Ergodic chaos in inventory oscillations: An example

This study reveals the dynamic structure of a nonlinear Metzlerian inventory cycle model that can generate complex dynamics. As a first step, it focuses on how the rational behavior of the supply side is responsible for the emergence of complex inventory fluctuations. To this end, the supply side is constructed on the basis of intertemporal optimization representing the short-run profit maximizing behavior of a firm. On the other hand, the demand side is constructed to be as simple as possible (i.e. the linear expenditure function). The firm is assumed to have a piecewisely flexible inventory-expected sales ratio. The resultant inventory adjustment system becomes piecewise-linear. This study demonstrates that the inventory accumulation may follow a wide range of complex dynamics including stable cycles, window phenomenon and ergodic chaos. It also shows that various types of dynamic behavior depend on the relationship between the marginal propensity to consume and marginal propensity to invest. Thus it can be concluded that the source of such a wide spectrum of complex inventory dynamics is an interaction between demand and supply.     

Relaxation oscillations and diffusion chaos in the Belousov reaction

Asymptotic and numerical analysis of relaxation self-oscillations in a three-dimensional system of Volterra ordinary differential equations that models the well-known Belousov reaction is carried out. A numerical study of the corresponding distributed model-the parabolic system obtained from the original system of ordinary differential equations with the diffusive terms taken into account subject to the zero Neumann boundary conditions at the endpoints of a finite interval is attempted. It is shown that, when the diffusion coefficients are proportionally decreased while the other parameters remain intact, the distributed model exhibits the diffusion chaos phenomenon; that is, chaotic attractors of arbitrarily high dimension emerge.     

Local chaos induced by spatial oscillations of a perturbation

We study motion of an one-dimensional Hamiltonian oscillator driven by an external force which is periodic in time and in coordinate as well. It is shown that dynamics of the oscillator is strongly affected by the resonance between spatial and temporal oscillations of the perturbation imposed. In particular, this resonance can induce strong but bounded chaotic diffusion in certain areas of phase space. The model of the Duffing oscillator is used as an example for the numerical simulation.     

Using invariants to determine change detection in dynamical system with chaos

The problem of change detection in dynamical systems originated from ordinary differential equations and real world phenomena is covered. Until now suitable methods for detecting changes for linear systems and nonlinear systems have been elaborated but there are no such method for chaotic systems. In this paper we propose the method of change detection based on the fractal dimension, which is the one of characteristics dynamical system invariants. The application of the method is illustrated with simulations.     

Using digital stories to reduce misconceptions and mistakes about fractions: an action study

The aim of this study is to determine the mistakes and misconceptions about fractions of the fourth-grade students in a primary school and to eliminate these mistakes and misconceptions through digital stories. The study has been conducted with a total of 25 students in the fourth-grade of a private primary school in Turkey. An action research method based on qualitative research approach has been used in the study. In the preparation of activities related to digital stories, the mistakes and misconceptions that students have regarding fractions are taken into account. As a data collection tool, the Determination of Misconceptions Form consisting of 30 open ended questions about the fractions prepared by the researchers and the observation notes taken during the implementation process have been used. In addition, semi-structured interviews have been conducted to identify views of teachers and students about activities designed with digital stories. The implementation has been carried out for 5 weeks. Qualitative data analysis techniques have been used in analysing the data. As a result of the research, it has emerged that teaching implementations designed with digital stories have eliminated the mistakes and misconceptions that a large majority of students have about fractions. Most of the students who had a limited perception about the concept of fraction before implementation have fully perceived the definition of fractions after the activities designed with digital stories. Likewise, most of the students who incorrectly modelled the operations with fractions before the implementation, have been able to use the correct model for the operations with fractions after the-implementation fractions. When opinions about the activities designed with digital stories have been examined, it has been found that most students’ express positive opinions on digital story use in lessons and found digital stories fun, instructive and constructive.     

Experimental study of quantum chaos with cold atoms

We investigate experimentally the quantum behavior of laser-cooled atoms in a pulsed standing wave, a system that is an atomic analog of the quantum kicked rotor. In particular, it may display the well-known phenomenon of “dynamical localization”, when the standing wave is driven periodically. Furthermore, we study some interesting properties of a quasi-periodically driven kicked rotor, which presents resonances that are shown to be sharper than the inverse of the driven–excitation duration, thus presenting a sub-Fourier character.     

Using repair priorities to reduce stock investment in spare part networks

In this paper, we examine the impact of repair priorities in spare part networks. Several heuristics for assigning priorities to items as well as optimising stock levels are developed, extending the well-known VARI-METRIC method. We model repair shops by multi-class, multi-server priority queues. A proper priority setting may lead to a significant reduction in the inventory investment required to attain a target system availability (usually 10–20%). The saving opportunities are particularly high if the utilisation of the repair shops is high and if the item types sharing the same repair shop have clearly different characteristics (price, repair time). For example, we find an investment reduction of 73% for a system with single server repair shops with an utilisation of 0.90 that handle five different item types.     

Coupling leads to chaos

The main goal of this paper is to prove analytically the existence of strange attractors in a family of vector fields consisting of two Brusselators linearly coupled by diffusion. We will show that such a family contains a generic unfolding of a 4-dimensional nilpotent singularity of codimension 4. On the other hand, we will prove that in any generic unfolding Xμ of an n-dimensional nilpotent singularity of codimension n there are bifurcation curves of (n−1)-dimensional nilpotent singularities of codimension n−1 which are in turn generically unfolded by Xμ. Arguments conclude recalling that any generic unfolding of the 3-dimensional nilpotent singularity of codimension 3 exhibits strange attractors.     

Disorder of particle chains as a dynamical problem of transition to chaos: Analogy to simulation induced chaos

The problem of the equilibria of particle chains with nearest and next-to-nearest neighbor interaction has been reduced to the dynamical system, given by 4D or 2D web-maps. It is shown that at the same time these maps can represent difference schemes for differential equations used in computational simulation. An analogy between particle disordering, dynamical chaos, and simulation induced chaos is established. © 1994 John Wiley & Sons, Inc.     

On the occurrence of chaos via different routes to chaos: period doubling and border-collision bifurcations

This paper introduces a new 2D piecewise smooth discrete-time chaotic mapping with rarely observed phenomenon – the occurrence of the same chaotic attractor via different and distinguishable routes to chaos: period doubling and border-collision bifurcations as typical futures. This phenomenon is justified by the location of system equilibria of the proposed mapping, and the possible bifurcation types in smooth dissipative systems. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

An algebraic approach to chaos

The aim of this paper is to study chaotic actions on objects other than metric spaces, e.g. locales and commutative rings. To do so, point-free versions of topologically transitivity and the density of periodic points are obtained for actions on a locale, and then generalized to a category which includes the desired objects of study.Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.     

Using assignment examples to infer weights for ELECTRE TRI method: Some experimental results

Given a finite set of alternatives A, the sorting (or assignment) problem consists in the assignment of each alternative to one of the pre-defined categories. In this paper, we are interested in multiple criteria sorting problems and, more precisely, in the existing method ELECTRE TRI. This method requires the elicitation of preferential parameters (weights, thresholds, category limits,…) in order to construct a preference model which the decision maker (DM) accepts as a working hypothesis in the decision aid study. A direct elicitation of these parameters requiring a high cognitive effort from the DM (V. Mosseau, R. Slowinski, Journal of Global Optimization 12 (2) (1998) 174), proposed an interactive aggregation–disaggregation approach that infers ELECTRE TRI parameters indirectly from holistic information, i.e., assignment examples. In this approach, the determination of ELECTRE TRI parameters that best restore the assignment examples is formulated through a nonlinear optimization program.In this paper, we consider the subproblem of the determination of the weights only (the thresholds and category limits being fixed). This subproblem leads to solve a linear program (rather than nonlinear in the global inference model). Numerical experiments were conducted so as to check the behaviour of this disaggregation tool. Results showed that this tool is able to infer weights that restores in a stable way the assignment examples and that it is able to identify “inconsistencies” in the assignment examples.