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.     

一种具有隐藏吸引子的分数阶混沌系统的动力学分析及有限时间同步

对于具有隐藏吸引子的混沌系统,既有文献大多只针对整数阶系统进行分析与控制研究.基于Sprott E系统,构建了仅有一个稳定平衡点的分数阶混沌系统,通过相位图、Poincare映射和功率谱等,分析了该系统的基本动力学特征.结果显示,该系统展现出了丰富而复杂的动力学特性,且通过随阶次变化的分岔图可知,系统在不同阶次下呈现出周期运动、倍周期运动和混沌运动等状态,这些动力学特征对于保密通信等实际工程领域有重要的研究价值.针对该具有隐藏吸引子的分数阶系统,应用分数阶系统有限时间稳定性理论设计控制器,对系统进行有限时间同步控制,并通过数值仿真验证了其有效性.     

Analysis of grazing bifurcations of quasiperiodic system attractors

This paper presents the first application of the discontinuity-mapping approach to the study of near-grazing bifurcations of originally quasiperiodic, co-dimension-two system attractors. The paper establishes an exact formulation for the discontinuity-mapping methodology under the assumption that a Poincaré section can be found that is everywhere transversal to the grazing attractor. In particular, it is shown that, while a reduced formulation may be employed successfully in the case of co-dimension-one attractors, it fails to capture dynamics in directions transversal to the original quasiperiodic attractor. This shortcoming necessitates the full machinery presented here. The generality of the proposed approach is illustrated through numerical analysis of two nonlinear dynamical systems of dimension three and four.     

Strange nonchaotic attractors in random dynamical systems

Whether strange nonchaotic attractors (SNAs) can occur typically in dynamical systems other than quasiperiodically driven systems has long been an open question. Here we show, based on a physical analysis and numerical evidence, that robust SNAs can be induced by small noise in autonomous discrete-time maps and in periodically driven continuous-time systems. These attractors, which are relevant to physical and biological applications, can thus be expected to occur more commonly in dynamical systems than previously thought.     

Return map structure and entrainment in a time-state-scale re-entrant system

From the perspective of a heterarchy, endo-physics, or internal measurement, a time-state-scale re-entrant system has been proposed [Y.-P. Gunji, K. Sasai, S. Wakisaka, BioSystems (submitted for publication)]. However, the dynamical structure of this system has yet to be estimated. Because the return map of the time-space re-entrant system results from the twisted coupling between the temporal and state-scale states, it is analytically determined. The attractors of entrainment in the interactive re-entrant system are also determined by a particular recursive rule, and it is also shown that a resulting return map can control the attractors of the interactive systems.     

Quantization of Maxwell’s Equations on Curved Backgrounds and General Local Covariance

We develop a quantization scheme for Maxwell??s equations without source on an arbitrary oriented four-dimensional globally hyperbolic spacetime. The field strength tensor is the key dynamical object and it is not assumed a priori that it descends from a vector potential. It is shown that, in general, the associated field algebra can contain a non-trivial centre and, on account of this, such a theory cannot be described within the framework of general local covariance unless further restrictive assumptions on the topology of the spacetime are taken.     

Bimetric renormalization group flows in quantum Einstein gravity

The formulation of an exact functional renormalization group equation for quantum Einstein gravity necessitates that the underlying effective average action depends on two metrics, a dynamical metric giving the vacuum expectation value of the quantum field, and a background metric supplying the coarse graining scale. The central requirement of “background independence” is met by leaving the background metric completely arbitrary. This bimetric structure entails that the effective average action may contain three classes of interactions: those built from the dynamical metric only, terms which are purely background, and those involving a mixture of both metrics. This work initiates the first study of the full-fledged gravitational RG flow, which explicitly accounts for this bimetric structure, by considering an ansatz for the effective average action which includes all three classes of interactions. It is shown that the non-trivial gravitational RG fixed point central to the asymptotic safety program persists upon disentangling the dynamical and background terms. Moreover, upon including the mixed terms, a second non-trivial fixed point emerges, which may control the theory’s IR behavior.     

Dynamical analysis of fractional-order Rössler and modified Lorenz systems

This Letter is devoted to the dynamical analysis of fractional-order systems, namely the Rössler and a modified Lorenz system. The work here described compares the dynamical regimes of such fractional-order systems to that of the corresponding standard systems. It turns out that most of the chaotic attractors are topologically equivalent to those found in the original integer-order systems, although in some particular (and apparently rare) cases unusual bifurcation patterns and attractors are found.     

Unstable attractors induce perpetual synchronization and desynchronization

Common experience suggests that attracting invariant sets in nonlinear dynamical systems are generally stable. Contrary to this intuition, we present a dynamical system, a network of pulse-coupled oscillators, in which unstable attractors arise naturally. From random initial conditions, groups of synchronized oscillators (clusters) are formed that send pulses alternately, resulting in a periodic dynamics of the network. Under the influence of arbitrarily weak noise, this synchronization is followed by a desynchronization of clusters, a phenomenon induced by attractors that are unstable. Perpetual synchronization and desynchronization lead to a switching among attractors. This is explained by the geometrical fact, that these unstable attractors are surrounded by basins of attraction of other attractors, whereas the full measure of their own basin is located remote from the attractor. Unstable attractors do not only exist in these systems, but moreover dominate the dynamics for large networks and a wide range of parameters.     

Effect of resonant-frequency mismatch on attractors

Resonant perturbations are effective for harnessing nonlinear oscillators for various applications such as controlling chaos and inducing chaos. Of physical interest is the effect of small frequency mismatch on the attractors of the underlying dynamical systems. By utilizing a prototype of nonlinear oscillators, the periodically forced Duffing oscillator and its variant, we find a phenomenon: resonant-frequency mismatch can result in attractors that are nonchaotic but are apparently strange in the sense that they possess a negative Lyapunov exponent but its information dimension measured using finite numerics assumes a fractional value. We call such attractors pseudo-strange. The transition to pesudo-strange attractors as a system parameter changes can be understood analytically by regarding the system as nonstationary and using the Melnikov function. Our results imply that pseudo-strange attractors are common in nonstationary dynamical systems.     

“Bulk-SQUID effect” in a complex system as a consequence of its nonergodicity

The dynamics of the phases in a discrete superconductor model has been studied both theoretically and in computer simulation for the case of the passage of a direct current higher than the total critical current of junctions. It has been shown that a bulk-SQUID phenomenon appears in the system in this case and the system is nonergodic. This means that the dynamics of the system certainly depends on the initial conditions and dynamical attractors are limit cycles each having an attracting domain in the configuration space of initial conditions. A mathematical technique for recovering ergodicity in the system under investigation has been proposed. It has also been demonstrated that the bulk-SQUID phenomenon is not observed when the ergodicity is recovered in the system. It has been shown that the results are quite general and describe the behavior of a class of dynamical systems.     

Stochastic climate dynamics: Random attractors and time-dependent invariant measures

This article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s: the nonlinear deterministic and the linear stochastic one. This unification, via the theory of random dynamical systems (RDS), allows one to consider the detailed geometric structure of the random attractors associated with nonlinear, stochastically perturbed systems. We report on high-resolution numerical studies of two idealized models of fundamental interest for climate dynamics. The first of the two is a stochastically forced version of the classical Lorenz model. The second one is a low-dimensional, nonlinear stochastic model of the El Niño-Southern Oscillation (ENSO). These studies provide a good approximation of the two models’ global random attractors, as well as of the time-dependent invariant measures supported by these attractors; the latter are shown to have an intuitive physical interpretation as random versions of Sinaï-Ruelle-Bowen (SRB) measures.     

Generalizations of SRB Measures to Nonautonomous,Random, and Infinite Dimensional Systems

We review some developments that are direct outgrowths of, or closely related to, the idea of SRB measures as introduced by Sinai, Ruelle and Bowen in the 1970s. These new directions of research include the emergence of strange attractors in periodically forced dynamical systems, random attractors in systems defined by stochastic differential equations, SRB measures for infinite dimensional systems including those defined by large classes of dissipative PDEs, quasi-static distributions for slowly varying time-dependent systems, and surviving distributions in leaky dynamical systems.     

A plethora of strange nonchaotic attractors

We show that it is possible to devise a large class of skew-product dynamical systems which have strange nonchaotic attractors (SNAs): the dynamics is asymptotically on fractal attractors and the largest Lyapunov exponent is non-positive. Furthermore, we show that quasiperiodic forcing, which has been a hallmark of essentially all hitherto known examples of such dynamics is not necessary for the creation of SNAs.     

Some comments on the Mindlin-Goodman method of solving vibrations problems with time dependent boundary conditions

Experimental evidence is presented for chaotic type non-periodic motions of a deterministic magnetoelastic oscillator. These motions are analogous to solutions in non-linear dynamic systems possessing what have been called “strange attractors”. In the experiments described below a ferromagnetic beam buckled between two magnets undergoes forced oscillations. Although the applied force is sinusoidal, nevertheless bounded, non-periodic, apparently chaotic motions result due to jumps between two or three stable equilibrium positions. A frequency analysis of the motion shows a broad spectrum of frequencies below the driving frequency. Also the distribution of zero crossing times shows a broad spectrum of times greater than the forcing period. The driving amplitude and frequency parameters required for these non-periodic motions are determined experimentally. A continuum model based on linear elastic and non-linear magnetic forces is developed and it is shown that this can be reduced to a single degree of freedom oscillator which exhibits chaotic solutions very similar to those observed experimentally. Thus, both experimental and theoretical evidence for the existence of a strange attractor in a deterministic dynamical system is presented.     

“Bulk-SQUID effect” in a complex system as a consequence of its nonergodicity

The dynamics of the phases in a discrete superconductor model has been studied both theoretically and in computer simulation for the case of the passage of a direct current higher than the total critical current of junctions. It has been shown that a bulk-SQUID phenomenon appears in the system in this case and the system is nonergodic. This means that the dynamics of the system certainly depends on the initial conditions and dynamical attractors are limit cycles each having an attracting domain in the configuration space of initial conditions. A mathematical technique for recovering ergodicity in the system under investigation has been proposed. It has also been demonstrated that the bulk-SQUID phenomenon is not observed when the ergodicity is recovered in the system. It has been shown that the results are quite general and describe the behavior of a class of dynamical systems.

     

Multistability in coupled different-dimensional dynamical systems

Multistability or coexistence of different chaotic attractors for a given set of parameters depending on the initial condition only is one of the most exciting phenomenon in dynamical systems. The schemes to design multistability systems via coupling two identical or non-identical but the same-dimensional systems have been proposed earlier. Coupled different-dimensional systems are very useful to describe the real-world physical and biological systems. In this paper, a scheme for designing a multistable system by coupling two different-dimensional dynamical systems has been proposed. Coupled Lorenz and Lorenz–Stenflo systems have been considered to illustrate the scheme. The efficiency of the scheme is shown numerically, by presenting phase diagrams, bifurcation diagrams and variation of maximum Lyapunov exponents.     

From Limit Cycles to Strange Attractors

We define a quantitative notion of shear for limit cycles of flows. We prove that strange attractors and SRB measures emerge when systems exhibiting limit cycles with sufficient shear are subjected to periodic pulsatile drives. The strange attractors possess a number of precisely-defined dynamical properties that together imply chaos that is both sustained in time and physically observable.     

Vulnerability to paroxysmal oscillations in delayed neural networks: a basis for nocturnal frontal lobe epilepsy?

Resonance can occur in bistable dynamical systems due to the interplay between noise and delay (τ) in the absence of a periodic input. We investigate resonance in a two-neuron model with mutual time-delayed inhibitory feedback. For appropriate choices of the parameters and inputs three fixed-point attractors co-exist: two are stable and one is unstable. In the absence of noise, delay-induced transient oscillations (referred to herein as DITOs) arise whenever the initial function is tuned sufficiently close to the unstable fixed-point. In the presence of noisy perturbations, DITOs arise spontaneously. Since the correlation time for the stationary dynamics is ～τ, we approximated a higher order Markov process by a three-state Markov chain model by rescaling time as t?→?2sτ, identifying the states based on whether the sub-intervals were completely confined to one basin of attraction (the two stable attractors) or straddled the separatrix, and then determining the transition probability matrix empirically. The resultant Markov chain model captured the switching behaviors including the statistical properties of the DITOs. Our observations indicate that time-delayed and noisy bistable dynamical systems are prone to generate DITOs as switches between the two attractors occur. Bistable systems arise transiently in situations when one attractor is gradually replaced by another. This may explain, for example, why seizures in certain epileptic syndromes tend to occur as sleep stages change.     

混沌吸引子的对称破缺激变

许多非线性动力系统都有某种对称性,在不同情形下可有不同的表现形式,但始终保持其对称的特点.不同对称形式间的转变导致对称破缺分岔或激变.关于非线性动力系统中相空间运动轨道的对称破缺分岔,已有大量研究工作,但绝大多数是指周期或拟周期相轨的对称破缺,偶尔提到对称系统中的混沌相轨也存在“对偶性”.最近,在简谐外激Duffing系统周期轨道对称破缺引发鞍-结分岔的研究中,得到了分岔后由Poincaré映射点间断流构成的图像,其中包括两个稳定周期结点、一个周期鞍点,及其稳定流形与不稳定流形,均较规则.本工作研究了正弦 关键词： 对称破缺 混沌 激变 分形吸引域