Nonlinear dynamics in a Cournot duopoly with relative profit delegation

This paper analyses the dynamics of a nonlinear Cournot duopoly with managerial delegation and homogeneous players. We assume that the owners of both firms hire a manager and delegate output decisions to him or her. Each manager receives a fixed salary plus a bonus based on relative (profit) performance. Managers of both firms may collude or compete. In cases of both collusion and a low degree of competition, we find that synchronised dynamics take place. However, when the degree of competition is high, the dynamics may undergo symmetry-breaking bifurcations, which can cause significant global phenomena. Specifically, there is on–off intermittency and blow-out bifurcations for several parameter values. In addition, several attractors may coexist. The global behaviour of the noninvertible map is investigated through studying a transverse Lyapunov exponent and the folding action of the critical curves of the map. These phenomena are impossible under profit maximisation.     

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.     

Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

The nonlinear reaction‐diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor ?? in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ?‐entropy. Upper and lower bounds of this entropy are obtained. Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ?ⁿ. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional “time” if n > 1) acting on a phase space ??. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy. In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz‐continuous) embedding of this system into the spatial shifts on the attractor. Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.     

Competition analysis of a triopoly game with bounded rationality

A dynamic Cournot game characterized by three boundedly rational players is modeled by three nonlinear difference equations. The stability of the equilibria of the discrete dynamical system is analyzed. As some parameters of the model are varied, the stability of Nash equilibrium is lost and a complex chaotic behavior occurs. Numerical simulation results show that complex dynamics, such as, bifurcations and chaos are displayed when the value of speed of adjustment is high. The global complexity analysis can help players to take some measures and avoid the collapse of the output dynamic competition game.     

Three-dimensional discrete-time Lotka–Volterra models with an application to industrial clusters

We consider a three-dimensional discrete dynamical system that describes an application to economics of a generalization of the Lotka–Volterra prey–predator model. The dynamic model proposed is used to describe the interactions among industrial clusters (or districts), following a suggestion given by [23]. After studying some local and global properties and bifurcations in bidimensional Lotka–Volterra maps, by numerical explorations we show how some of them can be extended to their three-dimensional counterparts, even if their analytic and geometric characterization becomes much more difficult and challenging. We also show a global bifurcation of the three-dimensional system that has no two-dimensional analogue. Besides the particular economic application considered, the study of the discrete version of Lotka–Volterra dynamical systems turns out to be a quite rich and interesting topic by itself, i.e. from a purely mathematical point of view.     

On the periodic points of a two-parameter family of maps of the plane

We study the dynamics of a two-parameter family of noninvertible maps of the plane, derived from a model in population dynamics. We prove that, as one parameter varies with the other held fixed, the nonwandering set changes from the empty set to an unstable Cantor set on which the map is topologically equivalent to the shift endomorphism on two symbols. With the help of some numerical work, we trace the genealogies of the periodic points of the family of period 5, and describe their stability types and bifurcations. Among our results we find that the family has a fixed point which undergoes fold, flip and Hopf bifurcations, and that certain families of period five points are interconnected through a codimension-two cusp bifurcation.     

A dynamic Stackelberg duopoly model with different strategies

In this paper, a duopoly Stackelberg model of competition on output is formulated. The firms announce plan products sequentially in planning phase and act simultaneously in production phase. For the duopoly Stackelberg model, a nonlinear dynamical system which describes the time evolution with different strategies is analyzed. We present results on existence, stability and local bifurcations of the equilibrium points. Numerical simulations demonstrate that the system with varying model parameters may drive to chaos and the loss of stability may be caused by period doubling bifurcations. It is also shown that the state variables feedback and parameter variation method can be used to keep the system from instability and chaos.     

Chaos and fractal properties induced by noninvertibility of models in the form of maps

Many classes of discrete dynamical systems give rise to models in the form of noninvertible maps. With respect to invertible maps, noninvertible maps introduce a singularity of a different nature: the critical set of rank-one, as the geometrical locus of points having at least two coincident preimages. Such new singularities play a fundamental role in the definition of attractors, basins and their bifurcations. The purpose of this paper is a survey of some fundamental results related to two-dimensional noninvertible maps leading to specific chaotic behaviors, as fractal sets, characterizing irreversibility properties of a class of discrete systems.     

The role of constraints in a segregation model: The symmetric case

In this paper we study the effects of constraints on the dynamics of an adaptive segregation model introduced by Bischi and Merlone (2011) [3]. The model is described by a two dimensional piecewise smooth dynamical system in discrete time. It models the dynamics of entry and exit of two populations into a system, whose members have a limited tolerance about the presence of individuals of the other group. The constraints are given by the upper limits for the number of individuals of a population that are allowed to enter the system. They represent possible exogenous controls imposed by an authority in order to regulate the system. Using analytical, geometric and numerical methods, we investigate the border collision bifurcations generated by these constraints assuming that the two groups have similar characteristics and have the same level of tolerance toward the members of the other group. We also discuss the policy implications of the constraints to avoid segregation.     

Noninvertible maps and their embedding into higher dimensional invertible maps

The first part is devoted to a presentation of specific features of noninvertible maps with respect to the invertible ones. When embedded into a three-dimensional invertible map, the specific dynamical features of a plane noninvertible map are the germ of the three-dimensional dynamics, at least for sufficiently small absolute values of the embedding parameter. The form of the paper, as well as its contents, is approached from a non abstract point of view, in an elementary form from a simple class of examples.     

Global bifurcations and chaos in externally excited cyclic systems

The global bifurcations in mode interaction of a nonlinear cyclic system subjected to a harmonic excitation are investigated with the case of the primary resonance, the averaged equations representing the evolution of the amplitudes and phases of the interacting normal modes exhibit complex dynamics. The energy-phase method proposed by Haller and Wiggins is employed to analyze the global bifurcations for the cyclic system. The results obtained here indicate that there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the resonant case in both Hamiltonian and dissipative perturbations, which imply that chaotic motions occur for this class of systems. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found and the visualizations of these complicated structures are presented.     

Control of the Planar Takens–Bogdanov Bifurcation with Applications

It is well-known that on a versal deformation of the Takens–Bogdanov bifurcation is possible to find dynamical systems that undergo saddle-node, Hopf, and homoclinic bifurcations. In this document a nonlinear control system in the plane is considered, whose nominal vector field has a double-zero eigenvalue, and then the idea is to find under which conditions there exists a scalar control law such that be possible establish a priori, that the closed-loop system undergoes any of the three bifurcations: saddle-node, Hopf or homoclinic. We will say then that such system undergoes the controllable Takens–Bogdanov bifurcation. Applications of this result to the averaged forced van der Pol oscillator, a population dynamics, and adaptive control systems are discussed.     

Global qualitative analysis of a quartic ecological model

In this paper we complete the global qualitative analysis of a quartic ecological model. In particular, studying global bifurcations of singular points and limit cycles, we prove that the corresponding dynamical system has at most two limit cycles.     

Designing torus-doubling solutions to discrete time systems by hybrid projective synchronization

Doubling of torus occurs in high dimensional nonlinear systems, which is related to a certain kind of typical second bifurcations. It is a nontrivial task to create a torus-doubling solution with desired dynamical properties based on the classical bifurcation theories. In this paper, dead-beat hybrid projective synchronization is employed to build a novel method for designing stable torus-doubling solutions into discrete time systems with proper properties to achieve the purpose of utilizing bifurcation solutions as well as avoiding the possible conflict of physical meaning of the created solution. Although anti-controls of bifurcation and chaos synchronizations are two different topics in nonlinear dynamics and control, the results imply that it is possible to develop some new interdisciplinary methods between chaos synchronization and anti-controls of bifurcations.     

Towards a second cybernetics model for cognitive systems

We introduce an adaptive system for dynamics recognition. Thereby, an externally presented dynamics (stimulus) is mapped onto a mirror dynamics which is capable to simulate (simulus). A sudden change of the external dynamics leads to an surprisingly quick re-adaptation of the simulus, even if the presented dynamics is chaotic. The system consists of an internal pool of dynamical modules. The modules are forced to the latter dynamics in the sense of Pyragas' control mechanism by the stimulus. The control term, i.e. the strength of forcing, is used as a measure for which modules fit best to the external dynamics. In a sense, this defines a “dynamics-gradient” within the pool. The mirror dynamics now can be constructed by a linear combination of the best fitting modules with weights given by the control term amplitudes. If one adds the so-constructed mirror dynamics to the pool, one has a representation of the corresponding external dynamics within the pool. Later if the same external dynamics is presented again an even quicker adaptation is possible since a well-fitting module is already present. In order not to blow up the dimensionality of the pool, one can eliminate modules that have not been used for a long time. In principle, the modules can undergo an internal control. In addition, one principally can introduce evolution within the pool. Therefore, the system is able to show what sometimes is called a “second cybernetics”, i.e. a hyper-dynamics of the dynamics modules.     

A cobweb model with gradient adjustment mechanism: nonlinear dynamics and multistability

A cobweb model, characterized by boundedly rational producers with a production adjustment mechanism based on the gradient rule, is described by a nonlinear discrete time dynamical system of the plane. Firms do not have a complete knowledge of the demand function and try to infer how the market will respond to their production changes by an empirical estimates of the marginal profits. Analytical conditions for local stability of the market equilibrium are provided, showing that the stability loss of the market equilibrium may give rise to chaotic dynamic as well. When memory is introduced in the production adjustment mechanism, a locally stabilizing effect is revealed as well as a globally qualitatively destabilizing role for memory. This is related to the occurrence of period doubling and Neimark–Sacker bifurcations, the latter being of supercritical nature as analytically proved. Endogenous fluctuations and multistability, with consequent loss of predictability in the long run dynamics, are observed.     

Periodic motions and bifurcations of a vibro-impact system

Based on the analysis of a two-degree-of-freedom plastic impact oscillator, we introduce a three-dimensional map with dynamical variables defined at the impact instants. The non-linear dynamics of the vibro-impact system is analyzed by using the Poincaré map, in which piecewise property and singularity are found to exist. The piecewise property is caused by the transitions of free flight and sticking motions of two masses immediately after the impact, and the singularity of map is generated via the grazing contact of two masses and corresponding instability of periodic motions. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. Simulations of the free flight and sticking solutions are carried out, and regions of existence and stability of different impact motions are therefore presented in (δ, ω) plane of dimensionless clearance δ and frequency ω. The influence of non-standard bifurcations on dynamics of the vibro-impact system is elucidated accordingly.     

Nonlinear processes in Hamiltonian reconnection

The generation of small spatial scales and their interplay with large scale coherent structures is one of the outstanding phenomena of plasma physics and fluid mechanics. In high temperature space and laboratory plasmas dissipative effects become important at length scales that are much smaller than those where microscopic dynamical effects, related e.g., to electron inertia, come into play. Here we discuss the role of this dissipationless small scale dynamics on the nonlinear evolution of collisionless magnetic reconnection within the framework of the so called “two-field” and “four-field models”.     

Local expansion concepts for detecting transport barriers in dynamical systems

In the last two decades, the mathematical analysis of material transport has received considerable interest in many scientific fields such as ocean dynamics and astrodynamics. In this contribution we focus on the numerical detection and approximation of transport barriers in dynamical systems. Starting from a set-oriented approximation of the dynamics we combine discrete concepts from graph theory with established geometric ideas from dynamical systems theory. We derive the global transport barriers by computing the local expansion properties of the system. For the demonstration of our results we consider two different systems. First we explore a simple flow map inspired by the dynamics of the global ocean. The second example is the planar circular restricted three body problem with Sun and Jupiter as primaries, which allows us to analyze particle transport in the solar system.     

Clusters die hard: Time-correlated excitation in the Hamiltonian mean field model

The Hamiltonian mean field (HMF) model has a low-energy phase where N particles are trapped inside a cluster. Here, we investigate some properties of the trapping/untrapping mechanism of a single particle into/outside the cluster. Since the single particle dynamics of the HMF model resembles the one of a simple pendulum, each particle can be identified as a high-energy particle (HEP) or a low-energy particle (LEP), depending on whether its energy is above or below the separatrix energy. We then define the trapping ratio as the ratio of the number of LEP to the total number of particles and the “fully-clustered” and “excited” dynamical states as having either no HEP or at least one HEP. We analytically compute the phase-space average of the trapping ratio by using the Boltzmann–Gibbs stable stationary solution of the Vlasov equation associated with the N → ∞ limit of the HMF model. The same quantity, obtained numerically as a time average, is shown to be in very good agreement with the analytical calculation. Another important feature of the dynamical behavior of the system is that the dynamical state changes transitionally: the “fully-clustered” and “excited” states appear in turn. We find that the distribution of the lifetime of the “fully-clustered” state obeys a power law. This means that clusters die hard, and that the excitation of a particle from the cluster is not a Poisson process and might be controlled by some type of collective motion with long memory. Such behavior should not be specific of the HMF model and appear also in systems where itinerancy among different “quasi-stationary” states has been observed. It is also possible that it could mimick the behavior of transient motion in molecular clusters or some observed deterministic features of chemical reactions.