DISSIPATIVE OSCILLATORS’ POWER CHARACTERISTIC NON-SYMMETRY DYNAMIC EFFECT

The features of motion of a non-linear oscillator under the instantaneous force pulse loading are studied. The elastic characteristic of the oscillator is given by a polygonal chain consisting of two linear segments. The focus of the paper is on the influence of the dissipative forces on the possibility of occurrence of the elastic characteristic non-symmetry dynamic effect, studied previously without taking into account the influence of these forces. Four types of drag forces are considered, namely linear viscous friction, Coulomb dry friction, position friction, and quadratic viscous resistance. For the cases of linear viscous friction and Coulomb dry friction the analytical solutions of the differential equation of oscillations are found by the fitting method and the formulae for computing the swings are derived. The conditions on the parameters of the problem are determined for which the elastic characteristic non-symmetry dynamic effect occurs in the system. The conditions for the effect to occur in the system with the position friction are derived from the energy relations without solving the differential equation of motion. In the case of quadratic viscous friction the first integral of the differential equation of motion is given by the Lambert function of either positive or negative argument depending on the value of the initial velocity. The elastic characteristic non-symmetry dynamic effect is shown to occur for small initial velocities, whereas it is absent from the system when the initial velocities are sufficiently large. The values of the Lambert function are proposed to be computed by either linear interpolation of the known data or approximation of the Lambert function by elementary functions using asymptotic formulae which approximation error is less than 1%. The theoretical study presented in the paper is followed up by computational examples. The results of the computations by the formulae proposed in the paper are shown to be in perfect agreement with the results of numerical integration of the differential equation of motion of the oscillator using a computer.

Arnold tongue entrainment reveals dynamical principles of the embryonic segmentation clock

Living systems exhibit an unmatched complexity, due to countless and entangled interactions across scales. Here we aim to understand and gain control of a complex system, such as the segmentation timing of a developing mouse embryo, without a reference to these detailed interactions. To this end, we develop a coarse-grained approach in which theory guides the experimental identification of the system-level responses to entrainment, in the context of a network of coupled cellular oscillators that constitute the embryonic somite segmentation clock. We demonstrate period- and phase-locking of the embryonic system across a wide range of entrainment parameters, including higher-order coupling. These experimental quantifications allow to derive the phase response curve (PRC) and Arnold tongues of the system, revealing the essential dynamical properties of the embryonic segmentation clock. Our results indicate that at the macro-scale, the somite segmentation clock has characteristics of a highly non-linear oscillator close to a saddle-node on invariant cycle (SNIC) bifurcation and suggests the presence of long-term feedbacks. Combined, this coarse-grained theoretical-experimental approach reveals how we can derive simple, essential features of a highly complex dynamical system and hereby provides precise experimental control over the pace and rhythm of the somite segmentation clock.

Irreversibility in dynamical phases and transitions

Living and non-living active matter consumes energy at the microscopic scale to drive emergent, macroscopic behavior including traveling waves and coherent oscillations. Recent work has characterized non-equilibrium systems by their total energy dissipation, but little has been said about how dissipation manifests in distinct spatiotemporal patterns. We introduce a measure of irreversibility we term the entropy production factor to quantify how time reversal symmetry is broken in field theories across scales. We use this scalar, dimensionless function to characterize a dynamical phase transition in simulations of the Brusselator, a prototypical biochemically motivated non-linear oscillator. We measure the total energetic cost of establishing synchronized biochemical oscillations while simultaneously quantifying the distribution of irreversibility across spatiotemporal frequencies.

On the complex interplay between spectral harmonicity and different types of cross frequency couplings in non linear oscillators and biologically plausible neural network models

BackgroundCross-frequency coupling (CFC) refers to the non linear interaction between oscillations in different frequency bands, and it is a rather ubiquitous phenomenon that has been observed in a variety of physical and biophysical systems. In particular, the coupling between the phase of slow oscillations and the amplitude of fast oscillations, referred as phase-amplitude coupling (PAC), has been intensively explored in the brain activity recorded from animals and humans. However, the interpretation of these CFC patterns remains challenging since harmonic spectral correlations characterizing non sinusoidal oscillatory dynamics can act as a confounding factor.MethodsSpecialized signal processing techniques are proposed to address the complex interplay between spectral harmonicity and different types of CFC, not restricted only to PAC. For this, we provide an in-depth characterization of the Time Locked Index (TLI) as a novel tool aimed to efficiently quantify the harmonic content of noisy time series. It is shown that the proposed TLI measure is more robust and outperform traditional phase coherence metrics (e.g. Phase Locking Value, Pairwise Phase Consistency) in several aspects.ResultsWe found that a non linear oscillator under the effect of additive noise can produce spurious CFC with low spectral harmonic content. On the other hand, two coupled oscillatory dynamics with independent fundamental frequencies can produce true CFC with high spectral harmonic content via a rectification mechanism or other post-interaction nonlinear processing mechanisms. These results reveal a complex interplay between CFC and harmonicity emerging in the dynamics of biologically plausible neural network models and more generic non linear and parametric oscillators.ConclusionsWe show that, contrary to what is usually assumed in the literature, the high harmonic content observed in non sinusoidal oscillatory dynamics, is neither sufficient nor necessary condition to interpret the associated CFC patterns as epiphenomenal. There is mounting evidence suggesting that the combination of multimodal recordings, specialized signal processing techniques and theoretical modeling is becoming a required step to completely understand CFC patterns observed in oscillatory rich dynamics of physical and biophysical systems.HighlightsTime locked index efficiently quantifies the harmonic content of noisy time series.A non linear oscillator under the effect of additive noise can produce spurious cross frequency couplings (CFC) with low spectral harmonic content.Two coupled oscillatory dynamics with independent fundamental frequencies can produce true CFC with high spectral harmonic content via rectification mechanisms or other post-interaction nonlinear processing mechanisms.A non sinusoidal oscillatory dynamics with high harmonic content is neither sufficient nor necessary condition for spurious CFC.A complex interplay between CFC and harmonicity emerges from the dynamics of nonlinear, parametric and biologically plausible oscillators.

On the analytical approximation of the quadratic non-linear oscillator by modified extended iteration method

A modified analytical solution of the quadratic non-linear oscillator has been obtained based on an extended iteration method. In this study, truncated Fourier terms have been used in each step of iterations. The frequencies obtained by this technique show good agreement with the exact frequency. The percentage of error between the exact frequency and our third approximate frequency is as low as 0.001%. There is no algebraic complexity in our calculation, which is why this technique is very easy. The results have been compared with the exact and other existing results, which are both convergent and consistent.

Earth-like planetary magnetotails as non-linear oscillators

A non-linear oscillator model of a simple system analogous to Earth-like magnetotail plasmoid formation and release dynamics is presented. In this context, Earth-like refers to any magnetosphere with an upstream bow-shock and an elongated downstream tail that undergoes tail plasmoid formation and release. It includes, for the first time in such a model, separate drivers for the Dungey and Vasyliunas Cycles and the capacity to include stochastic and deterministic driving in varying relative and absolute terms. The effects of measurement noise on the model output can also be simulated. This makes the model suitable to investigate the magnetotail dynamics of Mercury, Earth, Jupiter, Saturn and hypothetical exoplanets with similar magnetospheric configurations. The capacity to predict, in general terms, the behavior of a wide range of stellar-wind – magnetosphere interactions has become even more important in the light of the discovery of thousands of exoplanets in recent years. This model represents the first step towards being able to make such predictions for a wide variety of cases without resorting to detailed modelling of individual cases. It is demonstrated that the model can exhibit limit cycle (periodic) and chaotic (long-term unpredictable) behavior. The effects of a sufficiently strong dynamical noise component (stochastic driving) are shown to be inherently different from the effects of an equivalent level of simulated observational noise (simulated Gaussian instrument error). The possibilities of chaotic behavior and of dynamical noise dominating the underlying determinism imply that often only short-term forecasting of magnetotail plasmoid formation is possible.

Eddy-Mean flow oscillations in the Southern Ocean

<p>Geostrophic eddies have a leading order effect on the dynamics of the Southern Ocean (SO), and numerous studies have shown that they are also key to the response of both the zonal transport and the meridional overturning circulation to wind stress changes. The role played by eddies in setting the intrinsic variability of the SO, however, is less well-understood. Here, inspired by recent work on the atmospheric jet, we investigate whether the eddy-mean flow interaction in the Antarctic Circumpolar Current can be described by a prey-predator nonlinear model.</p><p> </p><p>To this end, we analyse data from a high-resolution eddy-resolving configuration of the MIT general circulation model: an idealised “channel” model with mechanical and thermodynamical forcing at the surface, and plausible zonal and meridional circulations.</p><p> </p><p>Here, we show that a mechanism of eddy-mean flow interaction driving the intrinsic variability of the SO-like model is well described by a stochastic non-linear oscillator with damping. This model is a generalisation of the Ambaum-Novak oscillator, which has been successfully employed to describe the atmospheric storm track variability.</p><p> </p><p>We find that, on length scales similar to that of individual zonal jets, the eddy-mean flow interaction is characterised by a high-frequency oscillatory mode, and that the characteristic time scale of the oscillation is comparable with classical estimates of the baroclinic life-cycle. A Gaussian smoothing of the phase space diagram also reveals the damped oscillatory character of the oscillation: this is in contrast with the atmospheric case, where damping is negligible and orbits are confined to energy surfaces.</p><p> </p><p>This result may help inform the interpretation of the SO intrinsic and forced variability (such as, for example, the response to wind stress changes), and pave the way to further studies featuring more realistic model configurations.</p>

High-performance GPU computations in nonlinear dynamics: an efficient tool for new discoveries

The main aim of this paper is to demonstrate the benefit of the application of high-performance computing techniques in the field of non-linear science through two kinds of dynamical systems as test models. It is shown that high-resolution, multi-dimensional parameter scans (in the order of millions of parameter combinations) via an initial value problem solver are an efficient tool to discover new features of dynamical systems that are hard to find by other means. The employed initial value problem solver is an in-house code written in C++ and CUDA C software environments, which can exploit the high processing power of professional graphics cards (GPUs). The first test model is the Keller–Miksis equation, a non-linear oscillator describing the dynamics of a driven single spherical gas bubble placed in an infinite domain of liquid. This equation is important in the field of cavitation and sonochemistry. Here, the high-resolution parameter scans gave us the opportunity to lay down the basis of a non-feedback technique to control multi-stability in which direct selection of the desired attractor is possible. The second test model is related to a pressure relief valve that can exhibit a special kind of impact dynamics called grazing impact. A fine scan of the initial conditions revealed a second focal point of the grazing lines in the initial-condition space that was hidden in previous studies.