Faster network disruption from layered oscillatory dynamics Melvyn Tyloo1 1Theoretical Division Los Alamos National Laboratory Los Alamos NM 87545 United States of America

2025-04-27 0 0 729.86KB 7 页 10玖币

侵权投诉

Faster network disruption from layered oscillatory dynamics

Melvyn Tyloo*1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, United States of America

and Center for Nonlinear Studies (CNLS), Los Alamos National Laboratory, Los Alamos, NM 87545,

United States of America

(*Electronic mail: mtyloo@lanl.gov.)

(Dated: 8 June 2023)

Nonlinear complex network-coupled systems typically have multiple stable equilibrium states. Following perturbations

or due to ambient noise, the system is pushed away from its initial equilibrium and, depending on the direction and

the amplitude of the excursion, might undergo a transition to another equilibrium. It was recently demonstrated [M.

Tyloo, J. Phys. Complex. 303LT01 (2022)], that layered complex networks may exhibit ampliﬁed ﬂuctuations. Here I

investigate how noise with system-speciﬁc correlations impacts the ﬁrst escape time of nonlinearly coupled oscillators.

Interestingly, I show that, not only the strong ampliﬁcation of the ﬂuctuations is a threat to the good functioning of the

network, but also the spatial and temporal correlations of the noise along the lowest-lying eigenmodes of the Laplacian

matrix. I analyze ﬁrst escape times on synthetic networks and compare noise originating from layered dynamics, to

uncorrelated noise.

Complex networked systems are often made of different

layers of dynamics that somehow interact together. Due

to this intricate structure, noise or perturbations affect-

ing one layer are typically transferred to other layers with

additional statistical properties e.g. spatial and temporal

correlations. The latters might have serious consequences

on the desired functioning of some layers, one of them be-

ing the ampliﬁcation of ﬂuctuations.1Together with the

inherent multistability of nonlinearly coupled dynamical

systems, noise acting on one layer may be more prompt to

drive other layers outside their initial basin of attraction,

through a transition to another equilibrium, if such a state

exists. Here, I investigate how ﬁrst escape times are af-

fected by noise propagating through a layered system and

compare it to uncorrelated noise.

I. INTRODUCTION

Various systems in nature and engineered applications can

be modelled as individual dynamical units, interacting with

one another in a complex way such as neurons spiking to-

gether in the brain2or rotating masses of power generators

evolving at the same frequency in large-scale transmission

networks.3Collective phenomena taking place in these sys-

tems such as synchronization, are enabled by both the inter-

nal dynamics and the interaction within the individual units.4

Another remarkable feature of such systems is multistabil-

ity. Thanks to nonlinearities in the interaction, numerous

equilibria might exist, each of them having their own basin

of attraction.5,6 Due to external perturbations such as envi-

ronmental noise or faults occurring at some components, the

whole system might undergo transitions between equilibrium

points. In many instances, this is not desirable as both the

transient dynamics and the new equilibirum might threaten

the correct operation of the system or even cause damages.7

An important task is to predict such transition as they could

disrupt the desired functioning state of the networked system.

The latter question has been investigated mostly in coupled

systems made of a single layer of interaction, usually sub-

jected to spatially uncorrelated noise.8–11 However, many ap-

plications require more involved coupling structures in order

to correctly describe them.12 A natural extension of the single

layer framework, is to include multiple layers of dynamical

systems.13,14 In such layered dynamics, noise acting on one

sub-network propagates to other layers with system-speciﬁc

correlations.1In other words, when going through a layer,

the noise acquire some speciﬁc statistical properties, which

is then injected into other layers. Including such system spe-

ciﬁc correlation in the noise might increase the level of ﬂuctu-

ations and thus the risk of transitions between equilibria. This

is depicted in Fig. 1, where uncorrelated noise is acting on

the ﬁrst layer [blue in panel (a)], which is then transmitted to

the second one [red in panel (a)]. In this setting, the second

layer is subjected to correlated noise, which we describe be-

low. Both layers are made of a single cycle network which has

multiple equilibira as illustrated in Fig. 1(b). Due to the noise

injected in both of them, the two layers are pushed away from

their initial equilibrium and, depending on the noise amplitude

and direction, eventually leave their initial basin of attraction.

Transitions between basins of attraction can be detected by

changes in the winding numbers in each layer q1and q2[see

Fig. 1(c)]. One clearly sees that, while the ﬁrst layer do not

operate any transition, the second one exits its initial basin of

attraction multiple times. Similar transitions where observed

for a single layer system8,10 and related to the eigenvalues of

the network Laplacian matrix and the distance between the

initial ﬁxed point and the closest saddle point with a single

unstable direction.11 Here I consider layered systems where

each layer has its own individual units, nonlinearly coupled

on a complex network. The different layers interact together

via a coupling function. It was recently shown that noise act-

ing on layer might be strongly ampliﬁed in the other layers

depending on the network connectivities.1Besides being am-

pliﬁed, the noise structure seems to align with the lowest-lying

eigenmodes of the network Laplacian. Building on these re-

sults, I investigate the ﬁrst escape time from the initial basin

arXiv:2210.01180v3 [nlin.AO] 6 Jun 2023

FIG. 1. (a) Two-layer system with a single cycle in each layer. (b)

Three different ﬁxed points with q=−1,0,1 are illustrated for a

n=5 nodes system, where the orientation of the arrows represents

the degrees of freedom. (c) Time-evolution of the winding num-

bers q1and q2in each layer (c) of the two-layers system shown in

panel (a), corresponding to Eqs. (1) with parameters such that in-

jected noise amplitudes are the same in both layers. The two layers

have the same cyclic coupling network of size n=40 nodes. Transi-

tions between different basins of attraction are tracked by observing

changes in values of q1,2. Noise amplitudes are the same in both

layers, however their spatial and temporal correlations are different.

of attraction in one layer due to noise originating from con-

nected layers. Remarkably, correlations in space and time of

this type of noise seem more efﬁcient in driving the system

through a transition to another basin than uncorrelated noise.

The paper is organized as follows. In Sec. II, I introduce

the two-layer system of nonlinearly coupled oscillators con-

sidered, recalls previous results1and discuss escape from the

basin of attraction. Numerical results are presented in Sec. III

and conclusions in Sec. IV.

II. NOISY NONLINEAR LAYERED OSCILLATORS

A. Kuramoto dynamics and its linearization

In order to investigate the effect of noise in layered net-

works, I consider a simple two-layer system made of Ku-

ramoto oscillators15 as the one shown in Fig. 1(a). Note that

the results below may apply to a larger set of diffusively cou-

pled systems. Their dynamics is governed by 2ncoupled dif-

ferential equations,

˙xi=ω(1)

i−

∑

j=1

b(1)

i j sin(xi−xj) + ηii=1,...n,

˙yi=ω(2)

i−

∑

j=1

b(2)

i j sin(yi−yj) + fi({xk},{yk})i=1, ...n,

(1)

where degrees of freedom and natural frequencies respectively

in layer 1 and 2 are denoted {xk},{yk}and ω(1)

k,ω(2)

k. The

undirected coupling network in the l-th layer is given by the

adjacency matrix elements b(l)

i j ≥0, and fiis a coupling func-

tion between the two layers. The noise acting on the ﬁrst layer

is encoded in ηithat is taken as white and uncorrelated in

space, i.e. ⟨ηi(t)ηj(t′)⟩=δi j η2

0δ(t−t′). This model made

of Kuramoto oscillators, is a nonlinear version of the one used

in Ref. 1. Thanks to the sine coupling and provided that b(l)

i j ’s

are sufﬁciently large compared to the distributions of ω(l)

i’s,

both Eqs. (1) may have multiple stable or unstable ﬁxed points

({x(0)

k},{y(0)

k}), depending on the coupling networks. In both

cases they satisfy,

0=ω(1)

i−

∑

j=1

b(1)

i j sin(x(0)

i−x(0)

j)i=1,...n,

0=ω(2)

i−

∑

j=1

b(2)

i j sin(y(0)

i−y(0)

j) + fi({x(0)

k},{y(0)

k})i=1,...n.

(2)

In the following, I consider a simple coupling function

fi({xk},{yk}) = d(xi−n−1∑jxj)where degrees of freedom

in the ﬁrst layer tune the natural frequencies in the second

one, and dserves as tuning parameter. The dynamics of small

deviations close to a stable ﬁxed point is given by the Tay-

lor expansion of Eqs. (1) to the ﬁrst order in [δxi(t),δyi(t)] =

[xi(t)−x(0)

i,yi(t)−y(0)

i], and read,

δxi=−

∑

j=1

b(1)

i j cos(x(0)

i−x(0)

j)(δxi−δxj) + ηi

δyi=−

∑

j=1

b(2)

i j cos(y(0)

i−y(0)

j)(δyi−δyj)

+d(δxi−n−1∑

δxj),

(3)

which are diffusive linear systems. Indeed, one can deﬁne the

weighted Laplacian matrices,

L(1)

i j ({x(0)

i}) = (−b(1)

i j cos(x(0)

i−x(0)

j),i̸=j,

∑kb(1)

ik cos(x(0)

i−x(0)

k),i=j,

L(2)

i j ({y(0)

i}) = (−b(2)

i j cos(y(0)

i−y(0)

j),i̸=j,

∑kb(2)

ik cos(y(0)

i−y(0)

k),i=j,

which depend on both the initial coupling networks in each

layer and the considered equilibrium. Using these Laplacian

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

FasternetworkdisruptionfromlayeredoscillatorydynamicsMelvynTyloo*11TheoreticalDivision,LosAlamosNationalLaboratory,LosAlamos,NM87545,UnitedStatesofAmericaandCenterforNonlinearStudies(CNLS),LosAlamosNationalLaboratory,LosAlamos,NM87545,UnitedStatesofAmerica(*Electronicmail:mtyloo@lanl.gov.)(Dated:8Ju...

展开>> 收起<<

Faster network disruption from layered oscillatory dynamics Melvyn Tyloo1 1Theoretical Division Los Alamos National Laboratory Los Alamos NM 87545 United States of America.pdf

共7页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Faster network disruption from layered oscillatory dynamics Melvyn Tyloo1 1Theoretical Division Los Alamos National Laboratory Los Alamos NM 87545 United States of America

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: