on the configurations of bifurcation curves and symbolic dynamics techniques. Generically, the dynamics of these functions are classified in the following types: extinction, semi-stability, stability, period doubling, chaos, chaotic semistability and essential extinction. We obtain conditions on the parameter plane for the existence of a weak Allee effect region related to the appearance of cusp points. To support our results, we present fold and flip bifurcations curves and numerical simulations of several bifurcation diagrams.

Dosimetric effect by shallow air cavities in high energy electron beams

This study evaluates the dosimetric impact caused by an air cavity located at 2 mm depth from the top surface in a PMMA phantom irradiated by electron beams produced by a Siemens Primus linear accelerator. A systematic evaluation of the effect related to the cavity area and thickness as well as to the electron beam energy was performed by using Monte Carlo simulations (EGSnrc code), Pencil Beam algorithm and Gafchromic EBT2 films. A home-PMMA phantom with the same geometry as the simulated one was specifically constructed for the measurements. Our results indicate that the presence of the cavity causes an increase (up to 70%) of the dose maximum value as well as a shift forward of the position of the depthedose curve, compared to the homogeneous one. Pronounced dose discontinuities in the regions close to the lateral cavity edges are observed. The shape and magnitude of these discontinuities change with the dimension of the cavity. It is also found that the cavity effect is more pronounced (6%) for the 12 MeV electron beam and the presence of cavities with large thickness and small area introduces more significant variations (up to 70%) on the depthedose curves. Overall, the Gafchromic EBT2 film measurements were found in agreement within 3% with Monte Carlo calculations and predict well the fine details of the dosimetric change near the cavity interface. The Pencil Beam calculations underestimate the dose up to 40% compared to Monte Carlo simulations; in particular for the largest cavity thickness (2.8 cm).

Quantifying chaos for ecological stoichiometry

The theory of ecological stoichiometry considers ecological interactions among species with different chemical compositions. Both experimental and theoretical investigations have shown the importance of species composition in the outcome of the population dynamics. A recent study of a theoretical three-species food chain model considering stoichiometry [B. Deng and I. Loladze, Chaos 17, 033108 (2007)] shows that coexistence between two consumers predating on the same prey is possible via chaos. In this work we study the topological and dynamical measures of the chaotic attractors found in such a model under ecological relevant parameters. By using the theory of symbolic dynamics, we first compute the topological entropy associated with unimodal Poincareacute return maps obtained by Deng and Loladze from a dimension reduction. With this measure we numerically prove chaotic competitive coexistence, which is characterized by positive topological entropy and positive Lyapunov exponents, achieved when the first predator reduces its maximum growth rate, as happens at increasing delta(1). However, for higher values of delta(1) the dynamics become again stable due to an asymmetric bubble-like bifurcation scenario. We also show that a decrease in the efficiency of the predator sensitive to prey's quality (increasing parameter zeta) stabilizes the dynamics. Finally, we estimate the fractal dimension of the chaotic attractors for the stoichiometric ecological model.

Chaos and crises in a model for cooperative hunting: A symbolic dynamics approach

In this work we investigate the population dynamics of cooperative hunting extending the McCann and Yodzis model for a three-species food chain system with a predator, a prey, and a resource species. The new model considers that a given fraction sigma of predators cooperates in prey's hunting, while the rest of the population 1-sigma hunts without cooperation. We use the theory of symbolic dynamics to study the topological entropy and the parameter space ordering of the kneading sequences associated with one-dimensional maps that reproduce significant aspects of the dynamics of the species under several degrees of cooperative hunting. Our model also allows us to investigate the so-called deterministic extinction via chaotic crisis and transient chaos in the framework of cooperative hunting. The symbolic sequences allow us to identify a critical boundary in the parameter spaces (K, C-0) and (K, sigma) which separates two scenarios: (i) all-species coexistence and (ii) predator's extinction via chaotic crisis. We show that the crisis value of the carrying capacity K-c decreases at increasing sigma, indicating that predator's populations with high degree of cooperative hunting are more sensitive to the chaotic crises. We also show that the control method of Dhamala and Lai [Phys. Rev. E 59, 1646 (1999)] can sustain the chaotic behavior after the crisis for systems with cooperative hunting. We finally analyze and quantify the inner structure of the target regions obtained with this control method for wider parameter values beyond the crisis, showing a power law dependence of the extinction transients on such critical parameters.

Toplological Entropy in the Syncronization of Piecewise Linear and Monotone Maps. Coupled Duffing Oscillators

In this paper is presented a relationship between the synchronization and the topological entropy. We obtain the values for the coupling parameter, in terms of the topological entropy, to achieve synchronization of two unidirectional and bidirectional coupled piecewise linear maps. In addition, we prove a result that relates the synchronizability of two m-modal maps with the synchronizability of two conjugated piecewise linear maps. An application to the unidirectional and bidirectional coupled identical chaotic Duffing equations is given. We discuss the complete synchronization of two identical double-well Duffing oscillators, from the point of view of symbolic dynamics. Working with Poincare cross-sections and the return maps associated, the synchronization of the two oscillators, in terms of the coupling strength, is characterized.

Dynamical behaviour on the parameter space: new populational growth models proportional to beta densities

We present new populational growth models, generalized logistic models which are proportional to beta densities with shape parameters p and 2, where p > 1, with Malthusian parameter r. The complex dynamical behaviour of these models is investigated in the parameter space (r, p), in terms of topological entropy, using explicit methods, when the Malthusian parameter r increases. This parameter space is split into different regions, according to the chaotic behaviour of the models.

Femtosecond laser ablation of dentin

The surface morphology, structure and composition of human dentin treated with a femtosecond infrared laser (pulse duration 500 fs, wavelength 1030 nm, fluences ranging from 1 to 3 J cm(-2)) was studied by scanning electron microscopy, x-ray diffraction, x-ray photoelectron spectroscopy and Fourier transform infrared spectroscopy. The average dentin ablation threshold under these conditions was 0.6 +/- 0.2 J cm(-2) and the ablation rate achieved in the range 1 to 2 mu m/pulse for an average fluence of 3 J cm(-2). The ablation surfaces present an irregular and rugged appearance, with no significant traces of melting, deformation, cracking or carbonization. The smear layer was entirely removed by the laser treatment. For fluences only slightly higher than the ablation threshold the morphology of the laser-treated surfaces was very similar to the dentin fracture surfaces and the dentinal tubules remained open. For higher fluences, the surface was more porous and the dentin structure was partially concealed by ablation debris and a few resolidified droplets. Independently on the laser processing parameters and laser processing method used no sub-superficial cracking was observed. The dentin constitution and chemical composition was not significantly modified by the laser treatment in the processing parameter range used. In particular, the organic matter is not preferentially removed from the surface and no traces of high temperature phosphates, such as the beta-tricalcium phosphate, were observed. The achieved results are compatible with an electrostatic ablation mechanism. In conclusion, the high beam quality and short pulse duration of the ultrafast laser used should allow the accurate preparation of cavities, with negligible damage of the underlying material.

Finite element studies of the mechanical behaviour of the diaphragm in normal and pathological cases

The diaphragm is a muscular membrane separating the abdominal and thoracic cavities, and its motion is directly linked to respiration. In this study, using data from a 59-year-old female cadaver obtained from the Visible Human Project, the diaphragm is reconstructed and, from the corresponding solid object, a shell finite element mesh is generated and used in several analyses performed with the ABAQUS 6.7 software. These analyses consider the direction of the muscle fibres and the incompressibility of the tissue. The constitutive model for the isotropic strain energy as well as the passive and active strain energy stored in the fibres is adapted from Humphrey's model for cardiac muscles. Furthermore, numerical results for the diaphragmatic floor under pressure and active contraction in normal and pathological cases are presented.

Topological Complexity and Predictability in the Dynamics of a Tumor Growth Model with Shilnikov's Chaos

Dynamical systems modeling tumor growth have been investigated to determine the dynamics between tumor and healthy cells. Recent theoretical investigations indicate that these interactions may lead to different dynamical outcomes, in particular to homoclinic chaos. In the present study, we analyze both topological and dynamical properties of a recently characterized chaotic attractor governing the dynamics of tumor cells interacting with healthy tissue cells and effector cells of the immune system. By using the theory of symbolic dynamics, we first characterize the topological entropy and the parameter space ordering of kneading sequences from one-dimensional iterated maps identified in the dynamics, focusing on the effects of inactivation interactions between both effector and tumor cells. The previous analyses are complemented with the computation of the spectrum of Lyapunov exponents, the fractal dimension and the predictability of the chaotic attractors. Our results show that the inactivation rate of effector cells by the tumor cells has an important effect on the dynamics of the system. The increase of effector cells inactivation involves an inverse Feigenbaum (i.e. period-halving bifurcation) scenario, which results in the stabilization of the dynamics and in an increase of dynamics predictability. Our analyses also reveal that, at low inactivation rates of effector cells, tumor cells undergo strong, chaotic fluctuations, with the dynamics being highly unpredictable. Our findings are discussed in the context of tumor cells potential viability.

Complete synchronization and delayed synchronization in couplings

We consider a general coupling of two identical chaotic dynamical systems, and we obtain the conditions for synchronization. We consider two types of synchronization: complete synchronization and delayed synchronization. Then, we consider four different couplings having different behaviors regarding their ability to synchronize either completely or with delay: Symmetric Linear Coupled System, Commanded Linear Coupled System, Commanded Coupled System with delay and symmetric coupled system with delay. The values of the coupling strength for which a coupling synchronizes define its Window of synchronization. We obtain analytically the Windows of complete synchronization, and we apply it for the considered couplings that admit complete synchronization. We also obtain analytically the Window of chaotic delayed synchronization for the only considered coupling that admits a chaotic delayed synchronization, the commanded coupled system with delay. At last, we use four different free chaotic dynamics (based in tent map, logistic map, three-piecewise linear map and cubic-like map) in order to observe numerically the analytically predicted windows.

Cooperative effects in the detection of a nitroaliphatic liquid explosive and na explosive taggant in the vapor phase by calix[4]arene-based carbazole-containing conjugated polymers

Two fluorescent molecular receptor based conjugated polymers were used in the detection of a nitroaliphatic liquid explosive (nitromethane) and an explosive taggant (2,3-dimethyl-2,3-dinitrobutane) in the vapor phase. Results have shown that thin films of both polymers display remarkably high sensitivity and selectivity toward these analytes. Very fast, reproducible, and reversible responses were found. The unique behavior of these supramolecular host systems is ascribed to cooperativity effects developed between the calix[4] arene hosts and the phenylene ethynylene-carbazolylene main chains. The calix[4]-arene hosts create a plethora of host-guest binding sites along the polymer backbone, either in their bowl-shaped cavities or between the outer walls of the cavity, to direct guests to the area of the transduction centers (main chain) at which favorable photoinduced electron transfer to the guest molecules occurs and leads to the observed fluorescence quenching. The high tridimensional porous nature of the polymers imparted by the bis-calixarene moieties concomitantly allows fast diffusion of guest molecules into the polymer thin films.

Inherently chiral calix[4]arenes with planar chirality: two new entries to the family

The synthesis of two new inherently chiral calix[4]arenes (ICCs, 1 and 2), endowed with electron-rich concave surfaces, has been achieved through the desymmetrization of a lower rim distal-bridged oxacyclophane (OCP) macrocycle. The new highly emissive ICCs were resolved by chiral HPLC, and the enantiomeric nature of the isolated antipodes proved by electronic circular dichroism (CD). Using time-dependent density functional calculations of CD spectra, their absolute configurations were established. NMR studies with (S)-Pirkle's alcohol unequivocally showed that the host-guest interactions occur in the chiral pocket comprehending the calix-OCP exo cavities and the carbazole moieties.

Avoiding healthy cells extinction in a cancer model

We consider a dynamical model of cancer growth including three interacting cell populations of tumor cells, healthy host cells and immune effector cells. For certain parameter choice, the dynamical system displays chaotic motion and by decreasing the response of the immune system to the tumor cells, a boundary crisis leading to transient chaotic dynamics is observed. This means that the system behaves chaotically for a finite amount of time until the unavoidable extinction of the healthy and immune cell populations occurs. Our main goal here is to apply a control method to avoid extinction. For that purpose, we apply the partial control method, which aims to control transient chaotic dynamics in the presence of external disturbances. As a result, we have succeeded to avoid the uncontrolled growth of tumor cells and the extinction of healthy tissue. The possibility of using this method compared to the frequently used therapies is discussed. (C) 2014 Elsevier Ltd. All rights reserved.

Generalized models from Beta(p,2) densities with strong allee effect: dynamical approach

A dynamical approach to study the behaviour of generalized populational growth models from Bets(p, 2) densities, with strong Allee effect, is presented. The dynamical analysis of the respective unimodal maps is performed using symbolic dynamics techniques. The complexity of the correspondent discrete dynamical systems is measured in terms of topological entropy. Different populational dynamics regimes are obtained when the intrinsic growth rates are modified: extinction, bistability, chaotic semistability and essential extinction.