Deterministic Oscillations Research Articles

Emergent seesaw oscillations during cellular directional decision-making

Motile cells inside living tissues often encounter junctions, where their path branches into several alternative directions of migration. We present a theoretical model of cellular polarization for a cell migrating along a one-dimensional line, arriving at a symmetric Y junction and extending protrusions along the different paths that originate at the junction. The model predicts the spontaneous emergence of deterministic oscillations of growth and cellular polarization between competing protrusions during the directional decision-making process. The oscillations are modified by cellular noise but remain a dominant feature that affects the time it takes the cell to migrate across the junction. These predictions are confirmed experimentally for two different cell types (non-cancerous endothelial and cancerous glioma cells) migrating on a patterned network of thin adhesive lanes with junctions.

Kinematically excited beam vibrations under random perturbations

There is numerous information in the technical and scientific literature that often the sources of forced kinematically excited oscillations have a clearly expressed random nature. Kinematically excited vibrations of beams are considered. Their sources are random disturbances. Steady (in a probabilistic sense) vibrations of beams are studied. As a consequence, the transverse vibrations of the beam will also be random, and it becomes necessary and expedient to switch to stochastic motion models.
 Introduction. For a random perturbation process, a spectral matrix is specified. Initial conditions for the equations are not required. The boundary conditions are similar to those used for free and kinematically excited deterministic oscillations. The task is to find the spectral matrix of the random field of beam deviations and dispersion for a given spectral matrix. The question of determining the mathematical expectation is not raised. It is argued that it can be easily reduced to known deterministic problems.
 Methods. To carry out specific calculations, a model of kinematic disturbances is used in the form of a vector N-dimensional stationary random process with stationary connected components that have hidden periodicities (characteristic frequencies). A test example was completed to determine the dispersion and standard deviation of displacements. The numerical integration step was adopted after numerical experiments. When performing calculations, the parity of the integrand was taken into account. The lower limit of integration is taken to be zero, followed by doubling of the final result.
 Results. For small values of the broadband parameter, the results of the stochastic and deterministic problems differ little. Examples are presented that are stochastic analogs of harmonic oscillations. For beams under kinematic harmonic disturbances, test examples were performed to determine dispersions and standard deviations.
 The discussion of the results. The results of calculations in the article are presented by curves having numbers that coincide with the numbers of the matrices. The first matrix corresponds to the absolute correlation of disturbances. The second matrix corresponds to the absolute correlation between the first and third, second and fourth disturbances, while these pairs are absolutely uncorrelated. The third matrix describes random perturbations when the indicated pairs, being perfectly positively correlated within, are absolutely perfectly negatively correlated between pairs. The fourth curve corresponds to the case of ideal correlation of the first three disturbances with each other, while the movements of the fourth support are in antiphase with them. In the latter case, the movements of the outer supports are in antiphase with the movements of the middle ones, which favors the greatest bending of the beam. Therefore, the elastic line has a curvature that is significantly greater than the others.

Methods of periodically non-stationary random processes for vibrations monitoring of rolling bearing with damaged outer race

The vibrations of a rolling bearing with a damaged outer race are analyzed using a method based on periodically non-stationary random processes (PNRPs). PNRPs can be considered as a particular case of poly-periodically non-stationary random processes (PPNRs), which describe the interactions of vibration stochastic rhythms with different periods that are determined based on the characteristic bearing frequencies. A PNRP model for these vibrations is verified using methods for discovering the first- and second-order periodicities. By applying the least squares technique, it is shown that the mean function and variance are time periodic functions, and their basic frequencies are determined. This enables us to calculate the amplitude spectra for the deterministic oscillations and the time changes in the power for the stochastic part. It is shown that the dependencies of the Fourier coefficients for the covariance function on lag have the forms of slowly damped groups. Vibration covariance structures are compared for different fault widths; it is found that high-frequency modulations of the PNRP carrier modulations are narrow-band, and Rice representations are used to model them. The PRNP model of vibrations is then represented in the form of a superposition of the stationary high-frequency components that are jointly periodically non-stationary. The mutual periodical non-stationarity of these components results in variance time changes. Band-pass filtering of the vibration signal is conducted, and the influence of the filter bandwidth on the number and values of the amplitude of the variance harmonics is analyzed. It is shown that PNRP methods for time series processing enable fault detection in cases where the squared envelope and other familiar techniques are ineffective.

On the One‐Dimensional Transition State Theory and the Relation between Statistical and Deterministic Oscillation Frequencies of Anharmonic Energy Wells

AbstractThe transition state theory allows the development of approximated models useful to study the non‐equilibrium evolution of systems undergoing transformations between two states (e.g., chemical reactions). In a simplified 1D setting, the characteristic rate constants are typically written in terms of a temperature‐dependent characteristic oscillation frequency , describing the exploration of the phase space. As a particular case, this statistical oscillation frequency can be defined for an arbitrary convex potential energy well. This value is compared here with the deterministic oscillation frequency of the corresponding anharmonic oscillator. It is proved that there is a universal relationship between statistical and deterministic frequencies, which is the same for classical and relativistic mechanics. The independence of this relationship from the adopted physical laws gives it an interesting thermodynamic and pedagogical meaning. Several examples clarify the meaning of this relationship from both physical and mathematical viewpoints.

Distinguishing between deterministic oscillations and noise

AbstractTime-dependent dynamics is ubiquitous in the natural world and beyond. Effectively analysing its presence in data is essential to our ability to understand the systems from which it is recorded. However, the traditional framework for dynamics analysis is in terms of time-independent dynamical systems and long-term statistics, as opposed to the explicit tracking over time of time-localised dynamical behaviour. We review commonly used analysis techniques based on this traditional statistical framework—such as the autocorrelation function, power-spectral density, and multiscale sample entropy—and contrast to an alternative framework in terms of finite-time dynamics of networks of time-dependent cyclic processes. In time-independent systems, the net effect of a large number of individually intractable contributions may be considered as noise; we show that time-dependent oscillator systems with only a small number of contributions may appear noise-like when analysed according to the traditional framework using power-spectral density estimation. However, methods characteristic of the time-dependent finite-time-dynamics framework, such as the wavelet transform and wavelet bispectrum, are able to identify the determinism and provide crucial information about the analysed system. Finally, we compare these two frameworks for three sets of experimental data. We demonstrate that while techniques based on the traditional framework are unable to reliably detect and understand underlying time-dependent dynamics, the alternative framework identifies deterministic oscillations and interactions.

Detection of distributed and localized faults in rotating machines using periodically non-stationary covariance analysis of vibrations

The covariance structure for vibrations of the noising mechanism for a port crane and a wind turbine with gear tooth breakage are studied on the basis of their models in the form of periodically non-stationary random processes (PNRPs). The investigation is performed by the authors using PNRP statistical analysis methods. The estimations of the basic frequencies of the vibration’s deterministic components and the time changes of the power for the stochastic part (the PNRP variance) are the key issues of this analysis. In the present paper, the basic frequency estimators are obtained using the least squares technique which are characterized by the maximum of the signal-to-noise ratio as compared with techniques known in the literature. Using the basic frequency, values for the Fourier coefficients for the mean and covariance functions are calculated and the dependencies of the forenamed moment functions on time and lag are ascertained. The amplitude spectra of the deterministic oscillations and the time changes of the power of the stochastic part are presented. These spectra are considered as the symptomatic features of the mechanism’s state of health. It is revealed that the high-frequency modulation of PNRP carrier harmonics is narrow-band for both distributed and localized faults. Using bandpass filtering and Hilbert transform, the quadratures for each high-frequency component are separated and analyzed. It is shown that the auto- and cross-covariance structure of the high-frequency component quadratures characterizes the specific features of the different types of faults. Using the cross-covariance map for high-frequency components, the peculiarities of the bandpass filtering of vibrations for a damaged wind turbine’s gearbox as a PNRP’s is considered. It is shown that the filtering bandwidth cannot be narrower than 14 times the basic frequency. This bandwidth essentially exceeds that recommended in the literature (two to three times the basic frequency).

Exact potentials in multivariate Langevin equations.

Systems governed by a multivariate Langevin equation featuring an exact potential exhibit straightforward dynamics but are often difficult to recognize because, after a general coordinate change, the gradient flow becomes obscured by the Jacobian matrix of the mapping. In this work, a detailed analysis of the transformation rules for Langevin equations under general nonlinear mappings is presented. We show how to identify systems with exact potentials by understanding their differential-geometric properties. To demonstrate the power of our method, we use it to derive exact potentials for broadly studied models of nonlinear deterministic and stochastic oscillations. In selected examples, we visualize the identified potentials. Our results imply a broad class of exactly solvable stochastic models, which can be self-consistently defined from given deterministic gradient systems.

Exploring the mechanics of fundamental frequency variation during phonation onset.

Fundamental frequency patterns during phonation onset have received renewed interest due to their promising application in objective classification of normal and pathological voices. However, the associated underlying mechanisms producing the wide array of patterns observed in different phonetic contexts are not yet fully understood. Herein, we employ theoretical and numerical analyses in an effort to elucidate the potential mechanisms driving opposing frequency patterns for initial/isolated vowels versus vowels preceded by voiceless consonants. Utilizing deterministic lumped-mass oscillator models of the vocal folds, we systematically explore the roles of collision and muscle activation in the dynamics of phonation onset. We find that an increasing trend in fundamental frequency, as observed for initial/isolated vowels, arises naturally through a progressive increase in system stiffness as collision intensifies as onset progresses, without the need for time-varying vocal fold tension or changes in aerodynamic loading. In contrast, reduction in cricothyroid muscle activation during onset is required to generate the decrease in fundamental frequency observed for vowels preceded by voiceless consonants. For such phonetic contexts, our analysis shows that the magnitude of reduction in the cricothyroid muscle activation and the activation level of the thyroarytenoid muscle are potential factors underlying observed differences in (relative) fundamental frequency between speakers with healthy and hyperfunctional voices. This work highlights the roles of sometimes competing laryngeal factors in producing the complex array of observed fundamental frequency patterns during phonation onset.

Optimization of automatic power control of pulsed reactor IBR-2M in the presence of instability

The paper presents the main results of computational and experimental optimization of the automatic power control system (AС) of the IBR-2M pulsed reactor in the presence of a high level of oscillatory instability. Optimization of the parameters of the AС made it possible to significantly reduce the influence of random and deterministic oscillations of reactivity on the noise of the pulse energy, as well as to sharply reduce the manifestation of the oscillatory instability of the reactor. As a result, the safety and reliability of operation of the reactor has increased substantially.

Chaos and Its Degradation-Promoting-Based Control in an Antithetic Integral Feedback Circuit

This letter deals with a novel variant of antithetic integral feedback controller (AIFC) motifs which can feature robust perfect adaptation, a pervasive (desired) ability in natural (synthetic) biomolecular circuits, when coupled with a wide class of process networks to be regulated. Using the separation of time-scales in the proposed kind of AIFC, here we find a reduced-order controller that captures the governing slow part of the original solutions under suitable assumptions. Inspired by Rössler systems, we then make use of such a simpler controller to show that the antithetic circuit can exhibit chaotic behaviors with strange attractors, where the bifurcation from a homeostatic state to chaotic orbits can happen, e.g., when considering saturated Hill-type reactions for the actuation. Addition of degradation terms to the controller species, whether naturally due to dilution or exogenously using protein tags, is showcased by simulation results to be an effective solution in suppressing deterministic chaos and aperiodic oscillations. In the same vein, we recapitulate the recently introduced antithetic rein IFC motif and confirm that the promotion of degradation by a rein mechanism also can control chaos and improve the stability of closed-loop circuit.

Canonical Hyperchaotic Oscillators With Single Generalized Transistor and Generative Two-Terminal Elements

This paper describes systematic design process toward third-order autonomous deterministic hyperchaotic oscillators with two coupled generative two-terminal elements. These active devices represent alternative to conventional dissipative accumulation elements such as capacitors and inductors. Analyzed network structures contain generalized bipolar transistor as only active element. Three different networks are studied, depending on common-electrode configurations. In each case, transistor is modelled using two-port admittance parameters with the non-zero linear backward and polynomial forward trans-conductance. As proved in paper, scalar nonlinearity caused by amplification property of transistor can push circuit into chaotic and, more interestingly, hyperchaotic steady states. Existence of parameter spaces leading to robust chaotic and hyperchaotic solution is documented by using concept of one-dimensional Lyapunov exponents and colored high-resolution surface-contour plots of two largest numbers. Geometrical structural stability of generated strange attractors is proved via construction of the flow-equivalent chaotic oscillator and real measurement. Plane projections of interesting observed attractors are captured by oscilloscope.

Evaluation of the mechanism damage using model of vibration signal as a periodically correlated random process

The results of the processing of the wind turbine gearbox vibration signals are presented. The least square estimators of the periods of the deterministic part of the vibration signal and time variations power of stochastic part are obtained. The amplitude spectra of the deterministic and stochastic oscillations for different degrees of the gearbox damage are analyzed. The covariance structure of stochastic part of vibration was analyzed. The efficient indicator for practical evaluation of gear teeth damage, formed on the basis of amplitude spectra was proposed.

Methods of Hidden Periodicity Discovering for Gearbox Fault Detection.

It is shown that the models of gear pair vibration, proposed in literature, are particular cases of the bi-periodically correlated random processes (BPCRPs), which describe its stochastic recurrence with two periods. The possibility of vibration and analysis within the framework of BPCRP approximation, in the form of periodically correlated random processes (PCRPs), is grounded and the implementation of vibration processing procedures using PCRP techniques, which are worked out by the authors, is given. Searching for hidden periodicities of the first and the second orders was considered as the main issue of this approach. The estimation of the non-stationary period (basic frequency) allowed us to carry out a detailed analysis of the deterministic part, the covariance structure of the stochastic part, and to form, using their parameters, the sensitive indicators for fault detection. The results of the processing of the wind turbine gearbox vibration signals are presented. The amplitude spectra of the deterministic oscillations and the time changes of the stochastic part power for different fault stages are analyzed. The most efficient indicators, which are formed using the amplitude spectra for practical applications, are proposed. The presented approach was compared with known in literature cyclostationary analysis and envelope techniques, and its advantages are shown.

Asymptotic Phase and Amplitude for Classical and Semiclassical Stochastic Oscillators via Koopman Operator Theory

The asymptotic phase is a fundamental quantity for the analysis of deterministic limit-cycle oscillators, and generalized definitions of the asymptotic phase for stochastic oscillators have also been proposed. In this article, we show that the asymptotic phase and also amplitude can be defined for classical and semiclassical stochastic oscillators in a natural and unified manner by using the eigenfunctions of the Koopman operator of the system. We show that the proposed definition gives appropriate values of the phase and amplitude for strongly stochastic limit-cycle oscillators, excitable systems undergoing noise-induced oscillations, and also for quantum limit-cycle oscillators in the semiclassical regime.

Forced harmonic oscillators, waves on a forced string and changes of measure

There is a connection between deterministic Gaussian processes, that is, flows that leave a Gaussian measure invariant and waves on a string. The connection appears when the solution to the wave equation is expanded in the eigenvalues of the Laplacian. The Fourier components behave as one dimensional harmonic oscillators. It so happens that if the initial conditions of each deterministic oscillator have an appropriate Gaussian distribution, then the motion of the oscillator in its phase space leaves it invariant.When we consider oscillations in a string that is subject to an external force, another interesting connection appears. Fourier components can be thought of as forced harmonic oscillators. The motion of each oscillator does not generate a time homogeneous flow in its phase space, but if the initial Gaussian measure is changed in time by means of an analogue of the Cameron–Martin-Girsanov–Maruyama transformation, the time flow of the forced oscillator preserves the initial Gaussian measure.It is the aim of this work to make this construction explicit.

Controlling oscillator coherence by multiple delay feedback

We consider the implementation of a weak feedback with two delay times for controlling the coherence of both deterministic chaotic and stochastic oscillators. This control strategy is revealed to allow one to decrease or enhance the coherence, which is quantified by the phase diffusion constant, by 2–3 orders of magnitude without destruction of the chaotic regime, which is by an order of magnitude more than one can achieve with a single delay time. Within the framework of the phase reduction, which is a rough approximation for the chaotic oscillators and rigorous for the stochastic ones, an analytical theory of the effect is constructed.

Stochastic Bifurcations, Chaos and Phantom Attractors in the Langford System with Tori

The variability of stochastic dynamics for a three-dimensional dynamic model in a parametric zone with 2-tori is investigated. It is shown how weak Gaussian noise transforms deterministic quasiperiodic oscillations into noisy bursting. The phenomenon of stochastic generation of a phantom attractor and its shift with noise amplification is revealed. This phenomenon, accompanied by order-chaos transitions, is studied in terms of stochastic [Formula: see text]- and [Formula: see text]-bifurcations.

Phase sensitivity for coherence resonance oscillators

The phase reduction approach has manifested its power in analyzing the rhythmic behaviors for limit cycle oscillators. For coherent oscillation purely induced by noise, e.g., the coherence resonance oscillator, the stochastic dynamics exhibit almost deterministic limit cycle phenomenon which inspires the application of the phase reduction approach to this kind of systems. In this paper, the FitzHugh–Nagumo system in coherence resonance is modeled as a jump process. The phase sensitivity is obtained by applying the phase reduction approach for the hybrid system. A modified direct method is proposed to compare the theoretical results with those of Monte Carlo simulation for the stochastic system, which shows a relatively good agreement for the perturbation not being too small. The phase reduction results of the coherent oscillators are applied to two coupled FitzHugh–Nagumo neurons. It is interesting that the phase difference of the coupled coherent oscillators does not converge as the deterministic oscillators, but forms a distribution where the peaks and valleys correspond to the stable and unstable synchronizations predicted by the phase coupling functions. The idea in this paper could be applied to other coherent oscillators where the stochastic dynamics follow almost deterministic paths.

An Approximate Solution for a Nonlinear Duffing – Harmonic Oscillator

The equivalent linearization method introduced by Caughey is a powerful tool for analyzing random oscillations. The method is also easy to apply for deterministic oscillations. However, with strong nonlinear systems, the error of this method is usually quite large and even not acceptable. In conjunction with a weighted averaging, the equivalent linearization method has shown more accuracy than the classical one in which the conventional averaging value is used. Combining advantages of the classical equivalent linearization method and accuracy of the weighted averaging, the proposed method has shown that it is a useful tool for analyzing nonlinear oscillations including strong nonlinear systems. In this paper, the proposed method is applied to analyze a nonlinear Duffing – harmonic oscillator. The present results are compared with the results obtained by using other analytical methods, exact results and numerical results.

A hybrid Finite Element-Statistical Energy Analysis method for impulsive and transient loading

The hybrid approach coupling Statistical Energy Analysis (SEA) and the finite element method has become a prominent technique for analysing structures under steady-state loads in the ‘mid-frequency’ range where some components behave in a deterministic manner with low modal density and others in a statistical manner with high modal density and statistical overlap. In this paper, the method is extended from its current steady-state capability to provide calculation of structural responses under impulsive and time-varying loads. Similar to the steady-state method, a system is split into deterministic components that are modelled using the finite element approach and statistical components that are modelled as SEA subsystems.An evolutionary spectrum approach based on the Priestley description of random processes is applied to model the response of both the SEA and deterministic components which are coupled by considering a power balance between the SEA subsystems. The diffuse field reciprocity relationship that relates the reverberant forces generated by a subsystem to the energy within it is explored under transient conditions where it is found that it can be important to account for the build-up of a reverberant field following an impulse. Results from the method are compared against finite element simulations for a system of plates coupled by a beam and it is found to generate predictions with the accuracy expected of an SEA-based method, although deterministic oscillations of the deterministic system at early times are not accounted for.