Two-dimensional Discrete Dynamical System Research Articles

Bifurcation and Chaos in a Fractional-Order Cournot Duopoly Game Model on Scale-Free Networks

In this study, a Cournot duopoly model describing Caputo fractional-order differential equations with piecewise constant arguments is discussed. We have obtained two-dimensional discrete dynamical system as a result of applying the discretization process to the model. By using the center manifold theory and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip bifurcation about the Nash equilibrium point. Phase portraits, bifurcation diagrams, and Lyapunov exponents show the existence of many complex dynamical behaviors in the model such as the stable equilibrium point, period-2 orbit, period-4 orbit, period-8 orbit, period-16 orbit, and chaos according to changing the speed of the adjustment parameter [Formula: see text]. The discrete Cournot duopoly game model is also considered on two scale-free networks with different numbers of nodes. It is observed that the complex dynamical networks exhibit similar dynamical behaviors such as the stable equilibrium point, flip bifurcation, and chaos depending on changing the coupling strength parameter [Formula: see text]. Moreover, flip bifurcation and transition chaos take place earlier in more heterogeneous networks. Calculating the largest Lyapunov exponents guarantees the transition from nonchaotic to chaotic states in complex dynamical networks.

Quantifying Stability and Chaoticity of One-dimensional and Two-dimensional Discrete Dynamical Systems using Stability Analysis and Lyapunov Exponents

Discrete dynamical system is a system that evolve dynamically with discrete time. In this paper, we consider two discrete systems which exhibit chaotic behaviour. We show that the chaoticity of a system is depend on the values of parameter in the system. The objective of this paper is to investigate both stability and chaoticity of the systems using stability analysis and Lyapunov exponent, respectively. The results show that there is only one fixed point for the one-dimensional system, while for the two-dimensional system, there exist four possible fixed points. We have proved the stability conditions for each fixed point obtained. The Lyapunov results show that the one-dimensional system is stable when r<π/2 and chaotic when r>π/2. Whereas for two-dimensional systems considered, the system is chaotic for the whole range of parameter α∈(0,1]. This study shows that it is significant to consider various values of parameter to study the stability of a dynamical systems in particular to control the chaos in a system.

Phase-amplitude dynamics in terms of extended response functions: Invariant curves and arnold tongues

Phase response curves (PRCs) have been extensively used to control the phase of oscillators under perturbations. Their main advantage is the reduction of the whole model dynamics to a single variable (phase) dynamics. However, in some adverse situations (strong inputs, high-frequency stimuli, weak convergence,...), the phase reduction does not provide enough information and, therefore, PRC lose predictive power. To overcome this shortcoming, in the last decade, new contributions have appeared that allow to reduce the system dynamics to the phase plus some transversal variable that controls the deviations from the asymptotic behaviour. We call this setting extended response functions. In particular, we single out the phase response function (PRF, a generalization of the PRC) and the amplitude response function (ARF) that account for the above-mentioned deviations from the oscillating attractor. It has been shown that in adverse situations, the PRC misestimate the phase dynamics whereas the PRF-ARF system provides accurate enough predictions. In this paper, we address the problem of studying the dynamics of the PRF-ARF systems under periodic pulsatile stimuli. This paradigm leads to a two-dimensional discrete dynamical system that we call 2D entrainment map. By using advanced methods to study invariant manifolds and the dynamics inside them, we construct an analytico-numerical method to track the invariant curves induced by the stimulus as two crucial parameters of the system increase (the strength of the input and its frequency). Our methodology also incorporates the computation of Arnold tongues associated to the 2D entrainment map. We apply the method developed to study inner dynamics of the invariant curves of a canonical type II oscillator model. We further compare the Arnold tongues of the 2D map with those obtained with the map induced only by the PRC, which give already noticeable differences. We also observe (via simulations) how high-frequency or strong enough stimuli break up the oscillatory dynamics and lead to phase-locking, which is well captured by the 2D entrainment map.

Global dynamics of the two-dimensional dynamical systems of special type

We study parametic class of two-dimensional discrete dynamical systems, which is a generalization of renormalization group mapping in the hierarchical fermionic model. To analyze global dynamics we develop methods of computer geometry. We reduce original dynamical system to the quadratic projective map and realize two-dimensional projective space as a disk. All fixed points and some invariant sets are described explicitly. With the using of this information we describe the global dynamics of the mapping for selected domains of the parameter space.

Multiple bifurcations in an early brain tumor model with piecewise constant arguments

In this paper, a differential equation with piecewise constant arguments modeling an early brain tumor growth is considered. The discretization process in the interval [Formula: see text] leads to two-dimensional discrete dynamical system. By using the Schur–Cohn criterion, stability conditions of the positive equilibrium point of the system are obtained. Choosing appropriate bifurcation parameter, the existence of Neimark–Sacker and flip bifurcations is verified. In addition, the direction and stability of the Neimark–Sacker and flip bifurcations are determined by using the normal form and center manifold theory. Finally, the Lyapunov exponents are numerically computed to characterize the complexity of the dynamical behaviors of the system.

Impact of price adjustment speed on the stability of Bertrand–Nash equilibrium and profit of the retailers

PurposeThe purpose of this paper is to investigate the impact of price adjustment speed on the stability of Bertrand–Nash equilibrium in the context of a dual-channel supply chain competition.Design/methodology/approachThe paper considers a dual-channel supply chain comprising a manufacturer, a traditional retailer and an online retailer. A two-dimensional discrete dynamical system is used to examine the Bertrand competition between the retailers. The retailers are assumed to follow bounded rational expectations. Local stability of Bertrand–Nash equilibrium is investigated with respect to the price adjustment speed.FindingsAs the price adjustment speed increases, the stability of Bertrand–Nash equilibrium is lost, leading to complex chaotic dynamics. The results showed that chaotic dynamics deteriorates the profit of the retailers. The authors also found that the chaos can be controlled using an adaptive adjustment mechanism and the retailers enjoy higher profit when the chaos is controlled.Practical implicationsThis study helps retail managers to choose an appropriate price adjustment speed to maximize profit.Originality/valueThe heterogeneity of the retailers is not considered in the studies involving dynamics of retailer competition. This paper contributes to the literature by considering the operational difference between a traditional retailer and an online retailer, i.e. price adjustment speed. In addition, the study establishes a link between price adjustment speed and profit.

Bubbling, riddling, blowout and critical curves

This paper provides a survey of some recent results and examples concerning the use of the method of critical curves in the study of chaos synchronization in discrete dynamical systems with an invariant one-dimensional submanifold. Some examples of two-dimensional discrete dynamical systems, which exhibit synchronization of chaoti1c trajectories with the related phenomena of bubbling, on–off intermittency, blowout and riddles basins, are examined by the usual local analysis in terms of transverse Lyapunov exponents, whereas segments of critical curves are used to obtain the boundary of a two-dimensional compact trapping region containing the one-dimensional Milnor chaotic attractor on which synchronized dynamics occur. Thanks to the folding action of critical curves, the existence of such a compact region may strongly influence the effects of bubbling and blowout bifurcations, as it acts like a ‘trapping vessel’ inside which bubbling and blowout phenomena are bounded by the global dynamical forces of the dynamical system.

Box dimension of Neimark–Sacker bifurcation

In this article we show how a change of a box dimension of orbits of two-dimensional discrete dynamical systems is connected to their bifurcations in a non-hyperbolic fixed point. This connection is already shown in the case of one-dimensional discrete dynamical systems and Hopf bifurcation for continuous systems. Namely, at the bifurcation point the box dimension changes from zero to a certain positive value which is connected to the appropriate bifurcation. We study a two-dimensional discrete dynamical system with only one multiplier on the unit circle, and show a result for the box dimension of an orbit on the centre manifold. We also consider a planar discrete system undergoing a Neimark–Sacker bifurcation. It is shown that box dimension depends on the order of non-degeneracy at the non-hyperbolic fixed point and on the angle–displacement map. As it was expected, we prove that the box dimension is different in the rational and irrational case.

Fractal sets attached to homogeneous quadratic maps in two variables

The fractal set attached to the iteration of a “generic” homogeneous quadratic map from the plane to itself is studied and depicted, by using a two-parametric family of normal forms obtained from the theory of invariants of symmetric bilinear maps F:R×R→R under the full linear group of the plane. While invariant theory classifies maps F:C×C→C on the complex plane, we confine ourselves to consider maps on the real plane, in order to include the results obtained from the theory of two-dimensional discrete dynamical systems. A discrete number of “topological types” for such fractals is conjectured to exist.

Energy cascades for NLS on the torus

We consider the nonlinear Schrödinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special small initial data containing only five modes, we exhibit a countable set of time layers in which arbitrarily large modes are created. The proof relies on a reduction to multiphase weakly nonlinear geometric optics, and on the study of a particular two-dimensional discrete dynamical system.

GLOBAL ANALYSIS AND FOCAL POINTS IN A MODEL WITH BOUNDEDLY RATIONAL CONSUMERS

In this paper we present a contribution toward a dynamic theory of the consumer. Canonical economic theory assumes a perfectly rational agent that acts according to optimization principles of a utility function subject to budget constraints. Here we propose a two-dimensional discrete dynamical system that describes the choice problem of a consumer under the assumption of bounded rationality. The map representing this adaptive process is characterized by the presence of a denominator that can vanish. We use recent results on the global bifurcations of this kind of maps in order to explain the coexistence of different attractors and the structure of the corresponding basins of attraction. The stationary equilibria of the map represent the rational choices of the consumer in the static setting, i.e. solutions of the utility maximization problem under budget constraints. We use geometric and numerical methods to study the problem of coexistence of different attractors and the related problem of the basins of attraction and their global bifurcations.

Nonlinear duopoly games with positive cost externalities due to spillover effects

A Cournot duopoly game is proposed where the interdependence between the quantity-setting firms is not only related to the selling price, determined by the total production through a given demand function, but also on cost-reduction effects related to the presence of the competitor. Such cost reductions are introduced to model the effects of know-how spillovers, caused by the ability of a firm to take advantage, for free, of the results of competitors' Research and Development (R&D) results, due to the difficulties to protect intellectual properties or to avoid the movements of skilled workers among competing firms. These effects may be particularly important in the modeling of high-tech markets, where costs are mainly related to R&D and workers' training. The results of this paper concern the existence and uniqueness of the Cournot–Nash equilibrium, located at the intersection of non-monotonic reaction curves, and its stability under two different kinds of bounded rationality adjustment mechanisms. The effects of spillovers on the existence of the Nash equilibrium are discussed, as well as their influence on the kind of attractors arising when the Nash equilibrium is unstable. Methods for the global analysis of two-dimensional discrete dynamical systems are used to study the structure of the basins of attraction.

From bi-stability to chaotic oscillations in a macroeconomic model

In this paper a discrete-time economic model is considered where the savings are proportional to income and the investment demand depends on the difference between the current income and its exogenously assumed equilibrium level, through a nonlinear S-shaped increasing function. The model can be ultimately reduced to a two-dimensional discrete dynamical system in income and capital, whose time evolution is “driven” by a family of two-dimensional maps of triangular type. These particular two-dimensional maps have the peculiarity that one of their components (the one driving the income evolution in the model at study) appears to be uncoupled from the other, i.e., an independent one-dimensional map. The structure of such maps allows one to completely understand the forward dynamics, i.e., the asymptotic dynamic behavior, starting from the properties of the associated one-dimensional map (a bimodal one in our model). The equilibrium points of the map are determined, and the influence of the main parameters (such as the propensity to save and the firms' speed of adjustment to the excess demand) on the local stability of the equilibria is studied. More important, the paper analyzes how changes in the parameters' values modify both the asymptotic dynamics of the system and the structure of the basins of the different and often coexisting attractors in the phase-plane. Finally, a particular “global” (homoclinic) bifurcation is illustrated, occurring for sufficiently high values of the firms' adjustment parameter and causing the switching from a situation of bi-stability (coexistence of two stable equilibria, or attracting sets of different nature) to a regime characterized by wide chaotic oscillations of income and capital around their exogenously assumed equilibrium levels.

Synchronization, intermittency and critical curves in a duopoly game

The phenomenon of synchronization of a two-dimensional discrete dynamical system is studied for the model of an economic duopoly game, whose time evolution is obtained by the iteration of a noninvertible map of the plane. In the case of identical players the map has a symmetry property that implies the invariance of the diagonal x 1= x 2, so that synchronized dynamics is possible. The basic question is whether an attractor of the one-dimensional restriction of the map to the diagonal is also an attractor for the two-dimensional map, and in which sense. In this paper, a particular dynamic duopoly game is considered for which the local study of the transverse stability, in a neighborhood of the invariant submanifold in which synchronized dynamics takes place, is combined with a study of the global behavior of the map. When measure theoretic, but not topological, attractors are present on the invariant diagonal, intermittency phenomena are observed. The global behavior of the noninvertible map is investigated by studying of the critical manifolds of the map, by which a two-dimensional region is defined that gives an upper bound to the amplitude of intermittent trajectories. Global bifurcations of the basins of attraction are evidenced through contacts between critical curves and basin boundaries.

Iterated Differentiable Maps with Nowhere Differentiable Basin Boundaries

Abstract The fractal basin boundary of a two-dimensional discrete dynamical system modelling a chaotic forcing applied to bistability is shown to be identical to the graph of an infinite series F(x,t)= of weighted iterates of an ergodic unimodal interval function f. In the special case, when f is the logistic map in "full chaos", i.e. ƒ: x ↦ 4x(1 - x), F is a nowhere differentiable function of x for each t > exp(-λf) (even equal to the Weierstrass function), where λf >0 is denoting the Lyapunov exponent of f. For further chaotic functions f, nowhere-differentiability is shown to be obvious from computer simulations.

Chaos in a simple identification/counter model

The development of C 3I models is of importance in investigating decision theory as applied to tactical and logistic problems. A recent model due to Meyer can be viewed as a discrete dynamical system in a four-dimensional euclidean space. In this paper, a two-dimensional discrete dynamical system is analyzed retaining several basic features of Meyer's model using the tools of nonlinear dynamics in general, and chaos theory in particular. Such features as attractors and repellers are identified and certain values of the parameters which admit chaotic regimes including strange attractors or repellers are determined. A substanial dynamic characterization of a discrete system of dimension greater than one is achieved.