Kinds Of Attractors Research Articles

The coevolution of interconnected relationship strategies in supply networks

Supply network (SN) complexity is the main challenge for managers in predicting the consequences of their firm’s relationship strategy decisions. The number of relationships and their interconnectedness constitute SN complexity and moreover, a firm’s relationships inside an SN are coevolved. It is, therefore, important to investigate how a specific relationship strategy has connected and coevolved with other types of relationship strategies in the context of a specific business situation and the interaction policies of firms. What kind of attractors of interconnected relationships will emerge?To address these issues, we model firms’ interactions inside an SN using cellular automata and complex adaptive system perspectives. Several micro-rules that mimic the drivers and mechanisms of all archetypes of relationship strategies (i.e. cooperation, defection, competition and co-opetition) were programmed inside agents. Experiments using the prisoner’s dilemma game reward schemes, representing business situation factors, have been built and the results show that the coevolution of interconnected relationship strategies coevolution causes the emergence of interesting attractors.

Dynamic behavior of product and stock markets with a varying degree of interaction

We develop a macroeconomic behavioral model in order to analyze the interactions between product and financial markets. The real subsystem is represented by a simple Keynesian income–expenditure model, while the financial subsystem is represented by an equilibrium stock market with heterogeneous speculators, i.e., chartists and fundamentalists. The interactions between the two markets are modeled in the following way: the aggregate demand depends, among other variables, also on the stock market price, while the fundamental value used by speculators in their decisional process depends on the real sector economic conditions. In our model we introduce a parameter that represents the degree of interaction. With the aid of analytical and numerical tools we show that an increasing degree of interaction between markets tends to locally stabilize the system. This stabilization occurs via a sequence of period-halving bifurcations. Globally, we find that the stabilization process implies multistability, i.e., the coexistence of different kinds of attractors.

Dynamics in a nonlinear Keynesian good market model

In this paper, we show how a rich variety of dynamical behaviors can emerge in the standard Keynesian income-expenditure model when a nonlinearity is introduced, both in the cases with and without endogenous government spending. A specific sigmoidal functional form is used for the adjustment mechanism of income with respect to the excess demand, in order to bound the income variation. With the aid of analytical and numerical tools, we investigate the stability conditions, bifurcations, as well as periodic and chaotic dynamics. Globally, we study multistability phenomena, i.e., the coexistence of different kinds of attractors.

Dynamics in a Nonlinear Keynesian Good Market Model

In this paper we show how a rich variety of dynamical behaviors can emerge in the standard Keynesian income-expenditure model when a nonlinearity is introduced, both in the cases with and without endogenous government spending. A specific sigmoidal functional form is used for the adjustment mechanism of income with respect to the excess demand, in order to bound the income variation. With the aid of analytical and numerical tools, we investigate the stability conditions, bifurcations, as well as periodic and chaotic dynamics. Globally, we study multistability phenomena, i.e., the coexistence of different kinds of attractors.

Dynamic Behavior of Real and Stock Markets with a Varying Degree of Interaction

We develop a macroeconomic behavioral model in order to analyze the interactions between real and financial markets. The real subsystem is represented by a simple Keynesian income-expenditure model, while the financial subsystem is represented by an equilibrium stock market with heterogeneous speculators, i.e., chartists and fundamentalists. The interactions between the two markets are modeled in the following way: the aggregate demand depends, among other variables, also on the stock market price, while the fundamental value used by speculators in their decisional process depends on real economic conditions. In our model we introduce a parameter that represents the degree of interaction. With the aid of analytical and numerical tools we show that an increasing degree of interaction between markets tends to locally stabilize the system. This stabilization occurs via a sequence of period-halving bifurcations. Globally, we find that the stabilization process implies multistability, i.e., the coexistence of different kinds of attractors.

ROLES OF CHAOTIC SADDLE AND BASIN OF ATTRACTION IN BIFURCATION AND CRISIS ANALYSIS

This paper is devoted to the dynamical behavior of a parametrically driven double-well Duffing (PDWD) system. Despite the invariant property of symmetry, this simple model exhibits a large diversity of patterns which can be observed in different situations. The transitions between symmetric forms of system responses often lead to bifurcation or crisis and complicated behaviors, such as the coexistence of different kinds of attractors. The bifurcations and crises are discussed, especially those inside the main periodic window. In particular, the role of chaotic saddles and their intrinsic links with the basin of attraction and transient chaos is studied.

Emerging dynamics in neuronal networks of diffusively coupled hard oscillators

Oscillatory networks are a special class of neural networks where each neuron exhibits time periodic behavior. They represent bio-inspired architectures which can be exploited to model biological processes such as the binding problem and selective attention. In this paper we investigate the dynamics of networks whose neurons are hard oscillators, namely they exhibit the coexistence of different stable attractors. We consider a constant external stimulus applied to each neuron, which influences the neuron's own natural frequency. We show that, due to the interaction between different kinds of attractors, as well as between attractors and repellors, new interesting dynamics arises, in the form of synchronous oscillations of various amplitudes. We also show that neurons subject to different stimuli are able to synchronize if their couplings are strong enough.

Topological horseshoe analysis and the circuit implementation for a four-wing chaotic attractor

The paper first analyzes a newly reported three-dimensional four-wing chaotic attractor, and observes all kinds of attractors, including periodic and chaotic, by numerical simulation. Then, the chaotic characteristic of the system is proved by investigating the existence of a topological horseshoe in it, based on the topological horseshoe theory. At last, an electronic circuit is designed to implement the chaotic system. The results of circuit experiment coincided well with those of numerical simulation.

The unified chaotic system describing the Lorenz and Chua systems

This paper introduces the dynamical behaviors of a unified nonlinear chaotic system which describes a two-family chaotic system containing the original Lorenz and the original Chua systems as two extremes and some other systems as a transition in between via a new constructed joint function. This system can display two kinds of attractors, those with one scroll and those with two scrolls.

ATTRACTORS OF FOURTH-ORDER CHUA'S CIRCUIT AND CHAOS CONTROL

In this paper, in order to show some interesting phenomena of fourth-order Chua's circuit with a piecewise-linear nonlinearity and with a smooth cubic nonlinearity and compare dynamics between them, different kinds of attractors and corresponding Lyapunov exponent spectra of systems are presented, respectively. The frequency-domain condition for absolute stability of a class of nonlinear systems is transformed into linear matrix inequality (LMI) by using the celebrated Kalman–Yakubovich–Popov (KYP) lemma. A stabilizing controller based on LMI is designed so that chaos oscillations of fourth-order Chua's circuit with the piecewise-linear nonlinearity disappear and chaotic or hyperchaotic trajectories of the system are led to the origin. Simulation results are provided to demonstrate the effectiveness of the method.

Dissipative and weakly--dissipative regimes in nearly--integrable mappings

We consider a dissipative standard map--like system, which is governed by two parameters measuring the strength of the dissipation and of the perturbation. In order to investigate the dynamics, we follow a numerical and an analytical approach. The numerical study relies on the frequency analysis and on the computation of the <em> differential</em> fast Lyapunov indicators. The analytical approach is based on the computation of a suitable normal form for dissipative systems, which allows us to derive an analytic expression of the frequency. <br> We explore different kinds of attractors (invariant curves, periodic orbits, strange attractors) and their relation with the choice of the perturbing function and of the main frequency of motion (i.e., the frequency of the invariant trajectory of the unperturbed system). In this context we also investigate the occurrence of periodic attractors by looking at the relationship between their periods and the parameters ruling the mapping. Particular attention is devoted to the investigation of the weakly chaotic regime and its transition to the conservative case.

K-chameleon and the coincidence problem

In this paper we present a hybrid model of $k$-essence and chameleon, named as $k$-chameleon. In this model, due to the chameleon mechanism, the directly strong coupling between the $k$-chameleon field and matters (cold dark matters and baryons) is allowed. In the radiation-dominated epoch, the interaction between the $k$-chameleon field and background matters can be neglected; the behavior of the $k$-chameleon therefore is the same as that of the ordinary $k$-essence. After the onset of matter domination, the strong coupling between the $k$-chameleon and matters dramatically changes the result of the ordinary $k$-essence. We find that during the matter-dominated epoch, only two kinds of attractors may exist: one is the familiar $\mathbf{K}$ attractor and the other is a completely new, dubbed $\mathbf{C}$ attractor. Once the Universe is attracted into the $\mathbf{C}$ attractor, the fraction energy densities of the $k$-chameleon ${\ensuremath{\Omega}}_{\ensuremath{\phi}}$ and dust matter ${\ensuremath{\Omega}}_{m}$ are fixed and comparable, and the Universe will undergo a power-law accelerated expansion. One can adjust the model so that the $\mathbf{K}$ attractor does not appear. Thus, the $k$-chameleon model provides a natural solution to the cosmological coincidence problem.

Vanishing viscosity limits and long-time behavior for 2D quasi-geostrophic equations

We study the 2D surface quasi-geostrophic equation for thermal active scalars, that is an equation arising in the study of fast rotating fluids. This equation is a model problem for the 3D Euler equation. We consider the problem of convergence of solutions of the viscous problem to the ones of the inviscid problem. We also consider the long time behavior and we discuss the various kind of attractors that make sense for the quasi-geostrophic equation.

Frequency catastrophe and co-existing attractors in a cell Ca 2+ nonlinear oscillation model with time delay

Frequency catastrophe is found in a cell Ca2+ nonlinear oscillation model with time delay. The relation of the frequency transition to the time delay is studied by numerical simulations and theoretical analysis. There is a range of parameters in which two kinds of attractors with great frequency differences co-exist in the system. Along with parameter changes, a critical phenomenon occurs and the oscillation frequency changes greatly. This mechanism helps us to deepen the understanding of the complex dynamics of delay systems and might be of some meaning in cell signalling.

Measure attractors and random attractors for stochastic partial differential equations

In the theory of stochastic differential equations we can distinguish between two kinds of attractors. The first one is the attractor (measure attractor) with respect to the Markov semigroup generated by a stochastic differential equation. The second meaning of attractors (random attractors) is to be understood with respect to each trajectory of the random equation. The aim of this paper is to bring together the two meanings of attractors. In particular, we show the existence of measure attractors if random attractors exist. We can also show the uniqueness of the stationary distributions of the stochastic Navier-Stokes equation if the viscosity is large

Covariance management based RLS algorithm for phasor estimation in severely noisy systems

This paper presents a new algorithm, derived from the recursive least squares (RLS), to estimate power system voltage and current harmonics for digital relay purposes. The algorithm is based on the RLS covariance matrix management, by making use of the covariance resetting in a different way. The approach provides a kind of attractor that absorbs all nonmodeled dynamics. In order to demonstrate its effectiveness, the proposed algorithm is applied to simulated signals and its performance is compared with other conventional methods.

A discrete version of the dynamic link network

Learning in a dynamic link network (DLN) is a composition of two dynamics: neural dynamics inside layers and link dynamics between layers. Based upon a rigorous analysis of the neural dynamics, we find an algorithm for selecting the parameters of the DLN in such a way that the neural dynamics preferentially converges to any chosen attractor. This control is important because the attractors of the neural dynamics determine the link dynamics which is the main tool for pattern retrieval. Thus in terms of our constructive algorithm it is possible to explore the link dynamics using all kinds of attractors of the neural dynamics. In particular, we show how to get on-center activity patterns which have been extensively used in the application of the DLN to image recognition tasks as well as having an important role in the image processing of the retina. We propose also a Hopfield-like discretized version of the neural dynamics which converges to the attractors much faster than the original DLN.

Evolutionary structure in the informational environmental field of large-scale human systems

The collective consciousness or mind of humanity, especially that aspect that deals with the world “economic system,” is the environment of major interest in this paper. The meaning of environment is discussed, and the environment is considered to be a force field. Field theory is briefly summarized, and some examples of human induction and destabilization of fields given. An overview of the Kondratiev cycle/structure (“economic long wave”) including relevant modeling approaches is presented. A new conceptual model is initiated. The Kondratiev is conceptualized as an evolving informational field. The informational field is related to the various kinds of attractors and to the expansion and contraction of phase space. The model is then extended to incorporate self-replicating micro-units. The conceptual model, with an integration of several facets to the extent now possible, is summarized, and some comparison made with other models. Suggestions for further work are offered.

Chaotic attractors of a locally conservative hyperbolic map with overlap

We introduce a class of hyperbolic maps which are locally conservative but globally dissipative. The dissipation stems not from local contraction of phase space but from the overlap that arises from the noninvertibility of the map. The dynamics of these maps are chaotic, and the attractors are strange, although not necessarily fractal. Though several examples of this class are provided, our analysis concentrates on a single example which is obtained as a modification of the baker's transform. We discuss the morphology of the attractors as the map parameter is varied. We find that some attractors are “thin” fractals and others are “fat” sets of positive measure. We are able to characterize the parameter values that lead to these two kinds of attractors, and for certain parameter values, we can derive analytically exact values for the fractal dimension or for the fat fractal scaling exponent. Taken together, these results lead us to conjecture that the generic attractor of this map has zero measure and full dimension.

Migratory dynamics of interacting subpopulations: Regular and chaotic behavior

AbstractWe demonstrate that different kinds of dynamic behavior may occur in a general migration system. The deterministic nonlinear coupled mean value equations are introduced. The intergroup and intragroup interactions of the subpopulations determine the dynamics of the system. First, the route to chaotic motion in the case of two subpopulations migrating between three regions is briefly presented. In the special case of three interacting sub‐populations migrating between three regions, we then show in detail that all kinds of attractors (fixed points, limit cycles, and strange attractors) exist. Characteristics of the motion, namely the Fourier spectrum, Lyapunov exponents, and the fractal dimension of the attractors, are analyzed. Finally, we discuss the relevance of the results for real migration systems.