On some properties of nearly conservative dynamics of Ikeda map and its relation with the conservative case

The behavior of the well-known Ikeda map with very weak dissipation (so-called nearly conservative case) is investigated. The changes in the bifurcation structure of the parameter plane while decreasing the dissipation are revealed. It is shown that when the dissipation is very weak the system demonstrates an “intermediate” type of dynamics combining the peculiarities of conservative and dissipative dynamics. The correspondence between the trajectories in the phase space in the conservative case and the transformations of the set of initial conditions in the nearly conservative case has been obtained. The dramatic increase of the number of coexisting low-period attractors and the extraordinary growth of the transient time while the dissipation decreases have been revealed. The method of plotting a bifurcation tree for the set of initial conditions has been used to classify the existing attractors by their structure. Also it was shown that most of the coexisting attractors are destroyed by rather small external noise, and the transient time in noisy driven systems increases still more. The new method of two-parameter analysis for conservative systems was proposed.     

Compressible turbulence: Multi-fractal scaling in the transition to the dissipative regime

Multi-fractal scaling in the transition to the dissipative regime for fully-developed compressible turbulence is considered. The multi-fractal power law scaling behavior breaks down for very small length scales thanks to viscous effects. However, the effect of compressibility is found to extend the single-scaling multi-fractal regime further into the dissipative range. In the ultimate compressibility limit, thanks to the shock waves which are the appropriate dissipative structures, the single-scaling regime is found to extend indeed all the way into the full viscous regime. This result appears to be consistent with the physical fact that vortices become more resilient and stretch stronger in a compressible fluid hence postponing viscous intervention. The consequent generation of enhanced velocity gradients in a compressible fluid appears to provide an underlying physical basis for the previous results indicating that fully-developed compressible turbulence is effectively more dissipative than its incompressible counterpart.     

保守系统的混沌控制

保守系统的混沌控制是一个重要而富有挑战性的研究课题。由于Liouville定理的限制和初始条件的特殊作用，使得适用于耗散系统的混沌控制方法不能直接用于保守系统。本文通过对耗散系统和保守系统混沌运动的特征进行分析和比较，阐述了保守系统混沌运动的规律，总结了近期研究过程中一些典型的基本理论和方法，综述了近年来保守系统混沌控制的相关进展和我们在保守系统的混沌控制方面所做的工作，并对保守系统混沌控制的应用和发展方向进行了展望。     

Energy Profile Fluctuations in Dissipative Nonequilibrium Stationary States

The exact large deviation function (ldf) for the fluctuations of the energy density field is computed for a chain of Ising (or more generally Potts) spins driven by a zero-temperature (dissipative) Glauber dynamics and sustained in a nontrivial stationary regime by an arbitrary energy injection mechanism at the boundary of the system. It is found that this ldf is independent of the dynamical details of the energy injection, and that the energy fluctuations, unlike conservative systems in a nonequilibrium state, are not spatially correlated in the stationary regime.This revised version was published online in March 2005 with corrections to the page numbers.     

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

Some dynamical properties for a time dependent Lorentz gas considering both the dissipative and non dissipative dynamics are studied. The model is described by using a four-dimensional nonlinear mapping. For the conservative dynamics, scaling laws are obtained for the behavior of the average velocity for an ensemble of non interacting particles and the unlimited energy growth is confirmed. For the dissipative case, four different kinds of damping forces are considered namely: (i) restitution coefficient which makes the particle experiences a loss of energy upon collisions; and in-flight dissipation given by (ii) F=-ηV(2); (iii) F=-ηV(μ) with μ≠1 and μ≠2 and; (iv) F=-ηV, where η is the dissipation parameter. Extensive numerical simulations were made and our results confirm that the unlimited energy growth, observed for the conservative dynamics, is suppressed for the dissipative case. The behaviour of the average velocity is described using scaling arguments and classes of universalities are defined.     

Self-organized criticality with and without conservation

The self-organized sandpile models lose criticality if dissipation is introduced. Recently Christensen et al. have shown that dissipative automata based on the Burridge-Knopoff earthquake model exhibit critical behavior. Criticality is qualitatively different for the cases with and without conservation: A new characteristic length appears for the dissipative case which diverges slower than the system size. For all dissipative models we have found a characteristic frequency in the power spectrum of the released energy, which is absent for the conservative case. The exponents describing criticality change continuously as a function of the strength of dissipation and crossover phenomena occur in the vicinity of conservation. Disorder is irrelevant if conservation is present while it destroys criticality in the dissipative case.     

On the dissipation of mechanical energy in a noncontact dynamic mode of a scanning probe microscope under vacuum

This paper presents a critical review of experiments on the measurement of conservative and dissipative forces in dynamic contacts of nanotips of scanning probe microscopes with surfaces (under high-vacuum conditions). In particular, it is noted that, although the perpendicular and parallel oscillatory motions of tips in the vicinity of the surface of various materials have been investigated experimentally, the question as to the nature of noncontact damping forces at distances ≥0.5 nm remains open. A phenomenological relationship between arbitrary conservative and dissipative forces acting on the tip is proposed. This relationship includes a model parameter characterizing the rate or time of the dissipative process and offers a correct explanation of the observed damping. It is shown that, despite the large differences in the damping coefficients, geometric sizes of tips, and types of conservative interactions in different experiments, the dissipative forces in perpendicular and parallel oscillatory motions of tips are adequately described within the proposed theory.     

Effective dissipation and turbulence in spectrally truncated euler flows

A new transient regime in the relaxation towards absolute equilibrium of the conservative and time-reversible 3D Euler equation with a high-wave-number spectral truncation is characterized. Large-scale dissipative effects, caused by the thermalized modes that spontaneously appear between a transition wave number and the maximum wave number, are calculated using fluctuation dissipation relations. The large-scale dynamics is found to be similar to that of high-Reynolds number Navier-Stokes equations and thus obeys (at least approximately) Kolmogorov scaling.     

Quantitative measurement of tip-sample forces by dynamic force spectroscopy in ambient conditions

We introduce a dynamic force spectroscopy technique enabling the quantitative measurement of conservative and dissipative tip-sample forces in ambient conditions. In difference to the commonly detected force-vs-distance curves dynamic force microscopy allows to measure the full range of tip-sample forces without hysteresis effects caused by a jump-to-contact. The approach is based on the specific behavior of a self-driven cantilever (frequency-modulation technique). Experimental applications on different samples (Fischer-sample, silicon wafer) are presented.     

Dissipation-induced d-wave pairing of fermionic atoms in an optical lattice

We show how dissipative dynamics can give rise to pairing for two-component fermions on a lattice. In particular, we construct a parent Liouvillian operator so that a BCS-type state of a given symmetry, e.g., a d-wave state, is reached for arbitrary initial states in the absence of conservative forces. The system-bath couplings describe single-particle, number-conserving and quasilocal processes. The pairing mechanism crucially relies on Fermi statistics. We show how such Liouvillians can be realized via reservoir engineering with cold atoms representing a driven dissipative dynamics.     

Dynamical ensembles in stationary states

We propose, as a generalization of an idea of Ruelle's to describe turbulent fluid flow, a chaotic hypothesis for reversible dissipative many-particle systems in nonequilibrium stationary states in general. This implies an extension of the zeroth law of thermodynamics to nonequilibrium states and it leads to the identification of a unique distribution describing the asymptotic properties of the time evolution of the system for initial data randomly chosen with respect to a uniform distribution on phase space. For conservative systems in thermal equilibrium the chaotic hypothesis implies the ergodic hypothesis. We outline a procedure to obtain the distribution : it leads to a new unifying point of view for the phase space behavior of dissipative and conservative systems. The chaotic hypothesis is confirmed in a nontrivial, parameter-free, way by a recent computer experiment on the entropy production fluctuations in a shearing fluid far from equilibrium. Similar applications to other models are proposed, in particular to a model for the Kolmogorov-Obuchov theory for turbulent flow.     

Gradient catastrophes and sawtooth solutions for a generalized Burgers equation on an interval

The asymptotic behavior of solutions of the Burgers equation and its generalizations with initial value-boundary problem on a finite interval with constant boundary conditions is studied. Since it describes a dissipative medium, any initial profile will evolve to a time-invariant solution with the same boundary values. Yet there are three distinctive asymptotic processes: the initial profile may regularly decay to a smooth invariant solution; or a Heaviside-type gap develops through a dispersive shock and multi-oscillations; or an asymptotic limit is a stationary ‘sawtooth’ solution with periodical breaks of derivative.     

Dissipative particle dynamics study of velocity autocorrelation function and self-diffusion coefficient in terms of interaction potential strength

This research focuses on numerically investigating the self-diffusion coefficient and velocity autocorrelation function (VACF) of a dissipative particle dynamics (DPD) fluid as a function of the conservative interaction strength. Analytic solutions to VACF and self-diffusion coefficients in DPD were obtained by many researchers in some restricted cases including ideal gases, without the account of conservative force. As departure from the ideal gas conditions are accentuated with increasing the relative proportion of conservative force, it is anticipated that the VACF should gradually deviate from its normally expected exponentially decay. This trend is confirmed through numerical simulations and an expression in terms of the conservative force parameter, density and temperature is proposed for the self-diffusion coefficient. As it concerned the VACF, the equivalent Langevin equation describing Brownian motion of particles with a harmonic potential is adapted to the problem and reveals an exponentially decaying oscillatory pattern influenced by the conservative force parameter, dissipative parameter and temperature. Although the proposed model for obtaining the self-diffusion coefficient with consideration of the conservative force could not be verified due to computational complexities, nonetheless the Arrhenius dependency of the self-diffusion coefficient to temperature and pressure permits to certify our model over a definite range of DPD parameters.     

On conservative and dissipative solitons in media with a periodic spatial modulation

A comparative theoretical analysis of properties of conservative and dissipative optical solitons in media with a periodic spatial modulation of optical characteristics is performed. It is shown that, in the case of modulation in the longitudinal (with respect to the axis of predominant propagation) direction, the mechanism of decay of conservative solitons because of the delocalization of their Fourier harmonics takes place, whereas, for dissipative solitons, this mechanism is absent. In the case of modulation in the transverse direction, the presence of discrete dissipative solitons in a set of optical fibers with nonlinear (saturable) amplification and absorption is shown, which, to a considerable extent, are similar to conservative discrete solitons.     

Excitability mediated by localized structures in a dissipative nonlinear optical cavity

We find and characterize an excitability regime mediated by localized structures in a dissipative nonlinear optical cavity. The scenario is that stable localized structures exhibit a Hopf bifurcation to self-pulsating behavior, that is followed by the destruction of the oscillation in a saddle-loop bifurcation. Beyond this point there is a regime of excitable localized structures under the application of suitable perturbations. Excitability emerges from the spatial dependence since the system does not exhibit any excitable behavior locally. We show that the whole scenario is organized by a Takens-Bogdanov codimension-2 bifurcation point.     

Quantum magnetic impurities in magnetically ordered systems

We discuss the problem of a spin 1/2 impurity immersed in a spin S magnetically ordered background. We show that the problem maps onto a generalization of the dissipative two level system with two independent heat baths, associated with the Goldstone modes of the magnet, that couple to different components of the impurity spin operator. Using analytical perturbative renormalization group methods and accurate numerical renormalization group we show that contrary to other dissipative models there is quantum frustration of decoherence and quasiscaling even in the strong coupling regime. We make predictions for the behavior of the impurity magnetic susceptibility. Our results may also have relevance to quantum computation.     

Remarks on nonequilibrium correlations in a simple dynamic model

We study a deterministic conservative dynamic model inspired by the Kac ring model. We show that some initial probability distributions with long range correlation, which go to equilibrium asymptotically, can be transformed, through a transition to new dissipative and stochastic dynamics, into states with damped correlations.     

Bragg optical solitons beyond the coupled mode approximation

Properties of conservative and dissipative Bragg solitons formed in single-mode optical fibers with induced longitudinal modulation of the refractive index are analyzed beyond the standard approximation of coupled modes, or of slowly varying amplitudes. It is shown that, if the initial velocity of a Bragg soliton is smaller than a critical value, the soliton stops in the process of propagation.     

不连续不可逆二维映象的动力学特性

总结两个保守映象不可逆地分段连续链接(称为类耗散系统)以及一个保守映象与一个耗散映象不可逆地分段连续链接(称为半耗散系统)情况下得到的五项共同动力学特征：不连续边界象集构成的随机网成为唯一的混沌轨道；由于某些相点具有两个逆象而导致的相空间塌缩(类耗散)；由于系统的不连续不可逆性质而出现的胖分形禁区网；在具有吸引子共存时占据不连续边界象集随机网和胖分形禁区网区域的点滴状吸引域以及由此导致的吸引子不可预言性；即使在传统强耗散存在的情况下点滴状吸引域仍由类耗散机制主宰.以一个累积-触发电路为例，说明这五项系统动 关键词： 随机网 禁区网 点滴状吸引域