Observation of stochastic resonance near a subcritical bifurcation

A hysteretic Subcritical period-doubling bifurcation is observed in the nonlinear strain dynamics of a magnetostrictive oscillator. The dynamic strain response of the magnetostrictive oscillator was observed with a high-resolution fiber optic interferometer. The effects of low-frequency modulation and band-limited stochastic fluctuations on such a bifurcation are investigated. Power spectral density measurements show that for an optimal value of externally injected noise the signal-to-noise ratio of a low-frequency modulation signal is enhanced by greater than 14 dB, thus indicating the first experimental observation of stochastic resonance near a bistable period-doubling bifurcation.     

Measurement of effective temperatures in an aging colloidal glass

We study the thermal fluctuations of an optically confined probe particle, suspended in an aging colloidal suspension, as the suspension transforms from a viscous liquid into an elastic glass. The micron-sized bead forms a harmonic oscillator. By monitoring the equal-time fluctuations of the tracer, at two different laser powers we determine the temperature of the oscillator, T(o). In the ergodic liquid the temperatures of the oscillator and its environment are equal, while in contrast, in a nonequilibrium glassy phase we find that T(o) substantially exceeds the bath temperature.     

Frequency Switches at Transition Temperature in Voltage-Gated Ion Channel Dynamics of Neural Oscillators

Understanding of the mechanisms of neural phase transitions is crucial for clarifying cognitive processes in the brain. We investigate a neural oscillator that undergoes different bifurcation transitions from the big saddle homoclinic orbit type to the saddle node on an invariant circle type, and the saddle node on an invariant circle type to the small saddle homoclinic orbit type. The bifurcation transitions are accompanied by an increase in thermodynamic temperature that affects the voltage-gated ion channel in the neural oscillator. We show that nonlinear and thermodynamical mechanisms are responsible for different switches of the frequency in the neural oscillator. We report a dynamical role of the phase response curve in switches of the frequency, in terms of slopes of frequency-temperature curve at each bifurcation transition. Adopting the transition state theory of voltagegated ion channel dynamics, we confirm that switches of the frequency occur in the first-order phase transition temperature states and exhibit different features of their potential energy derivatives in the ion channel. Each bifurcation transition also creates a discontinuity in the Arrhenius plot used to compute the time constant of the ion channel.     

Transient chaos in smooth memristor oscillator

This paper presents a new smooth memristor oscillator, which is derived from Chua's oscillator by replacing Chua's diode with a flux-controlled memristor and a negative conductance. Novel parameters and initial conditions are dependent upon dynamical behaviours such as transient chaos and stable chaos with an intermittence period and are found in the smooth memristor oscillator. By using dynamical analysis approaches including time series, phase portraits and bifurcation diagrams, the dynamical behaviours of the proposed memristor oscillator are effectively investigated in this paper.     

一类耦合非线性振子同步混沌Hopf分岔及其电路仿真

对于一类同时存在扩散耦合和梯度耦合的非线性振子系统, 通过空间傅里叶变换，得到具有不同波矢的各运动模式的相互独立的运动方程. 计算各横截模的Lyapunov指数, 可在耦合参数平面上确定同步混沌的稳定区域. 在稳定区域边界, 一对共轭横截模式失稳，导致同步混沌的Hopf分岔. 对耦合Lorenz振子系统进行了数值模拟，并设计了耦合Lorenz振子系统的电路, 进行耦合振子系统同步混沌Hopf分岔的电路仿真实验. 计算和仿真的结果表明，Hopf分岔的特征频率等于失稳横截模式的振荡频率. 关键词： 耦合非线性振子 同步混沌 横截模式 电路仿真     

Infinitely many coexisting attractors of a dual memristive Shinriki oscillator and its FPGA digital implementation

In this paper, a novel chaotic oscillator is proposed, which is derived from the classical Shinriki oscillator by substituting the series-parallel diode loop with a flux-controlled memristor and connecting an active charge-controlled memristor in series with an inductor. The mathematical model of the circuit is established, and the stability distribution maps of three non-zero eigenvalues in the equilibrium plane are obtained. The basic dynamical behaviors depending on the variation of the circuit parameters and memristor initial conditions are investigated by standard nonlinear analysis tools, such as bifurcation diagrams, Lyapunov exponents and phase portraits. Particularly, some striking phenomena, including the routes to double-scroll chaotic attractors, coexisting periodic-chaotic bubbles and asymmetric coexisting behaviors are observed. Furthermore, extreme multistability of the new oscillator is revealed by attraction basins under the initial condition of different dynamic elements. Finally, the Shinriki oscillator with two memristors is realized through Field-Programmable Gate Array (i.e., FPGA) to verify the effectiveness of the numerical simulations.     

Experimental bifurcations and chaos in a modified self-sustained macro electromechanical system

A class of self-sustained Macro ElectroMechanical (MaEMS) Systems is made up of a Rayleigh-Duffing oscillator actuating a mechanical arm through a magnetic coupling. In this paper, to avoid experimental constraints, an audio amplifier is added to the device. Quenching phenomenon, bifurcation and chaos are predicted and shown to occur in a device of this class of MaEMS. Especially by using linear stability analysis, the condition for the quenching phenomenon is derived. Chaos and bifurcation are predicted using Lyapunov exponent and bifurcation diagram. A prototype of device is designed and fabricated. Experimental results for this device that are consistent with results from theoretical investigations are presented and convincingly show quenching phenomenon, bifurcation and chaos.     

Bursting phenomena as well as the bifurcation mechanism in a coupled BVP oscillator with periodic excitation

We explore the complicated bursting oscillations as well as the mechanism in a high-dimensional dynamical system.By introducing a periodically changed electrical power source in a coupled BVP oscillator, a fifth-order vector field with two scales in frequency domain is established when an order gap exists between the natural frequency and the exciting frequency.Upon the analysis of the generalized autonomous system, bifurcation sets are derived, which divide the parameter space into several regions associated with different types of dynamical behaviors. Two typical cases are focused on as examples,in which different types of bursting oscillations such as sub Hopf/sub Hopf burster, sub Hopf/fold-cycle burster, and doublefold/fold burster can be observed. By employing the transformed phase portraits, the bifurcation mechanism of the bursting oscillations is presented, which reveals that different bifurcations occurring at the transition between the quiescent states(QSs) and the repetitive spiking states(SPs) may result in different forms of bursting oscillations. Furthermore, because of the inertia of the movement, delay may exist between the locations of the bifurcation points on the trajectory and the bifurcation points obtained theoretically.     

Flicker Noise in a Locally Nonequilibrium Medium

A model implying that particles of a medium are subjected to external forces with a flicker noise spectrum has been proposed to describe fluctuations in locally nonequilibrium physical media. The Langevin equation with additional integral terms describing the action of the locally nonequilibrium medium has been derived. The spectral density of fluctuations of an electric current flowing in the locally nonequilibrium medium has been calculated. It has been found that this density in the low-frequency spectral range is flicker noise.     

Stable nonequilibrium probability densities and phase transitions for meanfield models in the thermodynamic limit

A nonlinear Fokker-Planck equation is derived to describe the cooperative behavior of general stochastic systems interacting via mean-field couplings, in the limit of an infinite number of such systems. Disordered systems are also considered. In the weak-noise limit; a general result yields the possibility of having bifurcations from stationary solutions of the nonlinear Fokker-Planck equation into stable time-dependent solutions. The latter are interpreted as non-equilibrium probability distributions (states), and the bifurcations to them as nonequilibrium phase transitions. In the thermodynamic limit, results for three models are given for illustrative purposes. A model of self-synchronization of nonlinear oscillators presents a Hopf bifurcation to a time-periodic probability density, which can be analyzed for any value of the noise. The effects of disorder are illustrated by a simplified version of the Sompolinsky-Zippelius model of spin-glasses. Finally, results for the Fukuyama-Lee-Fisher model of charge-density waves are given. A singular perturbation analysis shows that the depinning transition is a bifurcation problem modified by the disorder noise due to impurities. Far from the bifurcation point, the CDW is either pinned or free, obeying (to leading order) the Grüner-Zawadowki-Chaikin equation. Near the bifurcation, the disorder noise drastically modifies the pattern, giving a quenched average of the CDW current which is constant. Critical exponents are found to depend on the noise, and they are larger than Fisher's values for the two probability distributions considered.     

Period-doubling cascades of canards from the extended Bonhoeffer-van der Pol oscillator

This Letter investigates the period-doubling cascades of canards, generated in the extended Bonhoeffer-van der Pol oscillator. Canards appear by Andronov-Hopf bifurcations (AHBs) and it is confirmed that these AHBs are always supercritical in our system. The cascades of period-doubling bifurcation are followed by mixed-mode oscillations. The detailed two-parameter bifurcation diagrams are derived, and it is clarified that the period-doubling bifurcations arise from a narrow parameter value range at which the original canard in the non-extended equation is observed.     

Inhomogeneous Glauber dynamics and the process of crystallization of a lattice gas

The model under consideration is a hard-core lattice gas in an external potential on a Bethe lattice with nonequilibrium time evolution governed by Glauber dynamics. A hierarchical decoupling of nonequilibrium correlations, motivated by and asymptotically providing the exact form of equilibrium multisite correlations in the inhomogeneous potential regime, is proposed. Application is made to the process of lattice gas crystallization, at high activity, from a spatially homogeneous fluid phase to an equilibrium crystal phase with unequal sublattice densities. The first few levels of the hierarchical decoupling give a consistent picture of two kinds of nonequilibrium instabilities—one leading to a sublattice density bifurcation, the other associated with an abrupt increase in densities and correlations in time.     

Bifurcations in a system of two identical diffusion-coupled relaxational brusselators

We propose a two-parameter bifurcation analysis of the dynamics of a system of two identical asymmetrically coupled Brusselators. The stability boundaries of the inhomogeneous steady states and periodic attractors are calculated as the functions of the constraint force and one of the free parameters. The coexistence of different attractors giving rise to multirhythmicity of the dynamics is studied as well as the bifurcation transitions between them. It is shown that the relaxation ability of an oscillator plays an important role in the simplification of the phase diagram, since it removes the overlapping of the existence region for different solutions. We assume that the results primarily characterize the properties of the diffusion coupling and, therefore, they can be applied in the study of other oscillator systems.P. N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 5, pp. 373–401, May, 1995.     

Semiclassical shell-structure moment of inertia for equilibrium rotation of a simple Fermi system

Semiclassical shell-structure components of the collectivemoment of inertia are derived within the mean-field cranking model in the adiabatic approximation in terms of the free-energy shell corrections through those of a rigid body for the statistically equilibriumrotation of a Fermi system at finite temperature by using the nonperturbative extended Gutzwiller periodic-orbit theory. Their analytical structure in terms of the equatorial and 3-dimensional periodic orbits for the axially symmetric harmonic oscillator potential is in perfect agreement with the quantum results for different critical bifurcation deformations and different temperatures.     

Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach

This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron–phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.     

Calculations of the parameters of the air plasma produced by an electron beam in the channel of an MHD generator with a nonequilibrium conductivity

Approximate analytic expressions for calculating the electron density in both steady and unsteady plasmas produced by pulsed electron beams are derived and proved to agree well with numerical calculations. It is shown that the algorithm for calculating the parameters of a nonequilibrium plasma in the channel of an MHD plasma generator depends on the type of generator. The effect of the magnetic field strength on the electron density and electric conductivity of the air plasma produced by an electron beam in the channel of a Faraday MHD generator is investigated. The influence of the parameters of the flow and ionizer on the efficiency of an MHD generator with a nonequilibrium conductivity is analyzed.     

A general mass law for broadband energy harvesting

It is well known that the power absorbed by a linear oscillator when excited by white noise base acceleration depends only on the mass of the oscillator and the spectral density of the base motion. This places an upper bound on the energy that can be harvested from a linear oscillator under broadband excitation, regardless of the stiffness of the system or the damping factor. It is shown here that the same result applies to any multi-degree-of-freedom nonlinear system that is subjected to white noise base acceleration: for a given spectral density of base motion the total power absorbed is proportional to the total mass of the system. The only restriction to this result is that the internal forces are assumed to be a function of the instantaneous value of the state vector. The result is derived analytically by several different approaches, and numerical results are presented for an example two-degree-of-freedom-system with various combinations of linear and nonlinear damping and stiffness.     

STOCHASTIC AVERAGING OF STRONGLY NON-LINEAR OSCILLATORS UNDER BOUNDED NOISE EXCITATION

A stochastic averaging method for strongly non-linear oscillators under external and/or parametric excitation of bounded noise is proposed by using the so-called generalized harmonics functions. The method is then applied to study the primary resonance of Duffing oscillator with hardening spring under external excitation of bounded noise. The stochastic jump and its bifurcation of the system are observed and explained by using the stationary probability density of amplitude and phase. Subsequently, the method is applied to study the dynamical instability and parametric resonance of Duffing oscillator with hardening spring under parametric excitation of bounded noise. The primary unstable region is delineated by evaluating the Lyapunov exponent of linearized system, and the response and jump of non-linear system around the unstable region are examined by using the sample functions and stationary probability density of amplitude and phase.     

Properties of synchronization in the systems of non-identical coupled van der Pol and van der Pol-Duffing oscillators. Broadband synchronization

The particular properties of dynamics are discussed for dissipatively coupled van der Pol oscillators, non-identical in values of parameters controlling the Hopf bifurcation. Possibility of a special synchronization regime in an infinitively long band between oscillator death and quasiperiodic areas is shown for such system. Features of the bifurcation picture are discussed for different values of the control parameters and for the case of additional Duffing-type nonlinearity. Analysis of the slow-flow equations is presented.     

Lévy稳定噪声激励下的Duffing-van der Pol振子的随机分岔

研究了Lévy稳定噪声激励下的双稳Duffing-van der Pol振子,利用Monte Carlo方法,得到了振幅的稳态概率密度函数.分析了Lévy稳定噪声的强度和稳定指数对概率密度函数的影响,通过稳态概率密度的性质变化,讨论了噪声振子的随机分岔现象,发现了不仅系统参数和噪声强度可以视为分岔参数,Lévy噪声的稳定指数 α 的改变也能诱导系统出现随机分岔现象. 关键词： Lévy稳定噪声 Duffing-van der Pol振子 稳态概率密度函数 随机分岔