On the relationship between uniqueness and stability in sum-aggregative,symmetric and general differentiable games

This article explores the relationship between uniqueness and stability in differentiable regular games, with a major focus on the important classes of sum-aggregative, two-player and symmetric games. We consider three types of popular dynamics, continuous-time gradient dynamics as well as continuous- and discrete-time best-reply dynamics, and include aggregate-taking behavior as a non-strategic behavioral variant. We show that while in general games stability conditions are only sufficient for uniqueness, they are likely to be necessary as well in models with sum-aggregative or symmetric payoff functions. In particular, a unique equilibrium always verifies the stability conditions of all dynamics if strategies are equilibrium complements, and this also holds for both continuous-time dynamics if strategies are equilibrium substitutes with bounded slopes. These findings extend to the case of aggregate-taking equilibria. We further analyze the stability relations between the various dynamics, and demonstrate that the restrictive nature of the discrete dynamics originates from simultaneity of adjustments. Asynchronous decisions or heterogeneous forward thinking may stabilize the adjustment process.     

Coupled problems in transient fluid and structural dynamics in nuclear engineering: Part 1: Safety problems solved by flow singularity methods

Some important problems in coupled fluid-structural dynamics which occur in safety investigations of liquid metal fast breeder reactors (LMFBR), light water reactors and nuclear reprocessing plants are discussed and a classification of solution methods is introduced. A distinction is made between the step by step solution procedure, where available computer codes in fluid and structural dynamics are coupled, and advanced simultaneous solution methods, where the coupling is carried out at the level of the fundamental equations. Results presented include the transient deformation of a two-row pin bundle surrounded by an infinite fluid field, vapour explosions in a fluid container and containment distortions due to bubble collapse in the pressure suppression system of a boiling water reactor. A recently developed simultaneous solution method is presented in detail. Here the fluid dynamics (inviscid, incompressible fluid) is described by a singularity method which reduces the three-dimensional fluid dynamics problem to a two-dimensional formulation. In this way the three-dimensional fluid dynamics as well as the structural (shell) dynamics can be described essentially by common unknowns at the fluid-structural interface. The resulting equations for the coupled fluid-structural dynamics are analogous to the equations of motion of the structural dynamics alone.     

微循环系统动力学综合数学模型研究(Ⅰ)——理论基础

根据微循环系统的生理情况,建立了微循环系统动力学非定常、非线性分布参数模型,包括血液动力学、间质动力学、淋巴动力学、蛋白质传输动力学、氧动力学、热量传输动力学和肌原性与代谢性调控过程,综合反映了它们之间的相互作用,并考虑了微动脉自律运动和血液非线性粘弹性的影响.几何模型是一个包括微动脉、开放与储备毛细血管、微静脉、初始淋巴管和微动静脉吻合支的简单网络.这种综合模型有助于临床数据的分析研究和“数值实验手段”的建立.     

The Immediate Basin of Attraction of Infinity for Polynomial Semigroups of Finite Type

As a generalization of the dynamics of a single polynomial,Hinkkanen and Martin studied the dynamics of polynomial semigroupsof finite type. Among the issues addressed were possible generalizationsof the filled-in Julia set of a single polynomial and the basinof attraction for infinity. In the paper, these concepts arefurther refined, and the connections between the dynamics ofa single function and the dynamics of semigroups are strengthened.     

Nonlinear Dynamics of the 3D Pendulum

A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. 3D pendulum dynamics have been much studied in integrable cases that arise when certain physical symmetry assumptions are made. This paper treats the non-integrable case of the 3D pendulum dynamics when the rigid body is asymmetric and the center of mass is distinct from the pivot location. 3D pendulum full and reduced models are introduced and used to study important features of the nonlinear dynamics: conserved quantities, equilibria, relative equilibria, invariant manifolds, local dynamics, and presence of chaotic motions. The paper provides a unified treatment of the 3D pendulum dynamics that includes prior results and new results expressed in the framework of geometric mechanics. These results demonstrate the rich and complex dynamics of the 3D pendulum.     

Projective dynamics and classical gravitation

We show that there exists a projective dynamics of a particle. It underlies intrinsically the classical particle dynamics as projective geometry underlies Euclidean geometry. In classical particle dynamics a particle moves in the Euclidean space subjected to a potential. In projective dynamics the position space has only the local structure of the real projective space. The particle is subjected to a field of projective forces. A projective force is not an element of the tangent bundle to the position space, but of some fibre bundle isomorphic to the tangent bundle. These statements are direct consequences of Appell’s remarks on the homography in mechanics, and are compatible with similar statements due to Tabachnikov concerning projective billiards. When we study Euclidean geometry we meet some particular properties that we recognize as projective properties. The same is true for the dynamics of a particle. We show that two properties in classical particle dynamics are projective properties. The fact that the Keplerian orbits close after one turn is a consequence of a more general projective statement. The fact that the fields of gravitational forces are divergence free is a projective property of these fields.      

Automata in Groups and Dynamics and Induced Systems of PDE in Tropical Geometry

In this paper we develop a dynamical scaling limit from rational dynamics to automata in tropical geometry. We compare these dynamics and induce uniform estimates of their orbits. We apply these estimates to introduce a comparison analysis of theory of automata groups in geometric group theory with analysis of rational dynamics and some hyperbolic PDE systems. Frameworks of characteristic properties of automata groups are inherited to the corresponding rational or PDE dynamics. As an application we study the Burnside problem in group theory and translate the property as the infinite quasi-recursiveness in rational dynamics.     

Opinion convergence in the Krasnoshchekov model

In this paper, a rigorous mathematical analysis of the Krasnoshchekov model is presented. We have shown that in case a community does not contain any group of people having zero resistance to interpersonal influence, which are moreover isolated from the pressure of the rest of community, the Krasnoshchekov opinion readjustment procedure can be reduced to the Friedkin–Johnsen dynamics. In turn, if one repeats the Krasnoshchekov opinion updating rule, the corresponding dynamics forces individuals’ opinions to converge eventually to some terminal opinions, which are a consensus under the same conditions as in the French–Harary–DeGroot dynamics. Otherwise, the Krasnoshchekov dynamics exhibits patterns, which are much closer to the behavior of electrons in the superconductivity state.     

Hybrid systems: Modelling and analysis using emergent dynamics

In this paper, we discuss modelling and analysis of hybrid systems with physical interaction dynamics. Such systems are typically considered complex and they are modelled using abstractions. Abstractions may, however, unintentionally exclude critical details, leading to partial or false results. Therefore, we study here use of a particle system in modelling and analysis. The novelty of the particle system is that it is designed to reveal interaction dynamics as emergent dynamics; thus, supporting analysis of complex and intricate interaction dynamics with acceptable modelling effort. As the main contribution, we formalize the particle system, and use it to model and analyze hybrid systems, both mechanical and biological, with nontrivial interaction dynamics.     

Multi-period models for analyzing the dynamics of process improvement activities

Managing knowledge based resource capabilities has become very important in recent years and during a finite horizon it seems to be reasonable to develop the capabilities intensively at the beginning as one can utilize those over a longer period of time. With the help of multi-period models we check the validity of this idea and characterize the dynamics of development activities. The paper identifies the factors that shape these dynamics and from the behavior of these factors we conclude when the dynamics can be increasing or decreasing. We point out that in stable environment there is tendency for decreasing dynamics but future expectations can significantly modify this outcome. Relationships between the successful or less successful implementation of a business strategy and the dynamics of improvement activities are highlighted as well. For specific model structures explicit solutions are derived.     

Chaos and Hyperchaos in Coupled Antiphase Driven Toda Oscillators

The dynamics of two coupled antiphase driven Toda oscillators is studied. We demonstrate three different routes of transition to chaotic dynamics associated with different bifurcations of periodic and quasi-periodic regimes. As a result of these, two types of chaotic dynamics with one and two positive Lyapunov exponents are observed. We argue that the results obtained are robust as they can exist in a wide range of the system parameters.     

Neimark–Sacker bifurcation for the discrete-delay Kaldor–Kalecki model

Results of an extensive numerical study of the influence of additive white noise on the dynamics of a pair of delayed coupled FitzHugh–Nagumo neurons are presented. An intuitively clear simple method is utilized to predict the critical intensities of the noise. In general, the qualitative properties of the noiseless dynamics are stable under the influence of the noise of a reasonable magnitude. However, there are regions of the coupling and time-lag parameters where the noise of a physically acceptable magnitude does cause qualitative changes of the dynamics. These regions are studied in detail.     

Effective particle methods for Fisher–Kolmogorov equations: Theory and applications to brain tumor dynamics

Extended systems governed by partial differential equations can, under suitable conditions, be approximated by means of sets of ordinary differential equations for global quantities capturing the essential features of the systems dynamics. Here we obtain a small number of effective equations describing the dynamics of single-front and localized solutions of Fisher–Kolmogorov type equations. These solutions are parametrized by means of a minimal set of time-dependent quantities for which ordinary differential equations ruling their dynamics are found. A comparison of the finite dimensional equations and the dynamics of the full partial differential equation is made showing a very good quantitative agreement with the dynamics of the partial differential equation. We also discuss some implications of our findings for the understanding of the growth progression of certain types of primary brain tumors and discuss possible extensions of our results to related equations arising in different modeling scenarios.     

Multiple Attractors in Juvenile-adult Single Species Models

The role of age-structure and the Allee effect in generating multiple attractors in juvenile-adult single species single patch discrete-time models without dispersal are studied. In the presence of the Allee effect juvenile-adult single patch models support multiple attractors. However, in the absence of the Allee effect single attractors are supported when the dynamics are compensatory while multiple attractors are supported under overcompensatory dynamics. When the governing dynamics are compensatory, the boundaries of the basins of attraction have simple structure while complicated fractal basin boundaries are supported under overcompensatory dynamics.     

THE IMPACT OF SCARCITY AND ABUNDANCE IN FOOD CHAINS ON SPECIES POPULATION DYNAMICS

ABSTRACT. The population dynamics in a food chain are derived from a sequence of short‐run equilibria of an ecosystem where predator species demand prey biomass, supply own biomass to their predators and are assumed to behave as if they maximize net biomass intake. Introducing prices as scarcity indicators for the biomass of each species enables us to determine a short‐run ecosystem equilibrium guided by prices. Equilibrium regimes differ with respect to their mix of zero‐priced (= abundant) and positive‐priced (= scarce) species. The population dynamics turn out to vary with the prevailing equilibrium regime. Our analysis yields a richer and more complex population dynamics than the traditional predator‐prey dynamics of the Lotka‐Volterra type.     

Algoritmo compensador neuronal discreto de dinámica en robots móviles usando filtro de Kalman extendido

This paper presents the design of an algorithm based on neural networks in discrete time for its application in mobile robots. In addition, the system stability is analyzed and an evaluation of the experimental results is shown.The mobile robot has two controllers, one addressed for the kinematics and the other one designed for the dynamics. Both controllers are based on the feedback linearization. The controller of the dynamics only has information of the nominal dynamics (parameters). The neural algorithm of compensation adapts its behaviour to reduce the perturbations caused by the variations in the dynamics and the model uncertainties. Thus, the differences in the dynamics between the nominal model and the real one are learned by a neural network RBF (radial basis functions) where the output weights are set using the extended Kalman filter. The neural compensation algorithm is efficient, since the consumed processing time is lower than the one required to learning the totality of the dynamics. In addition, the proposed algorithm is robust with respect to failures of the dynamic controller. In this work, a stability analysis of the adaptable neural algorithm is shown and it is demonstrated that the control errors are bounded depending on the error of approximation of the neural network RBF. Finally, the results of experiments performed by using a mobile robot are shown to test the viability in practice and the performance for the control of robots.     

Gradient stability of numerical algorithms in local nonequilibrium problems of critical dynamics

The critical dynamics of a spatially inhomogeneous system are analyzed with allowance for local nonequilibrium, which leads to a singular perturbation in the equations due to the appearance of a second time derivative. An extension is derived for the Eyre theorem, which holds for classical critical dynamics described by first-order equations in time and based on the local equilibrium hypothesis. It is shown that gradient-stable numerical algorithms can also be constructed for second-order equations in time by applying the decomposition of the free energy into expansive and contractive parts, which was suggested by Eyre for classical equations. These gradient-stable algorithms yield a monotonically nondecreasing free energy in simulations with an arbitrary time step. It is shown that the gradient stability conditions for the modified and classical equations of critical dynamics coincide in the case of a certain time approximation of the inertial dynamics relations introduced for describing local nonequilibrium. Model problems illustrating the extended Eyre theorem for critical dynamics problems are considered.     

The rolling motion of a truncated ball without slipping and spinning on a plane

This paper is concerned with the dynamics of a top in the form of a truncated ball as it moves without slipping and spinning on a horizontal plane about a vertical. Such a system is described by differential equations with a discontinuous right-hand side. Equations describing the system dynamics are obtained and a reduction to quadratures is performed. A bifurcation analysis of the system is made and all possible types of the top’s motion depending on the system parameters and initial conditions are defined. The system dynamics in absolute space is examined. It is shown that, except for some special cases, the trajectories of motion are bounded.     

种群模型在生物进化研究中的应用

介绍了进化动力学的基本知识和研究现状,把表型特征引入种群动力学模型,进而推导出进化适应动力学模型;总结了如何建立适应度函数以及分析研究进化动力学行为的一般理论和方法,并列举实例,模拟分析验证前面所陈述的理论方法,模拟结果说明收获对生物进化产生重要影响,并有效解释了物种多样性。     

具有Monod-Haldane功能反应的一类食物链模型的动力学行为

该文研究了一个由食饵种群、捕食者种群和杂食者种群所构成的食物链系统, 其捕食功能反应为Monod-Haldane功能反应. 应用定性分析和Hopf分支理论, 得到了该系统边界平衡点的全局稳定性和周期解存在性的判别准则. 为了概括和归类这个系统的全局动力学行为,该文得到了具有不同动力学行为的参数区域. 应用MATLAB软件,该文提供了一个例子来展示这些结论, 并且表明: 这个系统能够产生非常复杂的动力学行为.