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We apply this procedure to a model metallic glass [MATH] described by an embedded-atom method(EAM) potential mendelev2009development . The molecular dynamics(MD) simulations were conducted by an open source classical MD software: LAMMPS plimpton1995fast The simulation samples contained 8000 atoms, and metallic glass sa... |
As shown in Fig. (a),(b), the probability distribution function of the LTEC is typically Gaussian, with a standard deviation that increases as the coarse graining size decreases, which is qualitatively consistent with similar calculations of the local elastic moduli PhysRevE.80.026112 PhysRevE.87.042306 As the coarse g... |
In the following we will be particularly interested in discussing the possibility that the stress field induced by the heterogeneous thermal expansion induces shear transformations (ST), i.e. localized irreversible plastic events that are recognized as the elementary building blocks of plasticity in amorphous materials... |
III Internal stress by thermal expansion heterogeneity When the metallic glasses are heated, due to the heterogeneity of LTEC, internal stresses are Extracting these thermal stresses directly from molecular dynamics simulations is not easy, in particular as the system can also evolve during thermal cycling due to local... |
With these assumptions, the effective external stress [MATH] where [MATH] is the bulk modulus, [MATH] is the temperature change and [MATH] is the LTEC at location [MATH] [MATH] the Kronecker symbol. At mechanical equilibrium, the elastic stress would balance the thermal stress [MATH] as |
[EQUATION] using small deformation conditions and linear elasticity, the equations for the displacement field are [EQUATION] where [MATH] is the [MATH] component of the displacement field at location [MATH] . Since the thermal expansion external force is a conservative vector field, we search the solution for [MATH] in... |
[EQUATION] Integrating in a finite system with periodic boundary conditions, equation can be written as [EQUATION] where [MATH] . Equation can be solved numerically using as an input the LTEC obtained from the MD simulations, and the resulting elastic stress can be calculated from the potential field [MATH] . We charac... |
[MATH] where [MATH] are the eigenvalues of the tensor. As shown in Fig. , the probability distribution function [MATH] shows a long tail distribution which we have fitted by the expression [MATH] (Lomax distribution wiki:lomax ). We do not have any particular justification for this fit, which is only used for convenien... |
IV Probability of triggering an atomic rearrangement If the internal stress is larger than the local yield stress, we expect that thermal cycling will trigger plastic activity. In order to estimate the corresponding density of events, we first need to estimate the distribution of the local yield stresses in the materia... |
The probability distribution function of the yield stress, shown in Fig. , is well fitted by an hyperbolic probability distribution function of the form: |
[EQUATION] Again we use this distribution for convenience, without any particular theoretical justification. Knowing the probability distribution of the yield stress and the one of the thermal stress, and assuming that these two quantities are statistically independent, we can estimate the probability that the thermal ... |
[EQUATION] [MATH] is Heaviside function, [MATH] is the joint probability of the thermal stress and yield stress within coarse grain size. We assume that the thermal stress and yield stress are independent variables, so that [MATH] , where [MATH] and [MATH] are the probability distribution functions of the thermal and y... |
[EQUATION] Using the parameters extracted from simulation data, we can then estimate this probability numerically. The results are shown in table , and discussed in the following section. |
discussion and conclusion The probability obtained in table is small. On the scale of a simulation box that contains a few thousands atoms (or equivalently, a few hundred possible shear transformations), the chance of actually observing a plastic event triggered by thermal stresses will be negligible. On the other hand... |
[MATH] , which we will refer to as the activation probability. This probability becomes larger than [MATH] for [MATH] . This corresponds to a system of lateral size that can be roughly estimated to 10 nm, assuming that the system is made of non overlapping zones that can undergo independent shear transformations. |
An important question concerns the robustness of our results with respect to the particular choice of coarse graining size. While our initial choice was motivated by the typical scale of shear transformations in metallic glasses, it remains somewhat arbitrary. Moreover, it is seen in figure (b), that the variance of th... |
Two factors will influence the efficiency of the cryogenic rejuvenation effect. One is the number of cycles, which will progressively increase the density of events until a significant rejuvenation is achieved, bringing the system into a higher energy state. Indeed, in the experiments ketov2015rejuvenation , it was obs... |
The second important factor is of course the amplitude [MATH] of temperature cycles. Following equation , the internal thermal stress increases in proportion of the amplitude of the temperature cycle and the local yield stress is insensitive with temperature at low temperature, thus [MATH] would increase with the tempe... |
VI acknowledgments This work is supported by (BSS and PGF) the NSF of China (Grant Nos.51601009,Nos.51571011),the MOST 973 Program (No.2015CB856800) and the NSAF joint program (No.U1530401). BSS and PFG acknowledges the computational support from the Beijing Computational Science Research Center (CSRC). J-L. Barrat is ... |
Appendix Following the coarse grain method in ref PhysRevE.80.026112 , we calculated local elastic modulus with [MATH] , and using the same spatial position as LTEC, the distribution and correlation are shown in Fig. A1 . Clearly, there is no correlation between local thermal expansion and local elastic moduli. |
Next, we show that, as a first order approximation for the heterogeneity, our uniform elastic constant assumption in the article is reasonable. |
We take equation in the article as an illustration, and introduce the possibility of heterogeneous elasticity in this equation. A similar reasoning can be extended to the complete calculation. In the presence of heterogeneity, we can rewrite equation as: |
[EQUATION] we use the variance and mean of the local moduli to represent the heterogeneity, as [MATH] For a uniform medium, the solution is obtained by inverting the linear operator [MATH] (which is conveniently done in Fourier space) and applying its inverse [MATH] to [MATH] . The presence of heterogeneities makes the... |
Hence, the approximation used in the manuscript can be considered as a first order approximation that ignores terms of order [MATH] or [MATH] |
# Source: arxiv 1807.00783 # Title: Variety of rotation modes in a small chain of coupled pendulums # Sections: all # Downloaded: 2026-03-02T08:51:07.780878+00:00 |
Variety of rotation modes in a small chain of coupled pendulums March 19, 2024 March 19, 2024 Abstract Abstract This article studies the rotational dynamics of three identical coupled pendulums. There exist two parameter areas where the in-phase rotational motion is unstable and out-of-phase rotations are realized. Asy... |
pacs: 05.45.Xt, 45.20.dc Introduction Study of collective dynamics in networks of coupled oscillatory units in different objects is an actively developing direction in nonlinear dynamics. This area is important both for theoretical understanding of complex processes and for the wide range of practical applications. Syn... |
Systems of coupled pendulums are ones of actual models in different fields of science and technics. Despite of relative simplicity of such models, they adequately describe not only mechanic objects, but also different processes in semiconductional structures Barone , molecular biology Yakushevich and in systems of phas... |
Cluster and chimera states in ensembles of different dimensions are of particular interest in the study of synchronization phenomena and symmetry breaking (see, e.g., PikovskyRosenblum2015 Kurths2016 PanaggioAbrams2015 YaoZheng2016 KemethKrischer2016 Omelchenko2018 ). Cluster state is two or more oscillators groups, wi... |
A significant progress in theoretical studies of chimera states was achieved due to the formulation of dynamics equations in terms of local complex order parameter PanaggioAbrams2015 YaoZheng2016 KemethKrischer2016 Omelchenko2018 . For this complex field, the setup becomes similar to pattern formation problems for nonl... |
Chimera states can be found in small ensembles, for example in the system of four elements, where two oscillators are synchronous, and the others two are asynchronous Ashwin2015 Panaggio2016 Bick2016 Hart2016 Kemeth2018 . In the article Ashwin2015 the so-called weak chimera state is defined, characterized by various me... |
In this paper we examine singularities found in rotational dynamics of three nonlinearly coupled pendulums. Our system is somewhat like small ensembles considered in Maistrenko2017 Dudkowski2016 Wojewoda2016 . We are interested in in-phase rotations and nontrivial out-of-phase ones. In Section II.1 we describe the mode... |
II Mechanism of symmetry breaking in a small chain of coupled pendulums II.1 Model and problem statement Let us consider the chain of three coupled identical pendulums described by the following system of ODEs |
[EQUATION] Here [MATH] is the damping coefficient responsible for all the dissipative processes in the system, [MATH] is a constant external force identical for all pendulums, [MATH] characterizes the nonlinear coupling strength between the elements. |
For certain values of the parameters [MATH] and [MATH] the system ( ) demonstrates non-trivial behavior. First, the system can demonstrates in-phase dynamics, i.e. [MATH] . We shall denote such regime as (3:0). All pendulums move synchronously and their dynamics is described by a single equation: |
[EQUATION] The dynamics of this system is well studied Andronov . The parameter plane [MATH] is divided into three domains Tricomi1933 Belykh1977 In one domain there are two steady states: a saddle and a stable foci (node). In second domain there exist a stable [MATH] -periodic in [MATH] motion and a stable foci (node)... |
It is obvious that the system ( ) has an in-phase rotation motion [MATH] . We have found that for certain parameter values the instability of this motion can be observed. Let us demonstrate this for some fixed parameters [MATH] [MATH] [MATH] under very close initial conditions [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] |
As can be seen from Fig. , for [MATH] the general velocities [MATH] [MATH] practically coincide. From the second part of Fig. , when [MATH] , one can already see asynchrony in the oscillations of the pendulums, i.e. the instability of their synchronous rotation mode has developed. The difference between [MATH] |
[MATH] is quite noticeable. A new type of limit rotations develops when [MATH] with [MATH] [MATH] changing out-of-phase. Thus, when the coupling parameter [MATH] reaches some values, the instability of the synchronous periodic motion develops: a new [MATH] -periodic limit rotations emerge in the ensemble of pendulums. ... |
II.2 Self-induced parametric instability of the in-phase (perfectly symmetric) rotation mode Let us investigate the case where a system has small dissipation (i.e. when [MATH] ). Let us also assume [MATH] characterizing the external force to be close to 1. In this case, one can build an asymptotic theory that would exp... |
Let us construct an asymptotic solution to Eq. ( ) using the Lindstedt-Poincaré method Nayfeh , the essence of which is to introduce a new dimensionless time [MATH] , where [MATH] and |
[EQUATION] is an unknown angular frequency of the sought-for solution allowing to avoid secular terms. Taking for simplicity [MATH] , we represent the solution in the form of the following asymptotic expansion: |
[EQUATION] where [MATH] are [MATH] -periodic functions of the variable [MATH] By substituting Eqs. ( ) and ( ) into Eq. ( ), expanding both sides in powers of [MATH] , equating the coefficients of the same powers of [MATH] and determining [MATH] from the condition of absence of secular terms, we obtain the in-phase rot... |
[EQUATION] where [EQUATION] Let us find the stability conditions for the in-phase rotation mode. First linearize the system ( ) around [MATH] , then [MATH] |
[MATH] . Next we get the corresponding equations for variations [MATH] [MATH] [EQUATION] To continue with, we introduce two detuning variables [MATH] . For [MATH] and [MATH] we obtain a closed system of equations |
[EQUATION] which admits two simple solutions. First of them [MATH] corresponds to the regime with pairwise different phases of the oscillators [MATH] We shall denote this regime as (1:1:1). Introducing for brevity [MATH] , we obtain equation |
[EQUATION] This equation belongs to the Mathieu-type equation. Hence, the parametric instability effects can be observed for some values of the parameter [MATH] depended on [MATH] and [MATH] |
Smirnov2016 To find the boundaries of the instability domain of the in-phase rotation mode, we determine the coupling parameter [MATH] values for which the Eq. ( ) admits a solution with [MATH] period or, equivalently, with [MATH] frequency. |
Using some aspects of perturbation theory, taking results ( ) and ( ) and searching for a solution of Eq. ( ) with [MATH] frequency, we get boundaries [MATH] for the first instability domain |
[EQUATION] Another solution [MATH] corresponds to the regime with [MATH] , then two oscillators form in-phase synchronous cluster and the third one rotates separately with some delay. It is regime ( [MATH] ). As it is mentioned in Sec. , such behavior of the system is indicated to a chimera-like dynamics. |
Introducing again [MATH] , we obtain equation for detuning [MATH] [EQUATION] Similarly to the previously examined case, we get boundaries [MATH] for the second instability domain |
[EQUATION] Thus, for a chain of three pendulums, there can exist two intervals of coupling strength [MATH] values, corresponding to the regimes (2:1) and (1:1:1), for which in-phase periodic rotation becomes parametrically unstable. |
III Out-of-phase symmetry-broken rotational states III.1 Numerical setup In this section, we present the results of the detailed numerical simulations which are performed directly within the framework of the discussed model ( ) of three pendulums for a wield range of the parameters [MATH] [MATH] and [MATH] . First of a... |
LyapunovExponents . Computations extend typically over [MATH] time units that seem to provide a stabilization of the Lyapunov exponents at a good level of accuracy. |
The theoretical analysis above allows us to describe the initial stage of the discussed instability of the synchronous rotation mode. One also can find all intervals of values of the coupling coefficient [MATH] , for which the development of the set-induced parametric instability is possible, and estimate the boundarie... |
BifurcationTheory . The main ideas of this method are discussed in Appendix. As a result, we can identify both stable and unstable limit cycles in our system and study in detail their bifurcations and a process of transition to chaos. This is one of the main goals of the presented paper. |
The linear stability of the ensuing periodic solutions is investigated by means of a Floquet analysis, chiefly relying on numerical calculations (see, e.g., BifurcationTheory and Appendix bellow for details). To this end, we add a small perturbation to the periodic motion. Stability analysis is performed by diagonalizi... |
As a characteristic of the degree of synchronization, we consider the value [MATH] , which is the frequency lag of pendulums: [EQUATION] |
where [MATH] is the period of rotational mode. It follows from the definition ( 13 ) that [MATH] takes non-negative values, and [MATH] only in the case of in-phase mode. In the case of a out-of-phase regime, when there exists such a pair of pendulums that [MATH] , where [MATH] and [MATH] are the numbers of pendulums, [... |
III.2 Regular dynamic and bistability of in-phase and out-of-phase rotational modes From the expressions ( 10 ) and ( 12 ) we see that in the case of small values of [MATH] for [MATH] , two regions of instability of the in-phase mode arise. Next we will investigate the case [MATH] Let us consider the situation when the... |
The horizontal segments [MATH] [MATH] [MATH] correspond to the synchronous in-phase regime ( [MATH] ). There are two regions [MATH] and [MATH] of the values of the parameter [MATH] , when this regime becomes unstable. As shown above, in the course of the asymptotic consideration (the expressions ( 10 ) and ( 12 )), it ... |
Let us consider processes occurring in a chain when [MATH] takes values from the [MATH] and when [MATH] escapes from it. As the parameter [MATH] increases, the in-phase periodic motion undergoes period doubling bifurcation ( [MATH] ), while from the stable in-phase [MATH] -periodic in [MATH] of motion, a stable [MATH] ... |
Similarly, for the instability zone [MATH] . As the parameter [MATH] decreases as a result of period doubling bifurcation ( [MATH] ), the in-phase periodic motion loses stability and [MATH] -periodic in [MATH] motion occurs ( [MATH] branch), corresponding to a completely out-of-phase regime [MATH] (1:1:1). For [MATH] ,... |
Thus, when the in-phase mode is unstable, out-of-phase [MATH] -periodic (2:1) and (1:1:1) regimes are realized in the system. The bifurcation diagram (see Fig. ) shows clearly that there are also two value ranges of the coupling strength, in which two stable (and one unstable) rotation limit cycles exist at the same ti... |
III.3 Chaotic dynamics and chaotic chimera states As the dissipation parameter increases, the regions of instability of the in-phase regime approach each other. At the same time, chaotic dynamics is possible in the chain of three pendulums. In the paragraph III.2 , the case of coexistence of two regions of instability ... |
Let us consider the stable [MATH] -periodic (1:1:1) regime (branch [MATH] ). As the [MATH] decreases, the [MATH] -periodic motion loses stability through the pitchfork bifurcation ( [MATH] ), and from it two stable [MATH] -periodic motions arise, which in Fig. (b) the branch [MATH] corresponds to, and the unstable [MAT... |
As the values of the dissipation parameter [MATH] increase, the regions of instability of the in-phase periodic motion [MATH] and [MATH] approach each other. At a critical value of the parameter [MATH] , the regions of instability touch each other, after that they begin to overlap (Fig. (a)). The [MATH] region disappea... |
[MATH] is irregular (see Fig. (a)). This regime can be interpreted as a chaotic chimera Maistrenko2017 . If the coupling strength parameter [MATH] takes the value from the region [MATH] the chaotic regime (1:1:1) is realized (see Fig. (b)). |
With a further increase of the parameter [MATH] , the regions of chaotic dynamics become closer. Chaotic dymanics is realized for [MATH] and [MATH] for values of parameters [MATH] [MATH] (see Fig. ). |
Further, the regions of chaotic dynamics merge into one. For [MATH] [MATH] (see Fig. ) with an increase in the coupling strength, when the critical value [MATH] is reached, a chaotic (2:1) chimera state arises as a result of the cascade of period doubling bifurcations (see Figs. (a) and ). Further, at [MATH] , the chao... |
IV Conclusion We have studied the dynamics of a chain of three identical coupled pendulums. A relatively simple model demonstrates a great variety of regular and chaotic in-phase and out-of-phase regimes. Self-induced parametric instability of the perfectly symmetric in-phase rotation mod is found and theoretically pro... |
Acknowledgements. Authors acknowledge A. Pikovsky, V. N. Belykh and A. O. Kazakov for valuable advices and fruitful discussion. Results presented in Section II were supported by the RSF grant No. 14-12-00811. Results presented in Section III were supported by the RFBR grant No. 17-32-50096. L. A. Smirnov thanks DAAD gr... |
Appendix A Methods of numerical calculation of periodic motions and their stability This appendix contains a description of the numerical methods for numerical calculation of nontrivial periodic motions in the ensembles of [MATH] globally coupled pendulums, and an analysis of linear stability of these motions. |
In order to calculate regular rotation modes of a chain of coupled pendulum, we apply a modification of a commonly used scheme to find closed limit cycles in nonlinear dynamical systems BifurcationTheory . The main idea of this method is as follows. Each of the solutions [MATH] (here and below [MATH] ) we are intereste... |
[EQUATION] where [MATH] is the solution to Eqs. ( ) with initial conditions [MATH] , i.e. [MATH] . Therefore, a periodic solution with period [MATH] of Eqs. ( ) will be a root to ( 14 ). Because of the translational invariance symmetry (in time), we note that one value from the set [MATH] can always be taken to be zero... |
To study the linear stability of arbitrary ( [MATH] -, [MATH] -, [MATH] - and etc.) periodic motions (on the cylinder) of Eqs. ( ), we introduce a small perturbation [MATH] to a given periodic solution [MATH] As a result, we obtain the following linearized equations for [MATH] |
[EQUATION] Due to the periodicity of the trajectory [MATH] one can performe the Floque analisis of Eqs. ( 15 ). Hence, the stability properties of the considered trajectories is given by the spectrum of the Floquet operator [MATH] |
[EQUATION] The eigenvalues [MATH] (here and below [MATH] ) of the monodromy matrix [MATH] are dubbed the Floquet multipliers of the periodic solution [MATH] In considered case, the Floquet multipliers [MATH] are real or appear in complex conjugated pairs, because of the existence of the external force and the damping i... |
# Source: arxiv 1807.01486 # Title: Empirical fixed point bifurcation analysis # Sections: all # Downloaded: 2026-03-03T05:16:50.394897+00:00 |
Empirical fixed point bifurcation analysis Abstract In a common experimental setting, the behaviour of a noisy dynamical system is monitored in response to manipulations of one or more control parameters. Here, we introduce a structured model to describe parametric changes in qualitative system behaviour via stochastic... |
Machine Learning, ICML Introduction Data, especially from biological systems, is often gathered through slow, noisy measurements, and as such standard modelling tools are often not suited to processing the raw experimental data. Time evolution of processes is frequently ignored, both on short time scales, such as durin... |
Here we wish to address two features of experiments that do not always make the cut. Firstly, we explicitly model the short term evolution of the system using a stochastic dynamical system. Secondly, we wish to understand, how the behaviour changes with respect to varying a parameter that the experimenter either has di... |
Unfortunately, providing simple summaries and comparisons of dynamical systems learned from data is no easy task, and is a very active research area (Sussillo & Barak, 2013 ; Nonnenmacher et al., 2017 ; Sussillo et al., 2016 . One popular approach has been to use a simple model to fit to the data, which enabled scienti... |
On the other hand, more complex models may provide very good description of almost arbitrary data, especially with the recent rise of various deep learning algorithms. Despite their success in predicting newly observed data, we are far from an understanding of how changes in fitted parameters of such systems relate to ... |
We take a middle approach here: We follow changes of dynamical systems via the bifurcations of their fixed points as the experimental parameter varies, a type of analysis mostly applied to known, continuous time and noiseless systems. In order to use it in the wild, we need to relax these constraints, and use a non-par... |
The paper is organised as follows: In section we review deterministic bifurcation analysis on a simple example, and discuss the implications of noise. In section |
, we first describe how to infer a map from time series data using Gaussian Processes, then add the fixed points and derivative structures to our model of the map. Finally, in section |
, we prove the feasibility of our approach on a one-dimensional pitchfork bifurcation, then examine a more realistic experimental scenario using a model of mutually inhibiting neural populations. |
To provide a concise description of the model with the added clarity of indexing, we employ the indicial notation with Einstein summation convention throughout the paper. In short, the number of lower indices describes the order of a tensor, and repeated indices in the same term induces summation, e.g. [MATH] , and [MA... |
Bifurcation analysis in discrete stochastic systems We introduce concepts in bifurcation analysis through the normal form of the pitchfork bifurcation, described by the map |
[EQUATION] Bifurcation analysis concerns the number and stability of fixed points of a system as a function of a parameter – here [MATH] . When these change, the system is said to go through a bifurcation, a qualitative change in long term behaviour. Equation eq. |
has a fixed point at [MATH] for all [MATH] values. We understand the stability of a fixed point in a map by the magnitude of the derivative (or the eigenvalues of the Jacobian in higher dimensional systems). If it is smaller in absolute value than [MATH] , the system will move towards the point, otherwise away from it,... |
A. Unfortunately even minuscule amounts of noise destroy this beautiful picture (Crauel & Flandoli, 1998 . Stochastic bifurcation analysis examines random dynamical systems, and distinguishes between phenomenological and dynamical bifurcations. The former examines profiles of stationary probability densities, whereas t... |
B and C, where a deterministic system already over the point of bifurcation falls back to its former behaviour with added noise. |
In known systems, the expected number of fixed points may be given, but in real world measured datasets we have little idea. Therefore we need a model that is capable of automatically determining the number and location of likely fixed points, given little data. In the following section we describe such an approach, ba... |
Model In this section, we first introduce the type of datasets we expect for a single value of the bifurcation parameter, which arise frequently in biological studies, and the assumptions that underlie the models used here. We then describe the general machinery of learning a Gaussian Process transition map from data. ... |
3.2 we describe our extension of this machinery, which enables the fixed points and the corresponding local linearisations to be optimised directly. |
Given a dataset of repeated measurements of time-series [MATH] of [MATH] measured variables, we aim to model the data using a discrete-time, latent, non-linear, stochastic dynamical system: |
[EQUATION] where [MATH] represents the state of the dynamical system at time [MATH] on trial [MATH] [MATH] is the transition function, with [MATH] additive noise; and [MATH] is the observation function, with [MATH] additive noise. |
For simplicity, we only discuss the case where [MATH] is a linear function, [MATH] , and [MATH] . We can thus write down the conditional probability |
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