一类含参数不确定性混沌系统的自适应控制

基于Lyaponov函数,对一类含参数不确定性混沌系统,当系统的不确定性参数具有未知界时,给出了自适应控制器的设计方法和构造控制器的解析式.构造混沌控制器直接代入控制器的参数化公式即可.方法简单,无需拼凑,能够使系统消除混沌并渐近稳定到任何期望的光滑轨道上.而整个设计过程无需预先知道未知参数的估计值和精确值.仿真验证了其有效性 关键词： 混沌系统 自适应混沌控制 Lyaponov函数 参数不确定性     

Entropy description of a cooperation-competition system

Understanding the cooperation-competition dynamics is a long-standing challenge in studying complex systems.Inspired by the idea of Shannon entropy,we define competition information entropy and propose an entropy evolution model.The analytic results of the model of the relation between competition gain distribution parameters and entropy,as well as the relation between entropy and time are compared with empirical results obtained in 14 real world systems.They are found to be in good agreement with each other.     

Non-Gaussian Fluctuations of Local Lyapunov Exponents at Intermittency

In intermittent dynamical systems, the distributions of local Lyapunov exponents are markedly non-Gaussian and tend to be asymmetric and fat-tailed. A comparative analysis of the different time-scales in intermittency provides a heuristic explanation for the origin of the exponential tails, for which we also obtain an analytic expression deriving from a more quantitative theory. Application is made to several examples of discrete dynamical systems displaying intermittent dynamics.     

On the use of slow manifolds in molecular and geophysical fluid dynamics

Constraints typically arise from the elimination of high frequency oscillations in mechanical systems. Examples are provided by bond constraints in molecular simulations and incompressibility constraints in fluid dynamics. A key issue is the accuracy of constrained dynamics with regard to the full dynamics. In this review we focus on the smooth solution components and discuss the concept of slow manifold and soft constraints in molecular and geophysical fluid dynamics. While the formal mathematical derivation of constraints is the same for both molecular and fluid dynamics, the predominant numerical techniques for dealing with constraints are different in the two fields. Semi-implicit time- stepping methods are often used in geophysical fluid dynamics while explicitly enforced constraints are more common in molecular dynamics.     

Periodic Orbit Quantization: How to make Semiclassical Trace Formulae Convergent

Periodic orbit quantization requires an analytic continuation of non-convergent semiclassical trace formulae. We propose two different methods for semiclassical quantization. The first method is based upon the harmonic inversion of semiclassical recurrence functions. A band-limited periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. The frequencies of the periodic orbit signal are the semiclassical eigenvalues, and are determined by either linear predictor, Padé approximant, or signal diagonalization. The second method is based upon the direct application of the Padé approximant to the periodic orbit sum. The Padé approximant allows the resummation of the, typically exponentially, divergent periodic orbit terms. Both techniques do not depend on the existence of a symbolic dynamics, and can be applied to bound as well as to open systems. Numerical results are presented for two different systems with chaotic and regular classical dynamics, viz. the three-disk scattering system and the circle billiard.     

Time-evolution of quantum systems via a complex nonlinear Riccati equation. I. Conservative systems with time-independent Hamiltonian

The sensitivity of the evolution of quantum uncertainties to the choice of the initial conditions is shown via a complex nonlinear Riccati equation leading to a reformulation of quantum dynamics. This sensitivity is demonstrated for systems with exact analytic solutions with the form of Gaussian wave packets. In particular, one-dimensional conservative systems with at most quadratic Hamiltonians are studied.     

Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments

We study complex damped and undamped dynamics and targeted energy transfers (TETs) in systems of coupled oscillators, consisting of single-degree-of-freedom primary linear oscillators (LOs) with vibro-impact attachments, acting, in essence, as vibro-impact nonlinear energy sinks (VI NESs). First, the complicated dynamics of such VI systems is demonstrated by computing the VI periodic orbits of underlying Hamiltonian systems and depicting them in appropriate frequency–energy plots (FEPs). Then, VI damped transitions and distinct ways of passive TETs from the linear oscillators to the VI attachments for various parameter ranges and initial conditions are investigated. As in the case of smooth stiffness nonlinearity [Y. Lee, G. Kerschen, A. Vakakis, P. Panagopoulos, L. Bergman, D.M. McFarland, Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment, Physica D 204 (1–2) (2005) 41–69], both fundamental and subharmonic TET can be realized in the VI systems under consideration. It is found that the most efficient mechanism for VI TET is through the excitation of highly energetic VI impulsive orbits (IOs), i.e., of periodic or quasiperiodic orbits corresponding to zero initial conditions except for the initial velocities of the linear oscillators. In contrast to NESs with smooth essential nonlinearities considered in previous works, VI NESs are capable of passively absorbing and locally dissipating significant portions of the energies of the primary systems to which they are attached, at fast time scale. This renders such devices suitable for applications, like seismic mitigation, where dissipation of vibration energy in the early, highly energetic regime of the motion is a critical requirement.     

Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems

This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce a simplicial Dirac structure as a discrete analogue of the Stokes–Dirac structure and demonstrate that it provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The spatial domain, in the continuous theory represented by a finite-dimensional smooth manifold with boundary, is replaced by a homological manifold-like simplicial complex and its augmented circumcentric dual. The smooth differential forms, in discrete setting, are mirrored by cochains on the primal and dual complexes, while the discrete exterior derivative is defined to be the coboundary operator. This approach of discrete differential geometry, rather than discretizing the partial differential equations, allows to first discretize the underlying Stokes–Dirac structure and then to impose the corresponding finite-dimensional port-Hamiltonian dynamics. In this manner, a number of important intrinsically topological and geometrical properties of the system are preserved.     

一类具有四翼吸引子的超混沌系统

构造超混沌系统在目前并非难事,在Chua系统基础上构造具有多涡卷拓扑结构的吸引子也有很多系统方法,但较少有文献讨论基于Lorenz系统族的拥有多翼结构吸引子的光滑混沌系统,而同时具备超混沌特性和多翼拓扑结构吸引子的光滑混沌系统则更少报道.在现有超混沌系统的基础上,分析其共性,并利用坐标变化方法将其转为具有四个翼的超混沌吸引子.该方法在保留了系统原有超混沌特性的基础上,增加了吸引子的拓扑结构复杂性,使之更适合保密通信等领域的应用. 关键词： 多涡卷吸引子 多翼吸引子 超混沌     

Presentation functions,fixed points,and a theory of scaling function dynamics

Presentation functions provide the time-ordered points of the forward dynamics of a system as successive inverse images. They generally determine objects constructed on trees, regular or otherwise, and immediately determine a functional form of the transfer matrix of these systems. Presentation functions for regular binary trees determine the associated forward dynamics to be that of a period doubling fixed point. They are generally parametrized by the trajectory scaling function of the dynamics in a natural way. The requirement that the forward dynamics be smooth with a critical point determines a complete set of equations whose solution is the scaling function. These equations are compatible with a dynamics in the space of scalings which is conjectured, with numerical and intuitive support, to possess its solution as a unique, globally attracting fixed point. It is argued that such dynamics is to be sought as a program for the solution of chaotic dynamics. In the course of the exposition new information pertaining to universal mode locking is presented.     

A statistical-mechanical theory of slow dynamics near the glass transition

A statistical-mechanical theory of slow dynamics near the glass transition in two kinds of glass-forming systems, (M) molecular systems and (S) suspensions of colloids, is presented from a unified point of view based on the Tokuyama-Mori projection operator method. The exact diffusion equations for the coherent- and the incoherent-intermediate scattering functions are first derived, whose memory functions are convolutionless in time and contain the correlation effects due to the hydrodynamic interactions in (S). The analytic expressions of the memory functions are then calculated within the mode-coupling theory (MCT) approximation and are shown to coincide with the conventional ones obtained by MCT. Alternative mode-coupling equations are thus obtained in (M) and (S) separately. Self-diffusion is also discussed. Alternative equations for the mean-square displacement and the non-Gaussian parameter are also derived within MCT approximation. All results in both the systems are compared with those obtained by MCT.     

The relaxation dynamics of a confined glassy simple liquid

We use molecular-dynamics computer simulations to study the relaxation dynamics of a confined simple liquid. Two types of confining walls are considered: A rough wall and a smooth wall. The simulation is set up in such a way that the static properties of the confined system are identical to the ones of the bulk. Nevertheless, we find that upon cooling the relaxation dynamics of the confined systems differ strongly from the one of the bulk. In particular, we find that close to the rough/smooth wall this dynamics is slowed down/accelerated by orders of magnitude. Using these results we are able to extract a dynamical length scale of the system and we show that this length shows an Arrhenius dependence.Received: 1 January 2003, Published online: 14 October 2003PACS: 64.70.Pf Glass transitions - 68.15. + e Liquid thin films - 02.70.Ns Molecular dynamics and particle methods     

Fermi-Pasta-Ulam beta model: boundary jumps, Fourier's law, and scaling

We examine the interplay of surface and volume effects in systems undergoing heat flow. In particular, we compute the thermal conductivity in the Fermi-Pasta-Ulam beta model as a function of temperature and lattice size, and scaling arguments are used to provide analytic guidance. From this we show that boundary temperature jumps can be quantitatively understood, and that they play an important role in determining the dynamics of the system, relating soliton dynamics, kinetic theory, and Fourier transport.     

Optical Bistability in V Medium by Squeezed Vacuum Input

In this paper we present the analytic expressions of partition lines in the parametric space of a piece-wise smooth mapping describing an electronic relaxation oscillator. The supercritical regions with permission of period-doubling bifurcation, prohibition of period-doubling bifurcation, and complete phase-locking are discussed. Among them the region, where period-doubling bifurcation is prohibited but chaos is permitted is reported for the first time to our knowledge. These kinds of phenomena and regions can be observed in a lot of similar systems.     

一个张弛振子的超临界子区域

解析导出了描述一种电子张弛振子的分段光滑圆映象在参数空间超临界区域内显示几种不同动力学行为的子区域分界线.这些子区域分别是：允许倍周期分岔发生的区域,禁止倍周期分岔发生的区域,以及完全锁相区域.类似的现象和子区域可以在许多分段光滑系统中观察到. 关键词：     

ADAPTIVE CONTROL AND IDENTIFICATION OF CHAOTIC SYSTEMS

A novel adaptive control and identification on-line method is proposed for a class of chaotic system with uncertain parameters. We prove that, using the presented method, a controller and identifier is developed which can remove chaos in nonlinear systems and make the system asymptotically stabilizing to an arbitrarily desired smooth orbit. And at the same time, estimates to uncertain parameters converge to their true values. The advantage of our method over the existing result is that the controller and identifier is directly constructed by analytic formula without knowing unknown bounds about uncertain parameters in advance. A computer simulation example is given to validate the proposed approach.     

Quantum corrections and bound-state effects in the energy relaxation of hot dense hydrogen

Simple analytic formulas for energy relaxation (ER) in electron-ion systems, with quantum corrections, ion dynamics, and RPA-type screening are presented. ER in the presence of bound electrons is examined in view of recent simulations for ER in hydrogen in the range 10{20}-10{24} electrons/cc.     

理解随机过程分子动力学中的真实/虚拟动力学

Molecular dynamics with the stochastic process provides a convenient way to compute structural and thermodynamic properties of chemical, biological, and materials systems. It is demonstrated that the virtual dynamics case that we proposed for the Langevin equation[J. Chem. Phys. 147, 184104 (2017)] in principle exists in other types of stochastic thermostats as well. The recommended "middle" scheme[J. Chem. Phys. 147, 034109 (2017)] of the Andersen thermostat is investigated as an example. As shown by both analytic and numerical results, while the real and virtual dynamics cases approach the same plateau of the characteristic correlation time in the high collision frequency limit, the accuracy and efficiency of sampling are relatively insensitive to the value of the collision frequency in a broad range. After we compare the behaviors of the Andersen thermostat to those of Langevin dynamics, a heuristic schematic representation is proposed for understanding efficient stochastic thermostatting processes with molecular dynamics.     

Effective Rates in Dilute Reaction-Advection Systems for the Annihilation Process A+A→∅

A dilute system of reacting particles transported by fluid flows is considered. The particles react as A+A→? with a given rate when they are within a finite radius of interaction. The system is described in terms of the joint n-point number spatial density that it is shown to obey a hierarchy of transport equations. An analytic solution is obtained in the dilute or, which is equivalent, the long-time limit by using a Lagrangian approach where statistical averages are performed along non-reacting trajectories. In this limit, it is shown that the moments of the number of particles have an exponential decay rather than the algebraic prediction of standard mean-field approaches. The effective reaction rate is then related to Lagrangian pair statistics by a large-deviation principle. A phenomenological model is introduced to study the qualitative behavior of the effective rate as a function of the interaction length, the degree of chaoticity of the dynamics and the compressibility of the carrier flow. Exact computations, obtained via a Feynman–Kac approach, in a smooth, compressible, random delta-correlated-in-time Gaussian velocity field support the proposed heuristic approach.     

The multiple quantum NMR dynamics in systems of equivalent spins with a dipolar ordered initial state

The multiple quantum (MQ) NMR dynamics in the system of equivalent spins with the dipolar ordered initial state is considered. The high symmetry of the Hamiltonian responsible for the MQ NMR dynamics (the MQ Hamiltonian) is used to develop analytic and numerical methods for the investigation of the MQ NMR dynamics in systems consisting of hundreds of spins from the “first principles.” We obtain the dependence of the intensities of the MQ NMR coherences on their orders (profiles of the MQ NMR coherences) for systems of 200–600 spins. It is shown that these profiles may be well approximated by exponential distribution functions. We also compare the MQ NMR dynamics in the systems of equivalent spins having two different initial states, the dipolar ordered state and the thermal equilibrium state in a strong external magnetic field.