The dynamics of a Bertrand duopoly with differentiated products: Synchronization,intermittency and global dynamics

We study the dynamics of a duopoly game à la Bertrand with horizontal product differentiation as proposed by Zhang et al. (2009) [35] by introducing opportune microeconomic foundations. The final model is described by a two-dimensional non-invertible discrete time dynamic system T. We show that synchronized dynamics occurs along the invariant diagonal being T symmetric; furthermore, we show that when considering the transverse stability, intermittency phenomena are exhibited. In addition, we discuss the transition from simple dynamics to complex dynamics and describe the structure of the attractor by using the critical lines technique. We also explain the global bifurcations causing a fractalization in the basin of attraction. Our results aim at demonstrating that an increase in either the degree of substitutability or complementarity between products of different varieties is a source of complexity in a duopoly with price competition.     

Complex nonlinear dynamics and controlling chaos in a Cournot duopoly economic model

Complex nonlinear economic dynamics in a Cournot duopoly model proposed by M. Kopel is studied in detail in this work. By utilizing the topological horseshoe theory proposed by Yang XS, the authors detect the topological horseshoe chaotic dynamics in the Cournot duopoly model for the first time, and also give the rigorous computer-assisted verification for the existence of horseshoe. In the process of the proof, the topological entropy of the Cournot duopoly model is estimated to be bigger than zero, which implies that this economic system definitely exhibits chaos. In particular, the authors observe two different types of economic intermittencies, including the Pomeau–Manneville Type-I intermittency arising near a saddle-node bifurcation, and the crisis-induced attractor widening intermittency caused by the interior crisis, which lead to the appearance of intermittency chaos. The authors also observe the transient chaos phenomenon which leads to the destruction of chaotic attractors. All these intermittency phenomena will help us to understand the similar dynamics observed in the practical stock market and the foreign exchange market. Besides, the Nash-equilibrium profits and the chaotic long-run average profits are analyzed. It is numerically demonstrated that both firms can have higher profits than the Nash-equilibrium profits, that is to say, both of the duopolists could be beneficial from a chaotic market. The controlled Cournot duopoly model can make one firm get more profit and reduce the profit of the other firm, and control the system to converge to an equilibrious state, where the two duopolists share the market equally.     

Nonlinear dynamics and intermittency in a long-term copepod time series

We consider the nonlinear dynamics of a long-term copepod (small crustaceans) time series sampled weekly in the Mediterranean sea from 1967 to 1992. Such population dynamics display a high variability that we consider here in an interdisciplinary study, using tools borrowed from the field of statistical physics. We analyse the extreme events of male and female abundances, and of the total population, and show that they both have heavy tailed probability density functions (pdf). We provide hyperbolic fits of the form p(x) ∼ 1/x^μ+1, and estimate the value of μ using Hill’s estimator. We then study the ratio of male to female abundances, compared to the female abundances. Using conditional probability density functions and conditional averages, we show that this ratio is independent of the female density, when the latter is larger than a given threshold. This property is very useful for modelization. We also consider the product of male to female abundances, which can be ecologically related to the encounters. We show that this product is extremely intermittent, and link its pdf to the female pdf.     

Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers

We consider 4-dimensional, real, analytic Hamiltonian systems with a saddle center equilibrium (related to a pair of real and a pair of imaginary eigenvalues) and a homoclinic orbit to it. We find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium. We also consider one-parameter families of reversible, 4-dimensional Hamiltonian systems. We prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points. We also present similar results for systems with heteroclinic orbits to saddle center equilibria. © 1997 John Wiley & Sons, Inc.     

Feedback saddle point solution of counterterror measures and economic growth game

In this paper, the feedback Nash equilibrium solutions of the differential game between counterterror measures and economic growth are investigated. The Hamilton–Jacobi–Isaacs (HJI) equation is used to obtain the feedback saddle point of this zero-sum game. Moreover, the characteristics of the feedback strategies for the government and terrorist organization as well as their relationships with both resource states are analyzed.     

Protein dynamics: Complex by itself

Biological functions are intimately rooted in biopolymer dynamics. It is commonly accepted that proteins can be considered as complex systems, but the origin of such complexity is still not fully understood. Moreover, it is still not really clear if proteins are true complex systems or complicated ones. Here, molecular dynamics simulations on a two helix bundle protein have been performed, and protein trajectories have been analyzed by using correlation functions in the frequency domain. We show that even a simple protein exhibits the hallmarks of complex systems. Moreover, the molecular bases of this complex behavior are possessed completely by the protein itself, because such complexity emerges without considering the solvent explicitly. © 2012 Wiley Periodicals, Inc. Complexity, 2012     

Chaotic population dynamics and biology of the top-predator

We study how the dynamics of a food chain depends on the biology of the top-predator. We consider two model food chains with specialist and generalist top-predators. Both types of food chains display same type of chaotic behavior, short-term recurrent chaos; but the generating mechanisms are drastically different. Food chains with specialist top-predators are dictated by exogenous stochastic factors. On the contrary, the dynamics of those with the generalist top-predator is governed by deterministic changes in system parameters. The study also suggests that robust chaos would be a rarity.     

一类金融系统行为的非线性混沌分析

利用相位随机化的替代数据方法对中国商品期货市场某些品种特性进行了判定,此方法用于随机时序与非线性混沌时序所得的判据值有明显差异．并应用混沌时序的奇异值分解技术对混沌时序的噪声进行了剥离,将相空间分解为值域空间和虚拟的噪声空间,在值域空间内重构了原混沌时序．进一步采用建立在改进的一般约束随机化方法基础之上强扰动的方法再次判定．根据计算结果对商品期货市场的走势进行了分析,结果表明中国商品期货市场是具有明显非线性混沌特性的一类复杂非线性混沌系统．     

Interactive economic dynamics and differential games

This survey reviews the applications of differential game theory in analyzing issues in the economic literature. The needs of the economic discipline are juxtaposed with the merits of various existing types of differential games. Suggestions are also made as to areas holding great promise for future research.The authors are grateful to Prof. George Leitmann for suggestions to their final draft, which improved the clarity of certain assumptions in keeping with the literature.     

复杂多卷波混沌系统的理论设计与电路实现

双卷波Chua电路的发明第一次在混沌理论与非线性电路之间建立了直接联系.复杂多卷波混沌系统在混沌理论与非线性电路之间架起了桥梁.复杂多卷波混沌系统具有明确的工程应用背景,它的理论设计与电路实现在过去三十年里得到迅猛发展.本文简要的回顾国内外过去三十年在复杂多卷波混沌系统的理论设计与电路实现上的主要研究进展,包括基本理论,设计方法与典型的工程应用,试图推进国内复杂多卷波混沌系统的研究.     

Method for Computer Simulation of Chaotic and Complex Periodic Oscillations

The possibility of constructing chaotic and complex periodic orbits of desired configurations is demonstrated on one-dimensional discontinuous maps. With appropriately located discontinuity, these maps can generate a rich selection of specific orbits with long laminar segments. A simple method is proposed to determine the features of the orbit obtained. This technique, applied to special maps with a horizontal linear branch, allows us to generate a great variety of stable periodic orbits with a specified future by only small variations of the map control parameter.     

Discretization and index-robust error analysis for constrained high-index saddle dynamics on the high-dimensional sphere

We develop and analyze numerical discretization to the constrained high-index saddle dynamics, the dynamics searching for the high-index saddle points confined on the high-dimensional unit sphere. Compared with the saddle dynamics without constraints, the constrained high-index saddle dynamics has more complex dynamical forms, and additional operations such as the retraction and vector transport are required due to the constraints, which significantly complicate the numerical scheme and the corr...     

Complex Ginzburg-Landau equations and dynamics of vortices,filaments, and codimension-2 submanifolds

Here we study the asymptotic behavior of solutions to the complex Ginzburg-Landau equations and their associated heat flows in arbitrary dimensions when the Ginzburg-Landau parameter tends to infinity. We prove that the energies of solutions in the flow are concentrated at vortices in two dimensions, filaments in three dimensions, and codimension-2 submanifolds in higher dimensions. Moreover, we show the dynamical laws for the motion of these vortices, filaments, and codimension-2 submanifolds. Away from the energy concentration sets, we use some measure-theoretic arguments to show the strong convergence of solutions in both static and heat flow cases. © 1998 John Wiley & Sons, Inc.     

Chaotic classical scattering and dynamics in oscillating 1-D potential wells

We study the motion of a classical particle interacting with one, two, and finally an infinite chain of 1-D square wells with oscillating depth. For a single well we find complicated scattering behavior even though there is no topological chaos due to the absence of hyperbolic periodic orbits. In contrast, for two coupled square wells there is chaotic scattering. The infinite oscillating chain yields the generic transition to chaos, with diffusion in energy and in space, as the separation between wells is increased. We briefly discuss the relevance of our results to solid state physics.     

Complex dynamics and control of arms race

The aim of this paper is to show that asymmetric, nonlinear armament strategies may lead to chaotic motion in a discrete-time Richardson-type model on the arms race between two rival nations. Local bifurcation analysis reveals that ‘complicated’ dynamics will only occur if neither nation has an absolute advantage over the other one with respect to its level of armament and its capability to keep up the expenditures on armament. The calculation of Lyapunov exponents supports the existence of chaos. Since transitions to chaos can be identified with transitions to war, we use the Ott-Grebogi-Yorke-algorithm to stabilize the arms race model in the chaotic regime and improve the system's performance by making very small time-dependent changes of a parameter under control.     

Graphene-based mass sensors: Chaotic dynamics analysis using the nonlocal strain gradient model

Nonlinear oscillations of graphene resonators are unavoidable due to enhancing the mass sensitivity of graphene-based mass sensors and the nonlinear behavior of the systems provides the route to chaos. In this paper, the nonlinear and chaotic behavior of the graphene-based mass sensor is investigated. The nano-mechanical sensor includes an electrostatically actuated fully clamped single-graphene sheet as a nano-resonator with an attached concentrated mass. By neglecting the rotary inertia, the equation of motion of the nano- resonator and the attached mass is derived using the nonlocal strain gradient theory of elasticity. The nano-resonator is modeled as a Kirchhoff nano-plate with the von Kármán-type geometric nonlinearity. Applying the Galerkin decomposition method to the partial differential equation of motion leads to the ordinary differential equation. Based on the Melnikov's integral method two analytical criteria are derived which provide necessary conditions that determine the chaotic region of the system. The chaotic dynamics of the system are also scrutinized and verified through plotting the Lyapunov exponent diagram, phase plane trajectories and Poincaré maps.