Noise-sensitive hysteresis loops around period-doubling bifurcation points

Summary The noise-sensitive hysteresis loops observed in the bouncing-ball model are described. The phenomenon is analysed within the formalisms of the square map and the dissipative standard mapping. The notion of steady-state paths is introduced. A linear approximation of the simplest steady-state path is found.

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Riassunto Si descrivono gli anelli d’isteresi sensibili al rumore osservati nel modello della palla che rimbalza. Il fenomeno è analizzato nell’ambito dei formalismi della mappa quadrata e della funzione dissipativa standard. S’introduce la nozione di percorsi dello stato stazionario. Si trova un’approssimazione lineare del percorso piú semplice dello stato stazionario.

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Резюме Описываются петли гистерезиса, чувствительные к шуму, которые наблюдаются в модели подскакивающего мяча. Это явление анализируется в рамках формализма отображения квадрата и формализма диссипативного стандартного отображения. Вводится понятие установившихся траекторий. Получается линейная аппроксимация для простейших установившихся траекторий.

     

Dynamical bifurcation with noise

It was shown by A. Neishtadt that dynamical bifurcation, in which the control parameter is varied with a small but finite speed , is characterized by adelay in bifurcation, here denoted _j and depending on . Here we study dynamical bifurcation, in the framework and with the language of Landau theory of phase transitions, in the presence of a Gaussian noise of strength . By numerical experiments at fixed = ₀, we study the dependence of _j on a for order parameters of dimension 3; an exact scaling relation satisfied by the equations permits us to obtain for this the behavior for general . We find that in the smallnoise regime _j() a(^–b ), while in the strong-noise regime _j() – ce(^–d); we also measure the parameters in these formulas.     

The Hopf bifurcation of twodimensional systems under the influence of one external noise source

The behaviour of the Hopf bifurcation under the influence of external noise is investigated by means of a twodimensional model which uses Gaussian white noise as input. The model includes the case of multiplicative and/or additive noise. Applying the Birkhoff transformation the model is transformed to the coordinates normally used to discuss the deterministic Hopf bifurcation. Then the stationary solution of the model is calculated as an expansion for weak noise: The Hopf bifurcation under the influence of noise exhibits a bifurcation interval with width and position depending on the noise power. Moreover, a class of the systems described by the model can perform noise driven bifurcations.     

Equilibrium-torus bifurcation in nonsmooth systems

Considering a set of two coupled nonautonomous differential equations with discontinuous right-hand sides describing the behavior of a DC/DC power converter, we discuss a border-collision bifurcation that can lead to the birth of a two-dimensional invariant torus from a stable node equilibrium point. We obtain the chart of dynamic modes and show that there is a region of parameter space in which the system has a single stable node equilibrium point. Under variation of the parameters, this equilibrium may disappear as it collides with a discontinuity boundary between two smooth regions in the phase space. The disappearance of the equilibrium point is accompanied by the soft appearance of an unstable focus period-1 orbit surrounded by a resonant or ergodic torus.Detailed numerical calculations are supported by a theoretical investigation of the normal form map that represents the piecewise linear approximation to our system in the neighbourhood of the border. We determine the functional relationships between the parameters of the normal form map and the actual system and illustrate how the normal form theory can predict the bifurcation behaviour along the border-collision equilibrium-torus bifurcation curve.     

Influence of frequency noise on nascent hysteresis in optical bistability

We study the influence of frequency noise on optical bistability in the neighborhood of the critical point where the hysteresis loop appears. We show that when the transmitted field evolves on a faster time scale than that of the noise, the hysteresis loop shifts toward lower values of the incident pumping field.     

Effects of non-Gaussian noise near supercritical Hopf bifurcation

We have studied the effects of non-Gaussian colored noise in a chemical oscillation system, the well-known Brusselator model, in the parameter region close to the supercritical Hopf bifurcation. With the variation of the parameter q, which quantifies the deviation from Gaussian character, the signal-to-noise ratio of noise induced oscillation exhibits a bell-shaped change, indicating the presence of resonant activity. The cooperative effects of q and the correlation time τ on the performance of noise induced oscillation are also investigated. Interestingly, resonance-like behavior can be induced by either q or τ when the other parameter is properly fixed. Stochastic normal form theory is used to analyze these nontrivial effects and the simulation results are well reproduced. This work provides us comprehensive understanding of how non-Gaussian noise influences the dynamics in chemical oscillation systems.     

Using qubits to measure fidelity in mesoscopic systems

We point out the similarities in the definition of the "fidelity" of a quantum system and the generating function determining the full counting statistics of charge transport through a quantum wire and suggest to use flux or charge qubits for their measurement. As an application we use the notion of fidelity within a first-quantized formalism in order to derive new results and insights on the generating function of the full counting statistics.     

Probabilistic metric spaces and hysteresis systems

A phenomenological theory of simple hysteresis is constructed with the aid of certain concepts from the theory of probabilistic metric spaces. The predicted forms of the dependence of average energy loss per hysteresis cycle on the maximum excursion of the hysteresis coordinate agree well with experimental results.     

非平稳性量度及其在齿轮箱噪声分析中的应用

从时变噪声过程的演变矢量描述出发,提出了非平稳性量度Q,同时包括了信号功率的不平稳性和谱形状的不平稳性。作为一个单参数为不同过程间非平稳性的比较,提供了一个可资比较的量。并将非平稳性量度应用到所谓的齿轮箱“杂声”分析中。     

Classification in bifurcation theory and reaction-diffusion systems

The fundamental open problem in bifurcation theory is to determine when linearization and construction are valid. This problem and the problem of matching a critical exponent with a bifurcation point are solved through the application of selection rules. Selection rules are also central for a classification theory which is a natural extension of the above problems. Solutions are classified by four equivalence relations differing in their coarseness. Canonical function bases and bifurcation points are studied. The direction of increased vanishing of integrals involved in existence theorems correspond to various interesting similar order relations, for example to increased phase symmetry and decreased solution symmetry. The symmetries of physical and phase spaces are correlated and the closure of the solutions under the symmetry group is shown and analyzed. The common group theoretical basis for equilibrium and nonequilibrium transitions is emphasized throughout. Thus in both settings the same selection rules determine if the transition is continuous or discontinuous. A theory of symmetry breaking for nonequilibrium-bifurcation systems is described. After discussion of stability and jumps, the theory and the history of potentials in chemical systems far from equilibrium are reviewed from the mathematical, thermodynamic and catastrophe theory points of view. Implications to biological control, morphogenesis and pattern formation are briefly indicated. Throughout, reaction-diffusion is of central importance and it also serves as a carrier for the general ideas in bifurcation theory .     

Use of resonating cavity to measure body volume

A technique to estimate the body volume of newborns has been developed using the principle of the Helmholtz resonator. The change in the resonance frequency of a small cavity after an infant is placed inside can be used to determine the body volume of the infant. A prototype system, termed an acoustic plethysmograph, was built and used to measure the volume of newborn miniature pigs. Results of the animal body volume measurements compare favorably (within an average of 1.1%) with body volumes measured by hydrostatic weighing.     

非线性动力系统分岔图中的暗线

基于非线性动力系统混沌运动的回归特性，构造了一种对分岔图中穿过混沌区的暗线进行研究的数值回归算法。运用该算法求得抛物线映射的暗线，并与通过暗线方程精确求得的暗线进行比较，验证了算法的有效性。对Brussel振子系统和分段线性单级齿轮动力系统的暗线进行了研究。通过对非线性动力系统分岔图中暗线的研究，由其切点可以得到嵌在混沌区中的周期窗口，由其交点可以得到混沌吸引子的激变点。研究结果表明该算法有助于分析系统的动力学行为和控制混沌运动。     

Pareto distribution in dynamical systems subjected to noise perturbation

Possible mechanisms for formation of the Pareto distribution in dynamical systems subjected to noise perturbation are studied. The effect of external noise characteristics on dimensions of the dynamic variable range where the system response can be approximated by this distribution with a good accuracy is discussed. Systems described by specific forms of the double-well potential are examined.     

1/ƒ noise spectrum on self-similar cascade of bifurcation processes

Using the analogy of an electric network we show that a self-similar process of the decay of macroscopic current fluctuation can give a 1/? like noise spectrum.     

Normalized noise energy as an acoustic measure to evaluate pathologic voice

In order to evaluate noise components included in pathologic voice signals, a novel acoustic measure, normalized noise energy (NNE), is proposed and its effectiveness for the detection of laryngeal pathologies is investigated with 250 vowel samples spoken by 64 control (normal) subjects and 186 patients with various laryngeal diseases. The NNE is automatically computed from the voice signals using an adaptive comb filtering method performed in the frequency domain. Experiments with the voice samples show that the NNE is especially effective for detecting glottic cancer, recurrent nerve paralysis, and vocal cord nodules. Specifically, when glottic cancer is represented in terms of the T classification adopted by the UICC (Union Internationale Contre le Cancer), glottic T2-T4 cancer can be perfectly discriminated from normal samples, but 22.6% of patients with glottic T1 cancer are incorrectly classified as normal, with an error rate of 9.4% for normal subjects.