Chaotic torsional vibration of imbalanced shaft driven by a limited power supply

In this paper, torsional vibrations of imbalanced shaft driven by a limited power supply are studied. It is shown that mutual interaction of shaft and power supply may in particular result in chaotic self-oscillations that correspond to the strange attractors in the phase space of the coupled dynamical system “shaft–power supply”. In this particular model, strange attractors represent classical Lorenz and Feigenbaum attractors. Rotation characteristic of the power supply and resonance characteristic of the shaft rotational motion in one of the resonance zones are studied. It is shown that at certain intervals, these characteristics may be non-unique, which corresponds to the case of chaotic dynamics. Such non-trivial properties of the coupled system “shaft–power supply” could be used for a better understanding of complex vibrational phenomena in real applied systems such as problems related to the damping of the torsional vibrations.     

Physical Complexity of Classical and Quantum Objects and Their Dynamical Evolution From an Information-Theoretic Viewpoint

Charles Bennett's measure of physical complexity for classical objects, namely logical-depth, is used to prove that a chaotic classical dynamical system is not physically complex. The natural measure of physical complexity for quantum objects, quantum logical-depth, is then introduced to prove that a chaotic quantum dynamical system too is not physically complex.     

Lagrangian mechanics and the geometry of configuration spacetime

Holonomic rheonomic systems having a finite number of degrees of freedom are considered in classical nonrelativistic mechanics. It is shown that the configuration spacetime manifold $\text{M}$ of such a system can be furnished with a linear symmetric connection (called the “dynamical connection”) in such a way that the worldline of the system is a geodesic on $\text{M}$. The connection is based upon a degenerate metric structure (called a “generalized Galilei structure”) which in turn is uniquely determined by the system and the forces acting on it. The connection is compatible with the generalized Galilei structure in the sense that the covariant derivatives of the latter vanish. Systems which can be described in terms of a Lagrangian give rise to a particularly interesting class of dynamical connections, called “Lagrange connections,” whose geometry is studied in some detail. Within the class of generalized Galilei connections they are characterized by a geometrical condition imposed on the affine curvature tensor. Noether symmetries of the dynamical system turn out to be equivalent to “isometries” of the generalized Galilei structure together with collineations of the Lagrange connection. They form a Lie group. Spacelike generators of Noether symmetries are linked to the existence of “conservors” (i.e., covectors with vanishing symmetrized covariant derivatives). Timelike generators of Noether symmetries give rise to (second rank) Killing tensors.     

Prequantum Classical Statistical Field Theory: Schrödinger Dynamics of Entangled Systems as a Classical Stochastic Process

The idea that quantum randomness can be reduced to randomness of classical fields (fluctuating at time and space scales which are essentially finer than scales approachable in modern quantum experiments) is rather old. Various models have been proposed, e.g., stochastic electrodynamics or the semiclassical model. Recently a new model, so called prequantum classical statistical field theory (PCSFT), was developed. By this model a “quantum system” is just a label for (so to say “prequantum”) classical random field. Quantum averages can be represented as classical field averages. Correlations between observables on subsystems of a composite system can be as well represented as classical correlations. In particular, it can be done for entangled systems. Creation of such classical field representation demystifies quantum entanglement. In this paper we show that quantum dynamics (given by Schrödinger’s equation) of entangled systems can be represented as the stochastic dynamics of classical random fields. The “effect of entanglement” is produced by classical correlations which were present at the initial moment of time, cf. views of Albert Einstein.     

Dynamical study of the necklace states

Based on finite-difference-time domain methods (FDTD), we have numerically directly investigated the dynamical effects of necklace states on the transmission for one-dimensional (1D) random systems with pulsed incidence in time domain. The necklace state propagation property, which is faster than the common localized modes, is demonstrated directly. From the instantaneous decay coefficient κ(t) and the instantaneous transmittance spectrum T(τ,ω), we have constructed a dynamical picture for the random systems with necklace states. In the picture, we have explained the high plateau on the κ(t) curves by the properties of necklace states, and then defined the time range of high plateau as the “effective time range” of necklace states effects. Further more, we have confirmed the dynamical picture by the ensemble study of random configurations. For the different length, we show that the effects of necklace states will be stronger if the system is longer. Besides these, we also introduce the instantaneous decay coefficient and the instantaneous transmittance spectrum to study the dynamical effects of necklace states. This theoretical study of necklace states can be helpful not only for the deeper physical understanding of necklace states, but also for the experimental observation of necklace states.     

Nanoelectromechanics and single-charge tunneling

The coupling of electronic and mechanical degrees of freedom has important consequences in nanoscale systems, as emphasized in recent theoretical and experimental work. In particular, the electrical properties of composite nanosystems containing elements with quite different abilities to conduct electricity and with different mechanical properties have been found to be strongly affected. Here we briefly review some of our recent work on the nanoelectromechanics of “heteroconducting” and “heteroelastic” Coulomb blockade systems, where single charge tunneling is the dominant conduction mechanism. We examplify nanoelectromechanical effects both in normal and superconducting systems by discussing (i) a self-assembled single-electron tunneling device exhibiting a dynamical instability leading to “shuttling” of electrons by a movable Coulomb dot and (ii) shuttling of Cooper pairs by a movable single-Cooper-pair box.     

Complexity for Extended Dynamical Systems

We consider dynamical systems for which the spatial extension plays an important role. For these systems, the notions of attractor, ϵ-entropy and topological entropy per unit time and volume have been introduced previously. In this paper we use the notion of Kolmogorov complexity to introduce, for extended dynamical systems, a notion of complexity per unit time and volume which plays the same role as the metric entropy for classical dynamical systems. We introduce this notion as an almost sure limit on orbits of the system. Moreover we prove a kind of variational principle for this complexity.     

Universal Hitting Time Statistics for Integrable Flows

The perceived randomness in the time evolution of “chaotic” dynamical systems can be characterized by universal probabilistic limit laws, which do not depend on the fine features of the individual system. One important example is the Poisson law for the times at which a particle with random initial data hits a small set. This was proved in various settings for dynamical systems with strong mixing properties. The key result of the present study is that, despite the absence of mixing, the hitting times of integrable flows also satisfy universal limit laws which are, however, not Poisson. We describe the limit distributions for “generic” integrable flows and a natural class of target sets, and illustrate our findings with two examples: the dynamics in central force fields and ellipse billiards. The convergence of the hitting time process follows from a new equidistribution theorem in the space of lattices, which is of independent interest. Its proof exploits Ratner’s measure classification theorem for unipotent flows, and extends earlier work of Elkies and McMullen.     

Spectra of regular quantum graphs

We consider a class of simple quasi-one-dimensional classically nonintegrable systems that capture the essence of the periodic orbit structure of general hyperbolic nonintegrable dynamical systems. Their behavior is sufficiently simple to allow a detailed investigation of both classical and quantum regimes. Despite their classical chaoticity, these systems exhibit a “nonintegrable analogue” of the Einstein-Brillouin-Keller quantization formula that provides their spectra explicitly, state by state, by means of convergent periodic orbit expansions.     

Predictability: a way to characterize complexity

Different aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov–Sinai entropy, Shannon entropy and algorithmic complexity is discussed. In particular, we emphasize how a characterization of the unpredictability of a system gives a measure of its complexity. Adopting this point of view, we review some developments in the characterization of the predictability of systems showing different kinds of complexity: from low-dimensional systems to high-dimensional ones with spatio-temporal chaos and to fully developed turbulence. A special attention is devoted to finite-time and finite-resolution effects on predictability, which can be accounted with suitable generalization of the standard indicators. The problems involved in systems with intrinsic randomness is discussed, with emphasis on the important problems of distinguishing chaos from noise and of modeling the system. The characterization of irregular behavior in systems with discrete phase space is also considered.     

一种识别混沌时间序列动力学异同性的方法

定义了两个新的统计量，分别记为Q和R，Q称为动力学自相关因子指数，R称为动力学互相关 因子指数.用Q和R对不同时间序列的动力学“距离”或同一时间序列中不同区段（窗口）的 动力学“距离”进行估算，可以鉴别这些时间序列的动力学属性或它们内在的动力学结构层 次以及复杂性.一些典型实例检验表明，这种方法是有效的. 关键词： 混沌时间序列 动力学自相关因子指数 动力学互相关因子指数     

The renormalization group and fractional Brownian motion

We find that in generic field theories the combined effect of fluctuations and interactions leads to a probability distribution function which describes fractional Brownian motion (fBM) and “complex behavior”. To show this we use the renormalization group as a tool to improve perturbative calculations, and check that beyond the classical regime of the field theory (i.e., when no fluctuations are present) the non-linearities drive the probability distribution function of the system away from classical Brownian motion and into a regime which to the lowest order is that of fBM. Our results can be applied to systems away from equilibrium and to dynamical critical phenomena. We illustrate our results with two selected examples: a particle in a heat bath, and the KPZ equation.     

Detecting dynamical complexity changes in time series using the base-scale entropy

Timely detection of dynamical complexity changes in natural and man-made systems has deep scientific and practical meanings. We introduce a complexity measure for time series： the base-scale entropy. The definition directly applies to arbitrary real-word data. We illustrate our method on a practical speech signal and in a theoretical chaotic system. The results show that the simple and easily calculated measure of base-scale entropy can be effectively used to detect qualitative and quantitative dynamical changes.     

Classical Limit and Quantum Logic

The analysis of the classical limit of quantum mechanics usually focuses on the state of the system. The general idea is to explain the disappearance of the interference terms of quantum states appealing to the decoherence process induced by the environment. However, in these approaches it is not explained how the structure of quantum properties becomes classical. In this paper, we consider the classical limit from a different perspective. We consider the set of properties of a quantum system and we study the quantum-to-classical transition of its logical structure. The aim is to open the door to a new study based on dynamical logics, that is, logics that change over time. In particular, we appeal to the notion of hybrid logics to describe semiclassical systems. Moreover, we consider systems with many characteristic decoherence times, whose sublattices of properties become distributive at different times.     

Dynamical process of complex systems and fractional differential equations

Behavior of dynamical process of complex systems is investigated. Specifically we analyse two types of ideal complex systems. For analysing the ideal complex systems, we define the response functions describing the internal states to an external force. The internal states are obtained as a relaxation process showing a “power law” distribution, such as scale free behaviors observed in actual measurements. By introducing a hybrid system, the logarithmic time, and double logarithmic time, we show how the “slow relaxation” (SR) process and “super slow relaxation” (SSR) process occur. Regarding the irregular variations of the internal states as an activation process, we calculate the response function to the external force. The behaviors are classified into “power”, “exponential”, and “stretched exponential” type. Finally we construct a fractional differential equation (FDE) describing the time evolution of these complex systems. In our theory, the exponent of the FDE or that of the power law distribution is expressed in terms of the parameters characterizing the structure of the system.     

A New Way of Looking at Nuclear Resonant Scattering

A new model for nuclear-resonant scattering of gamma radiation from resonant matter has been developed and is summarized here. This “coherent-path” model has lead to closed-form, finite-sum solutions for radiation scattered in the forward direction. The solution provides a unified microscopic picture of nuclear-resonant scattering processes. The resonant absorber or scatterer is modeled as a one-dimensional chain of “effective” nuclei or “effective” planes. The solution is interpreted as showing that the resonant radiation undergoes sequential scattering from one absorber “nucleus” or “plane” to another before reaching the detector. For recoil-free processes the various “paths” to the detector contribute coherently. The solution for this case gives calculated results that are indistinguishable from those using the classical optical model approach, although the forms of the solutions are completely different. The coherent-path model shows that the “speed-up” and “dynamical beating” effects are primarily a consequence of the fact that the single “effective” nuclear scattering processes are 180° out of phase with the incident radiation while the double nuclear scattering processes are in phase with the incident radiation. All multiple scattering paths are, and must be, included. The model can also treat the incoherent processes, i.e., processes involving gamma emission with recoil or conversion-electron emission. The source of the resonant gamma radiation can be from a radioactive source or from synchrotron radiation: both cases are treated. The model is used to explain and understand the results when each of the following experimental procedures is applied: time-differential Mössbauer spectroscopy, time-differential synchrotron radiation spectroscopy, enhanced-resolution resonant-detector Mössbauer spectroscopy, and the “gamma echo”.