Shadowing Property, Weak Mixing and Regular Recurrence

We show that a non-wandering dynamical system with the shadowing property is either equicontinuous or has positive entropy and that in this context uniformly positive entropy is equivalent to weak mixing. We also show that weak mixing together with the shadowing property imply the specification property with a special kind of regularity in tracing (a weaker version of periodic specification property). This in turn implies that the set of ergodic measures supported on the closures of orbits of regularly recurrent points is dense in the space of all invariant measures (in particular, invariant measures in such a system form the Poulsen simplex, up to an affine homeomorphism).     

Universal Systems for Entropy Intervals

A topological system is universal for a class of ergodic measure-theoretic systems if its simplex of invariant measures contains, up to an isomorphism, all elements of this class and no elements from outside the class. We construct universal systems for classes given by the combination of three properties: measure-theoretic entropy belonging to a nondegenerate interval of the extended nonnegative real halfline, invertibility and aperiodicity. For classes consisting of aperiodic systems the universal system can be made minimal.     

On the uniqueness of symmetric algebraic consequences implied by a system of conservation laws consistent with the additional conservation equation

In mathematical physics, one often encounters systems of conservation laws which are consistent with an additional conservation equation. Such systems are of particular interest from the point of view of phenomenological thermodynamics where the additional conservation equation is often interpreted as the entropy law. The systems of conservation laws which imply the additional conservation law are strongly related to symmetric systems. These relations are exploited in thermodynamical theories where the system of field equations consistent with the balance of entropy is often assumed to be symmetric.In this paper we use an invariant definition of symmetric system in order to show that the system of balance laws implies the additional balance law if and only if it implies a symmetric system of a certain kind (see Section 2) and that such a symmetric system is uniquely defined.This property is interesting in the context of a more general question; what conditions for a given system of conservation laws are necessary and/or sufficient to ensure the existence of the additional conservation law.     

Periodic Solutions of an Autonomous Reaction-Diffusion System—A Model System

We describe the dynamics of an autonomous system of two reaction-diffusion equations which can be looked at as a model system for more general reaction-diffusion systems. In our system all solutions tend to zero or to (finitely many) periodic orbits which can be fully described—including their stability properties. Furthermore, we construct invariant sets for the period map and show how a new invariant called torsion number is related to our model system.     

Invariant Measures for Dissipative Systems and Generalised Banach Limits

Inspired by a theory due to Foias and coworkers (see, for example, Foias et al. Navier–Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001) and recent work of Wang (Disc Cont Dyn Sys 23:521–540, 2009), we show that the generalised Banach limit can be used to construct invariant measures for continuous dynamical systems on metric spaces that have compact attracting sets, taking limits evaluated along individual trajectories. We also show that if the space is a reflexive separable Banach space, or if the dynamical system has a compact absorbing set, then rather than taking limits evaluated along individual trajectories, we can take an ensemble of initial conditions: the generalised Banach limit can be used to construct an invariant measure based on an arbitrary initial probability measure, and any invariant measure can be obtained in this way. We thus propose an alternative to the classical Krylov–Bogoliubov construction, which we show is also applicable in this situation.     

Spatial and Dynamical Chaos Generated by Reaction–Diffusion Systems in Unbounded Domains

We consider in this article a nonlinear reaction–diffusion system with a transport term (L,∇ _x )u, where L is a given vector field, in an unbounded domain Ω. We prove that, under natural assumptions, this system possesses a locally compact attractor in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov’s ɛ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in . In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584–637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on . As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of .     

Pesin’s Formula for Random Dynamical Systems on \mathbf {R}^d

Pesin’s formula relates the entropy of a dynamical system with its positive Lyapunov exponents. It is well known, that this formula holds true for random dynamical systems on a compact Riemannian manifold with invariant probability measure which is absolutely continuous with respect to the Lebesgue measure. We will show that this formula remains true for random dynamical systems on $\mathbf {R}^d$ which have an invariant probability measure absolutely continuous to the Lebesgue measure on $\mathbf {R}^d$ . Finally we will show that a broad class of stochastic flows on $\mathbf {R}^{d}$ of a Kunita type satisfies Pesin’s formula.     

Fractional generalized Hamiltonian equations and its integral invariants

In this paper, we present a new kind of fractional dynamical equations, i.e. the fractional generalized Hamiltonian equations, and study variation equations and the method of the construction of integral invariants of the system. Based on the definition of Riemann–Liouville fractional derivatives, fractional generalized Hamiltonian equations and its variation equations are established. Then, the relation between first integral and integral invariant of the system is studied, and it is proved that, using a first integral, we can construct an integral invariant of the system. As deductions of above results, a construction method of integral invariants of a traditional generalized Hamiltonian system are given. Further, one example of fractional generalized Hamiltonian system is given to illustrate the method and results of the application. Finally, we study the first integral and integral invariant of the Euler equation of a rigid body which rotates with respect to a fixed-point.     

Bifurcations of Invariant Curves of a Difference Equation

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Entropy conservation in the control of human action

The human motor system is highly adaptable with the ability to adjust its movement patterns under constantly changing task and environmental constraints. In this paper we develop the position that the probabilistic nature of human action can be characterized by entropies at the level of the organism, task, and environment. Systematic changes in motor adaptation are characterized as task-organism and environment-organism tradeoffs in entropy. Such compensatory adaptations lead to a view of goal-directed motor control as the product of an underlying conservation of entropy across the task-organism-environment system. The conservation of entropy supports the view that context dependent adaptations in human goal-directed action are guided fundamentally by natural law and provides a novel means of examining human motor behavior.     

Entropy measures for biological signal analyses

Entropies are among the most popular and promising complexity measures for biological signal analyses. Various types of entropy measures exist, including Shannon entropy, Kolmogorov entropy, approximate entropy (ApEn), sample entropy (SampEn), multiscale entropy (MSE), and so on. A fundamental question is which entropy should be chosen for a specific biological application. To solve this issue, we focus on scaling laws of different entropy measures and introduce an ensemble forecasting framework to find the connections among them. One critical component of the ensemble forecasting framework is the scale-dependent Lyapunov exponent (SDLE), whose scaling behavior is found to be the richest among all the entropy measures. In fact, SDLE contains all the essential information of other entropy measures, and can act as a unifying multiscale complexity measure. Furthermore, SDLE has a unique scale separation property to aptly deal with nonstationarity and characterize high-dimensional and intermittent chaos. Therefore, SDLE can often be the first choice for exploratory studies in biology. The effectiveness of SDLE and the ensemble forecasting framework is illustrated by considering epileptic seizure detection from EEG.     

Properties of thermocapillary fluids and symmetrization of motion equations

The equations of fluid motions are considered in the case of internal energy depending on mass density, volume entropy and their spatial derivatives. The model corresponds to domains with large density gradients in which the temperature is not necessarily uniform. The new general representation is written in symmetric form with respect to the mass and entropy densities. For conservative motions of perfect thermocapillary fluids, Kelvin's circulation theorems are always valid. Dissipative cases are also considered; we obtain the balance of energy and we prove that equations are compatible with the second law of thermodynamics. The internal energy form allows to obtain a Legendre transformation inducing a quasi-linear system of conservation laws which can be written in a divergence form and the stability near equilibrium positions can be deduced. The result extends classical hyperbolicity theory for governing-equations' systems in hydrodynamics, but symmetric matrices are replaced by Hermitian matrices.     

Birkhoff系统的一类积分不变量的构造

分别建立了自由Birkhoff系统和约束Birkhoff系统的非等时变分方程,并且利用系统的Birkhoff方程及其非等时变分方程证明,可由第一积分直接构造该系统基于非等时变分的一类积分不变量。文中,举例说明结果的应用。     

Kinetic solutions of the Boltzmann-Peierls equation and its moment systems

The evolution of heat in crystalline solids is described at low temperatures by the Boltzmann-Peierls equation, which is a kinetic equation for the phase density of phonons.In this study, we solve initial value problems for the Boltzmann-Peierls equation in relation to the following issues: In thermodynamics, a given kinetic equation is usually replaced by a truncated moment system, which in turn is supplemented by a closure principle so that a system of PDEs results for some moments as thermodynamic variables. A very popular closure principle is the maximum entropy principle, which yields a symmetric hyperbolic system. In recent times, this strategy has led to serious studies on two problems that might arise: 1. Do solutions of the maximum entropy principle exist? 2. Is the physics that is embodied by the kinetic equation more or less equivalently displayed by the truncated moment system? It was Junk who proved for the BOLTZMANN equation of gases that maximum entropy solutions do not exist. The same failure appears for the Fokker-Planck equation, which was proved by means of explicit solutions by Dreyer, Junk, and Kunik.This study has two main objectives:1. We give a positive existence result for the maximum entropy principle if the underlying kinetic equation is the Boltzmann-Peierls equation. In other words we show that the maximum entropy principle can be used here to establish a closed hyperbolic moment system of PDEs. However, the intent of the paper is by no means a general justification of the maximum entropy principle.2. We develop an approximative method that allows the solutions of the kinetic equations to be compared with the solutions of the hyperbolic moment systems. To this end we introduce kinetic schemes that consists of free flight periods and two classes of update rules. The first class of rules is the same for the kinetic equation as well as for the maximum entropy system, while the second class of update rules contains additional rules for the maximum entropy system. It is shown that if a sufficient number of moments are taken into account, the two solutions converge to each other. However, in terms of numerical effort, the presented solver for the kinetic equation clearly outperforms the one for the maximum entropy principle.Received: 15 August 2003, Accepted: 8 November 2003, Published online: 11 February 2004PACS: 02.30.Jr, 02.60.Cb, 05.30.Jp, 44.10. + i, 63.20.-e, 66.70. + f, 65.40.Gr Correspondence to: M. Herrmann     

The center of a system of non-parallel forces

From a discrete system F of applied forces given by a collection of vectors F_k applied to corresponding points P_k, a new system QF can be obtained through a rotation by Q of all F_k without changing P_k. In this note we examine invariant properties of F under arbitrary rotations. We also examine invariant properties of the family QF when all rotations share a fixed axis, giving a coordinate-free approach to the results of Kolosov (1927) .     

近哈密顿系统的Hopf分岔

简化了Wiggins提出的关于近哈密顿系统的Hopf分岔条件,并结合硬弹簧Duffing系统,研究了该类系统的Hopf分岔行为,并用数值积分的方法验证了结果的正确性。     

Chaotification of nonlinear discrete systems via immersion and invariance

This paper is concerned with the chaotification of nonlinear discrete systems. A novel method based on (system) immersion and (manifold) invariance (I&I) is introduced to chaotify nonlinear discrete systems. Its basic idea is to immerse an ideal system which holds chaotic properties and may be a lower dimension into the plant system, and then control trajectories of the plant system to converge toward the invariant manifold where the ideal system is immersed. For a class of linearizable systems, we present the immersion and the control law such that these systems can be chaotified through I&I design. An illustrative example with simulation is presented to validate the proposed chaotification scheme.     

On the theory of one-dimensional unsteady motions of an ideal compressible fluid

A transformation is constructed of the independent variables and the unknown functions for the momentum and continuity equations of which one-dimensional unsteady motions of a perfect gas, relative to which the governing system of equations is invariant.When this transformation is used, the governing equation of state of the gas is transformed into a new equation which contains arbitrary parameters. This may enable approximation of the complex equation of state of a given medium to be carried out by selection of the parameters (in particular, for gases with respect of the equilibrium reactions taking place therein), and the use of this transformation may make it possible to reduce the problem to one with a simpler equation of state, for which the corresponding problem is more easily solved.The transformations investigated do not have singularities and do not impose any significant limitations on the hydrodynamic quantitiesthey are applicable both for variable entropy and for flows with shock waves.     

An invariant manifold approach for studying waves in a one-dimensional array of non-linear oscillators

We consider a one-dimensional linear spring-mass array coupled to a one-dimensional array of uncoupled pendula. The principal aim of this study is to investigate the non-linear dynamics of this large-scale system in the limit of weak non-linearities, i.e. when the (fast) non-linear pendulum effects are small compared to the underlying (slow) linear dynamics of the linear spring-mass chain. We approach the dynamics in the context of invariant manifolds of motion. In particular, we prove the existence of an invariant manifold containing the (predominantly) slow dynamics of the system, with the fast pendulum dynamics providing small perturbations to the motions on the invariant manifold. By restricting the motion on the slow invariant manifold and performing asymptotic analysis we prove that the non-linear large-scale system possesses propagation and attenuation zones (PZs and AZs) in the frequency domain, similarly to the corresponding zones of the linearized system. Inside PZs non-linear travelling wave solutions exist, whereas in AZs only attenuating waves are permissible.     

FIRST INTEGRALS AND INTEGRAL INVARIANTS FOR VARIABLE MASS NONHOLONOMIC SYSTEM IN NONINERTIAL REFERENCE FRAMES

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