DOCSLIB.ORG

Recurrence Plots for the Analysis of Complex Systems Norbert Marwan∗, M

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Physics Reports 438 (2007) 237–329

www.elsevier.com/locate/physrep

Recurrence plots for the analysis of complex systems

Norbert Marwan∗, M. Carmen Romano, Marco Thiel, Jürgen Kurths

Nonlinear Dynamics Group, Institute of Physics, University of Potsdam, Potsdam 14415, Germany

Accepted 3 November 2006 Available online 12 January 2007 editor: I. Procaccia

Abstract

Recurrence is a fundamental property of dynamical systems, which can be exploited to characterise the system’s behaviour in phase space. A powerful tool for their visualisation and analysis called recurrence plot was introduced in the late 1980’s. This report is a comprehensive overview covering recurrence based methods and their applications with an emphasis on recent developments. After a brief outline of the theory of recurrences, the basic idea of the recurrence plot with its variations is presented. This includes the quantification of recurrence plots, like the recurrence quantification analysis, which is highly effective to detect, e. g., transitions in the dynamics of systems from time series. A main point is how to link recurrences to dynamical invariants and unstable periodic orbits. This and further evidence suggest that recurrences contain all relevant information about a system’s behaviour. As the respective phase spaces of two systems change due to coupling, recurrence plots allow studying and quantifying their interaction. This fact also provides us with a sensitive tool for the study of synchronisation of complex systems. In the last part of the report several applications of recurrence plots in economy, physiology, neuroscience, earth sciences, astrophysics and engineering are shown. The aim of this work is to provide the readers with the know how for the application of recurrence plot based methods in their own field of research. We therefore detail the analysis of data and indicate possible difficulties and pitfalls. © 2006 Elsevier B.V. All rights reserved.

PACS: 05.45; 07.05.Kf; 07.05.Rm; 91.25.−r; 91.60.Pn Keywords: Data analysis; Recurrence plot; Nonlinear dynamics

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 2. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 3. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
3.1. Trajectories in phase space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 3.2. Recurrence plot (RP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
3.2.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 3.2.2. Selection of the threshold ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 3.2.3. Structures in RPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 3.2.4. Influence of embedding on the structures in RPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 3.2.5. Modifications and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
3.3. Cross recurrence plot (CRP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

∗

Corresponding author.

E-mail address: [email protected] (N. Marwan).

0370-1573/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2006.11.001
238

N. Marwan et al. / Physics Reports 438 (2007) 23 7–329

3.4. Joint recurrence plot (JRP). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 3.5. Measures of complexity (recurrence quantification analysis, RQA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
3.5.1. Measures based on the recurrence density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 3.5.2. Measures based on diagonal lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 3.5.3. Measures based on vertical lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
3.6. Dynamical invariants derived from RPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
3.6.1. Correlation entropy and correlation dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 3.6.2. Recurrence times and point-wise dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 3.6.3. Generalised mutual information (generalised redundancies) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 3.6.4. Influence of embedding on the invariants estimated by RPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
3.7. Extension to spatial data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 3.8. Synchronisation analysis by means of recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
3.8.1. Synchronisation of chaotic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 3.8.2. Detection of synchronisation transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 3.8.3. Detection of PS by means of recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 3.8.4. Detection of GS by means of recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 3.8.5. Comparison with other methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 3.8.6. Onset of different kinds of synchronisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
3.9. Information contained in RPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 3.10. Recurrence-based surrogates to test for synchronisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 3.11. Localisation of unstable periodic orbits by RPs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 3.12. Influence of noise on RPs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
4. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
4.1. RQA analysis in neuroscience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 4.2. RQA analysis of financial exchange rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 4.3. Damage detection using RQA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 4.4. Time scale alignment of geophysical borehole data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
4.4.1. Time scale alignment of geological profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 4.4.2. Dating of a geological profile (magneto-stratigraphy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
4.5. Finding of nonlinear interrelations in palaeo-climate archives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

4.6. Automatised computation of K applied to the stability of extra-solar planetary systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

2

4.7. Synchronisation analysis of experimental data by means of RPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
4.7.1. Synchronisation of electrochemical oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 4.7.2. Synchronisation analysis of cognitive data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Appendix A. Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Appendix B. Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
B.1. Algorithm to fit the LOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 B.2. Algorithm for the reconstruction of a time series from its RP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 B.3. Twin surrogates algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 B.4. Automatisation of the K estimation by RPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

2

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

List of Abbreviations and Symbols

ꢀxꢁ x¯

mean of series x, expectation value of x series x normalised to zero mean and standard deviation of one estimator for x

xˆ

ꢀ(·) ꢀ(·, ·) jt (·) ꢀt

delta function (ꢀ(x) = {1 | x = 0; 0 | x = 0}) Kronecker delta function (ꢀ(x, y) = {1 | x = y; 0 | x = y})

d

dt

  • derivative with respect to time (jt (x) =
  • )

sampling time

ε

radius of neighbourhood (threshold for RP computation) Lyapunov exponent

ꢁ

N. Marwan et al. / Physics Reports 438 (2007) 23 7–329

239

ꢂ(·)

ꢃ

probability measure frequency

ꢄꢅ

order pattern phase

ꢆ

standard deviation

(·)

Heaviside function ( (x) = {1 | x > 0; 0 | x ꢀ0})

ꢇ

time delay (index-based units)

ꢈ

A

white noise a measurable set

Bxꢃ (ε)

the ε-neighbourhood around the point xꢃi on the trajectory

i

cov(·)

Cd(ε) CC(ε)

CR(ε)

CPR

CRP CS
(auto)covariance function correlation sum for a system of dimension d and by using a threshold ε cross correlation sum for two systems using a threshold ε cross recurrence matrix between two phase space trajectories by using a neighbourhood size ε synchronisation index based on recurrence probabilities cross recurrence plot complete synchronisation

D

D

distance matrix between phase space vectors distance

D1 D2

information dimension correlation dimension

DP

point-wise dimension

DET

measure for recurrence quantification: determinism measure for recurrence quantification: determinism of the ꢇth diagonal in the RP measure for recurrence quantification: divergence dimension of the system

DET DIV

d

ꢇ

ENTR

FAN FNN GS measure for recurrence quantification: entropy fixed amount of nearest neighbours (neighbourhood criterion) false nearest neighbour generalised synchronisation

Hq

Renyi entropy of order q

hD

topological dimension

Iq

generalised mutual information (redundancy) of order q indices joint correlation sum for two systems using thresholds ε1 and ε2 joint Renyi entropy of 2nd order

i, j, k

JC(ε1, ε2) JK2 JPR

JRP

K2

joint probability of recurrence joint recurrence plot Renyi entropy of 2nd order (correlation entropy) measure for recurrence quantification: average line length of diagonal lines

LL

measure for recurrence quantification: average line length of diagonal lines of the ꢇth diagonal in the

ꢇ

RP

Lmax

Lp-norm

lmin

LAM

LOI LOS

MFNN

m

measure for recurrence quantification: length of the longest diagonal line vector norm, e.g. the Euclidean norm (L2-norm), Maximum norm (L -norm)

∞

predefined minimal length of a diagonal line measure for recurrence quantification: laminarity line of identity (the main diagonal line in a RP, Ri,i = 1) line of synchronisation (the distorted main diagonal line in a CRP) Mutual false nearest neighbours embedding dimension

N

length of a data series

Nl, Nv

number of diagonal/vertical lines

240

N. Marwan et al. / Physics Reports 438 (2007) 23 7–329

Nn

N

number of (nearest) neighbours set of natural numbers
OPRP

Pi,j

order patterns recurrence plot probability to find a recurrence point at (i, j) histogram histogram or frequency distribution of line lengths probability probability that the trajectory recurs after ꢇ time steps probability to find a line of exactly length l probability to find a line of at least length l phase synchronisation

P (·) P (ε, l) p(·)

p(ε, ꢇ) p(ε, l) pc(ε, l)

PS

Q(ꢇ) q(ꢇ) q

CRP symmetry measure CRP asymmetry measure order

R

set of real numbers

R

set of recurrence points

R(ε)

R

recurrence matrix of a phase space trajectory by using a neighbourhood size ε mean resultant length of phase vectors (synchronisation measure) measure for recurrence quantification: ratio between DET and RR recurrence plot
RATIO RP RQA

RR, REC

recurrence quantification analysis measure for recurrence quantification: recurrence rate (percent recurrence) measure for recurrence quantification: recurrence rate of the ꢇth diagonal in the RP synchronisation index based on joint recurrence recurrence time of 1st type recurrence time of 2nd type phase period

RR

ꢇ

S

T (1) T (2)

Tph Trec

TREND TT

Vmax vmin

X

recurrence period measure for recurrence quantification: trend measure for recurrence quantification: trapping time measure for recurrence quantification: length of the longest vertical line predefined minimal length of a vertical line a measurable set

1. Introduction

El poeta tiene dos obligaciones sagradas: partir y regresar. (The poet has two holy duties: to set out and to return.)

Pablo Neruda

If we observe the sky on a hot and humid day in summer, we often “feel” that a thunderstorm is brewing. When children play, mothers often know instinctively when a situation is going to turn out dangerous. Each time we throw a stone, we can approximately predict where it is going to hit the ground. Elephants are able to find food and water during times of drought. These predictions are not based on the evaluation of long and complicated sets of mathematical equations, but rather on two facts which are crucial for our daily life:

(1) similar situations often evolve in a similar way; (2) some situations occur over and over again.

The first fact is linked to a certain determinism in many real world systems. Systems of very different kinds, from very large to very small time–space scales can be modelled by (deterministic) differential equations. On very large scales we might think of the motions of planets or even galaxies, on intermediate scales of a falling stone and on small scales of the firing of neurons. All of these systems can be described by the same mathematical tool of differential equations.

N. Marwan et al. / Physics Reports 438 (2007) 23 7–329

241

They behave deterministically in the sense that in principle we can predict the state of such a system to any precision and forever once the initial conditions are exactly known. Chaos theory has taught us that some systems—even though deterministic—are very sensitive to fluctuations and even the smallest perturbations of the initial conditions can make a precise prediction on long time scales impossible. Nevertheless, even for these chaotic systems short-term prediction is practicable.
The second fact is fundamental to many systems and is probably one of the reasons why life has developed memory.
Experience allows remembering similar situations, making predictions and, hence, helps to survive. But remembering similar situations, e.g., the hot and humid air in summer which might eventually lead to a thunderstorm, is only helpful if a system (such as the atmospheric system) returns or recurs to former states. Such a recurrence is a fundamental characteristic of many dynamical systems.
They can indeed be used to study the properties of many systems, from astrophysics (where recurrences have actually beenintroduced)overengineering, electronics, financialmarkets, populationdynamics, epidemicsandmedicinetobrain dynamics. The methods described in this review are therefore of interest to scientists working in very different areas of research.
The formal concept of recurrences was introduced by Henri Poincaré in his seminal work from 1890 [1], for

which he won a prize sponsored by King Oscar II of Sweden and Norway on the occasion of his majesty’s 60th birthday. Therein, Poincaré did not only discover the “homoclinic tangle” which lies at the root of the chaotic behaviour of orbits, but he also introduced (as a by-product) the concept of recurrences in conservative systems. When speaking about the restricted three body problem he mentioned: “In this case, neglecting some exceptional trajectories, the occurrence of which is infinitely improbable, it can be shown, that the system recurs infinitely many times as close as one wishes to its initial state.” (translated from [1]) Even though much mathematical work was carried out in the following years, Poincaré’s pioneering work and his discovery of recurrence had to wait for more than 70 years for the development of fast and efficient computers to be exploited numerically. The use of powerful computers boosted chaos theory and allowed to study new and exciting systems. Some of the tedious computations needed to use the concept of recurrence for more practical purposes could only be made with this digital tool.
In 1987, Eckmann et al. introduced the method of recurrence plots (RPs) to visualise the recurrences of dynamical

Recommended publications
  • Communications with Chaotic Optoelectronic Systems Cryptography and Multiplexing

    Communications with Chaotic Optoelectronic Systems Cryptography and Multiplexing

    COMMUNICATIONS WITH CHAOTIC OPTOELECTRONIC SYSTEMS CRYPTOGRAPHY AND MULTIPLEXING A Thesis Presented to The Academic Faculty by Damien Rontani In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Electrical and Computer Engineering Georgia Institute of Technology December 2011 Copyright c 2011 by Damien Rontani COMMUNICATIONS WITH CHAOTIC OPTOELECTRONIC SYSTEMS CRYPTOGRAPHY AND MULTIPLEXING Approved by: Professor Steven W. McLaughlin, Professor Erik Verriest Committee Chair School of Electrical and Computer School of Electrical and Computer Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Professor David S. Citrin, Advisor Adjunct Professor Alexandre Locquet School of Electrical and Computer School of Electrical and Computer Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Professor Marc Sciamanna, Co-advisor Professor Kurt Wiesenfeld Department of Optical School of Physics Communications Georgia Institute of Technology Ecole Sup´erieure d'Electricit´e Professor William T. Rhodes Date Approved: 30 August 2011 School of Electrical and Computer Engineering Georgia Institute of Technology To those who have made me who I am today, iii ACKNOWLEDGEMENTS The present PhD research has been prepared in the framework of collaboration between the Georgia Institute of Technology (Georgia Tech, USA) and the Ecole Sup´erieured'Electricit´e(Sup´elec,France), at the UMI 2958 a joint Laboratory be- tween Georgia Tech and the Centre National de la Recherche Scientifique (CNRS, France). I would like to acknowledge the Fondation Sup´elec,the Conseil R´egionalde Lorraine, Georgia Tech, and the National Science Foundation (NSF) for their financial and technical support. I would like to sincerely thank my \research family" starting with my two advisors who made this joint-PhD project possible; Prof.
  • Correlated Brownian Motion and Diffusion of Defects in Spatially Extended Chaotic Systems

    Correlated Brownian Motion and Diffusion of Defects in Spatially Extended Chaotic Systems

    Correlated Brownian motion and diffusion of defects in spatially extended chaotic systems Cite as: Chaos 29, 071104 (2019); https://doi.org/10.1063/1.5113783 Submitted: 07 June 2019 . Accepted: 29 June 2019 . Published Online: 16 July 2019 S. T. da Silva, T. L. Prado, S. R. Lopes , and R. L. Viana ARTICLES YOU MAY BE INTERESTED IN Intricate features in the lifetime and deposition of atmospheric aerosol particles Chaos: An Interdisciplinary Journal of Nonlinear Science 29, 071103 (2019); https:// doi.org/10.1063/1.5110385 Non-adiabatic membrane voltage fluctuations driven by two ligand-gated ion channels Chaos: An Interdisciplinary Journal of Nonlinear Science 29, 073108 (2019); https:// doi.org/10.1063/1.5096303 Symmetry induced group consensus Chaos: An Interdisciplinary Journal of Nonlinear Science 29, 073101 (2019); https:// doi.org/10.1063/1.5098335 Chaos 29, 071104 (2019); https://doi.org/10.1063/1.5113783 29, 071104 © 2019 Author(s). Chaos ARTICLE scitation.org/journal/cha Correlated Brownian motion and diffusion of defects in spatially extended chaotic systems Cite as: Chaos 29, 071104 (2019); doi: 10.1063/1.5113783 Submitted: 7 June 2019 · Accepted: 29 June 2019 · Published Online: 16 July 2019 View Online Export Citation CrossMark S. T. da Silva, T. L. Prado, S. R. Lopes, and R. L. Vianaa) AFFILIATIONS Departament of Physics, Federal University of Paraná, 81531-990 Curitiba, Paraná, Brazil a)Author to whom correspondence should be addressed: viana@fisica.ufpr.br ABSTRACT One of the spatiotemporal patterns exhibited by coupled map lattices with nearest-neighbor coupling is the appearance of chaotic defects, which are spatially localized regions of chaotic dynamics with a particlelike behavior.
  • A Translation Of

    A Translation Of

    2011 A Translation of NJV 72^ dsgecr 1 2011 NJV 72^ dsgecr 2 2011 American Mathematical Society Providence, Rhode Island USA ISSN 0077-1554 TRANSACTIONS OF THE MOSCOW MATHEMATICAL SOCIETY Translation edited by Frances H. Goldman with the assistance of AMS staff A translation of TRUDY MOSKOVSKOGO MATEMATIQESKOGO OBWESTVA Editorial Board A. G. Sergeev (Editor in Chief) V. M. Buchstaber E.` B. Vinberg Yu. S. Ilyashenko A. A. Shkalikov A. V. Domrin (Corresponding Secretary) Advisory Board R. A. Minlos S. P. Novikov N. Kh. Rozov Library of Congress Card Number 65-4713 SUBSCRIPTION INFORMATION. Transactions of the Moscow Mathematical Society for 2011 will consist of one issue. Beginning in 2004, Transactions of the Moscow Mathematical Society is accessible from www.ams.org/journals/. The subscription prices for 2011 are US$532.00 list; US$425.00 institutional member. Upon request, subscribers to paper delivery of this journal are also entitled to receive electronic delivery. For paper delivery, subscribers outside the United States and India must pay a postage surcharge of US$6.00; subscribers in India must pay a postage surcharge of US$13.00. Expedited delivery to destinations in North America US$8.00; elsewhere US$17.00. Subscription renewals are subject to late fees. See www.ams.org/journal-faq for more journal subscription information. BACK NUMBER INFORMATION. For back issues see www.ams.org/bookstore. Subscriptions and orders should be addressed to American Mathematical Society, P.O. Box 845904, Boston, MA 02284-5904, USA. All orders must be accompanied by payment. Other cor- respondence should be addressed to 201 Charles St., Providence, RI 02904-2294, USA.
  • Arxiv:1708.06571V1 [Nlin.AO] 22 Aug 2017 the Generating Unit, Which Is the Pressure Reservoir Be- Coupling: Neath the Pipe at a Basically Constant Rate

    Arxiv:1708.06571V1 [Nlin.AO] 22 Aug 2017 the Generating Unit, Which Is the Pressure Reservoir Be- Coupling: Neath the Pipe at a Basically Constant Rate

    SAW16, dated September 28, 2018 Synchronization of organ pipes 1, 2 1 Jakub Sawicki, ∗ Markus Abel, and Eckehard Schöll 1Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany 2Department of Physics and Astronomy, Potsdam University, Karl-Liebknecht-Straße 24, 14476 Potsdam Germany We investigate synchronization of coupled organ pipes. Synchronization and reflection in the organ lead to undesired weakening of the sound in special cases. Recent experiments have shown that sound interaction is highly complex and nonlinear, however, we show that two delay-coupled Van-der-Pol oscillators appear to be a good model for the occurring dynamical phenomena. Here the coupling is realized as distance-dependent, or time-delayed, equivalently. Analytically, we investigate the synchronization frequency and bifurcation scenarios which occur at the boundaries of the Arnold tongues. We successfully compare our results to experimental data. PACS numbers: 05.45.Xt, 05.45.-a Introduction - The physics of organ pipes is an in- we investigate the bifurcation scenarios in the context of terdisciplinary topic where many fields of science meet. two delay-coupled Van-der-Pol oscillators as a represen- It is highly interesting as it includes elements of non- tation of the system of two coupled organ pipes, such linear dynamical system theory [4, 6, 11], aeroacoustic as in the experimental setup of Bergweiler et al. [5]. modeling [13] and synchronization theory [17]. The fo- In extension of previous work, we study the dependence cus of these different research areas is the “queen of in- of Arnold tongues under variation of the time delay τ struments” which captivates through the grandeur of her and the coupling strength κ, to explore how undesired sight and majesty of her sound.
  • Recurrence Quantification of Fractal Structures

    Recurrence Quantification of Fractal Structures

    METHODS ARTICLE published: 01 October 2012 doi: 10.3389/fphys.2012.00382 Recurrence quantification of fractal structures Charles L. Webber Jr.* Department of Cell and Molecular Physiology, Stritch School of Medicine, Loyola University Chicago Health Sciences Division, Maywood, IL, USA Edited by: By definition, fractal structures possess recurrent patterns. At different levels repeating Michael A. Riley, University of patterns can be visualized at higher magnifications. The purpose of this chapter is three- Cincinnati, USA fold. First, general characteristics of dynamical systems are addressed from a theoretical Reviewed by: Rick Dale, University of California mathematical perspective. Second, qualitative and quantitative recurrence analyses are Merced, USA reviewed in brief, but the reader is directed to other sources for explicit details. Third, Adam Kiefer, Brown University, USA example mathematical systems that generate strange attractors are explicitly defined, giv- *Correspondence: ing the reader the ability to reproduce the rich dynamics of continuous chaotic flows or Charles L. Webber Jr., Department of discrete chaotic iterations. The challenge is then posited for the reader to study for them- Cell and Molecular Physiology, Stritch School of Medicine, Loyola University selves the recurrent structuring of these different dynamics. With a firm appreciation of the Chicago Health Sciences Division, power of recurrence analysis, the reader will be prepared to turn their sights on real-world 2160 South First Avenue, Maywood, systems (physiological, psychological, mechanical, etc.). IL 60153-3328, USA. e-mail: [email protected] Keywords: dynamical systems, recurrence analysis, mathematical fractals, homeodynamics, dimensionality DYNAMICAL SYSTEMS IN N-DIMENSIONAL SPACE are attracted to so-called constant or static values compatible with HOMEOSTASIS VERSUS HOMEODYNAMICS life.
  • Stochastic Analysis of Recurrence Plots with Applications to the Detection of Deterministic Signals

    Stochastic Analysis of Recurrence Plots with Applications to the Detection of Deterministic Signals

    Stochastic analysis of recurrence plots with applications to the detection of deterministic signals Gustavo K. Rohde,∗yJonathan M. Nichols, Bryan M. Dissinger, Frank Bucholtz Naval Research Laboratory, Washington, DC September 17, 2007 Abstract Recurrence plots have been widely used for a variety of purposes such as analyzing dy- namical systems, denoising, as well as detection of deterministic signals embedded in noise. Though it has been postulated previously that recurrence plots contain time correlation in- formation here we make the relationship between unthresholded recurrence plots and the covariance of a random process more precise. Computations using examples from harmonic processes, autoregressive models, and outputs from nonlinear systems are shown to illustrate this relationship. Finally, the use of recurrence plots for detection of deterministic signals in the presence of noise is investigated and compared to traditional signal detection meth- ods based on the likelihood ratio test. Results using simulated data show that detectors based on certain statistics derived from recurrence plots are sub-optimal when compared to well-known detectors based on the likelihood ratio. Keywords: Recurrence plots, stochastic processes, detection PACS codes: 05.45.Tp, 02.50.Fz ∗Currently at Center for Bioimage Informatics, Biomedical Engineering Department, Carnegie Mellon University. yCorresponding author: HH C 122, Center for Bioimage Informatics, Biomedical Engineering Department, Carnegie Mellon University. 5000 Forbes Ave., Pittsburgh, PA. 15213. Email: [email protected] 1 1 Introduction Since its introduction by Eckmann and Ruelle [1] the recurrence plot has emerged as a useful tool in the analysis of nonlinear, non-stationary time series. As the name suggests, a recurrence plot provides a graphical picture of the times at which a process will return (recur) to a given state.
  • Modulational Instability in Optical Fibers with Randomly-Kicked Normal

    Modulational Instability in Optical Fibers with Randomly-Kicked Normal

    Modulational instability in optical fibers with randomly-kicked normal dispersion G Dujardin, A Armaroli, Simona Nodari, A Mussot, A Kudlinski, S Trillo, M Conforti, Stephan de Bièvre To cite this version: G Dujardin, A Armaroli, Simona Nodari, A Mussot, A Kudlinski, et al.. Modulational instability in optical fibers with randomly-kicked normal dispersion. Physical Review A, American Physical Society, 2021, 103 (5), pp.053521. 10.1103/PhysRevA.103.053521. hal-03157350v2 HAL Id: hal-03157350 https://hal.archives-ouvertes.fr/hal-03157350v2 Submitted on 10 Jun 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PHYSICAL REVIEW A 103, 053521 (2021) Modulational instability in optical fibers with randomly kicked normal dispersion G. Dujardin ,1 A. Armaroli ,2 S. Rota Nodari ,3 A. Mussot,2 A. Kudlinski,2 S. Trillo,4 M. Conforti ,2,* and S. De Bièvre 1,† 1Univ. Lille, CNRS UMR 8524 - Laboratoire Paul Painlevé, Inria, F-59000 Lille, France 2Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France 3Institut de Mathématiques de Bourgogne (IMB), CNRS, UMR 5584, Université Bourgogne Franche Comté, F-21000 Dijon, France 4Department of Engineering, University of Ferrara, I-44122 Ferrara, Italy (Received 3 March 2021; accepted 7 May 2021; published 24 May 2021) We study modulational instability (MI) in optical fibers with random group-velocity dispersion (GVD) generated by sharply localized perturbations of a normal GVD fiber that are either randomly or periodically placed along the fiber and that have random strength.
  • DIFFERENCE EQUATION of LORENZ SYSTEM Wanqing Song1

    DIFFERENCE EQUATION of LORENZ SYSTEM Wanqing Song1

    International Journal of Pure and Applied Mathematics Volume 83 No. 1 2013, 101-110 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v83i1.9 ijpam.eu DIFFERENCE EQUATION OF LORENZ SYSTEM Wanqing Song1 §, Jianru Liang2 1,2College of Electronic and Electrical Engineering Shanghai University of Engineering Science Shanghai, 201620, P.R. CHINA Abstract: This paper uses difference equation to explore some of the more obvious properties of the Lorenz equation. We will investigate changes in the behavior of solutions of the Lorenz equation as the parameter r is varied. A trajectory in phase space is analyzed when iterative equation is magnified. The distance between two states i and j in the phase space is described by recurrence plots which may reveal the motion feature of chaos system trajectory with time change. AMS Subject Classification: 65P20 Key Words: discrete Lorenz, phase space trajectories, recurrence plot 1. Introduction The Lorenz equation is a model of thermally induced fluid convection in the atmosphere and published [1, 2] by E.N Lorenz (an atmospheric scientist) of M.I.T. in 1963. In Lorenz’s mathematical model of convection, three state vari- ables are used (x, y, z). they are not spatial variables but are more abstract. The variable x is proportional to the amplitude of the fluid velocity circula- tion in the fluid ring, positive representing clockwise and negative representing counterclockwise motion. The variable y is temperature difference between up c 2013 Academic Publications, Ltd. Received: October 23, 2012 url: www.acadpubl.eu §Correspondence author 102 W.
  • Bifurcations, Mode Locking, and Chaos in a Dissipative System

    Bifurcations, Mode Locking, and Chaos in a Dissipative System

    The Bogdanov Map: Bifurcations, Mode Locking, and Chaos in a Dissipative System David K. Arrowsmith, School of Mathematical Sciences Queen Mary and Westfield College University of London Mile End Road London E1 4NS, UK Julyan H. E. Cartwright, Departament de F´ısica Universitat de les Illes Balears 07071 Palma de Mallorca, Spain Alexis N. Lansbury, Department of Physics Brunel, The University of West London Uxbridge, Middlesex UB8 3PH, UK &ColinM.Place.∗ Int. J. Bifurcation and Chaos, 3,803–842,1993 We investigate the bifurcations and basins of attraction in the Bog- danov map, a planar quadratic map which is conjugate to the Henon´ area-preserving map in its conservative limit. It undergoesaHopf bifurcation as dissipation is added, and exhibits the panoply of mode locking, Arnold tongues, and chaos as an invariant circle grows out, finally to be destroyed in the homoclinic tangency of the manifolds of aremotesaddlepoint.TheBogdanovmapistheEulermapofatwo- dimensional system of ordinary differential equations firstconsidered by Bogdanov and Arnold in their study of the versal unfolding of the double-zero-eigenvalue singularity, and equivalently of avectorfield invariant under rotation of the plane by an angle 2π.Itisauseful system in which to observe the effect of dissipative perturbations on Hamiltonian structure. In addition, we argue that the Bogdanov map provides a good approximation to the dynamics of the Poincaremaps´ of periodically forced oscillators. ∗Formerly: Department of Mathematics, Westfield College, University of London, UK. 1 1. Introduction Tisnowwellknownthatthestudyofsystemsofordinarydifferential equations, which commonly arise in dynamical systems investigated in I many fields of science, can be aided by utilizing the surface-of-section technique of Poincaremaps.The´ Poincar´e or return map of a system of or- dinary differential equations reduces the dimension of the problem, replac- ing an n-dimensional set of ordinary differential equations with an (n 1)- dimensional set of difference equations.
  • Autoassociative Memory and Pattern Recognition in Micromechanical

    Autoassociative Memory and Pattern Recognition in Micromechanical

    www.nature.com/scientificreports OPEN Autoassociative Memory and Pattern Recognition in Micromechanical Oscillator Received: 30 August 2016 Accepted: 28 February 2017 Network Published: xx xx xxxx Ankit Kumar1 & Pritiraj Mohanty2 Towards practical realization of brain-inspired computing in a scalable physical system, we investigate a network of coupled micromechanical oscillators. We numerically simulate this array of all-to-all coupled nonlinear oscillators in the presence of stochasticity and demonstrate its ability to synchronize and store information in the relative phase differences at synchronization. Sensitivity of behavior to coupling strength, frequency distribution, nonlinearity strength, and noise amplitude is investigated. Our results demonstrate that neurocomputing in a physically realistic network of micromechanical oscillators with silicon-based fabrication process can be robust against noise sources and fabrication process variations. This opens up tantalizing prospects for hardware realization of a low-power brain- inspired computing architecture that captures complexity on a scalable manufacturing platform. Inspired by studies that have indicated that subsystems of the brain involved in associative learning exhibit syn- chronization dynamics by which pattern recognition emerges from the frequency entrainment of the constituent oscillating neurons1, significant recent interest has developed around the prospect of constructing analogous systems using artificial, physical oscillators. Such systems of coupled physical oscillators are capable of autoasso- ciative memory operation and other forms of parallel, non-Boolean and neuromorphic computing, and suitably engineered, offer the advantages of far higher operating frequencies than their biological counterparts, and far lower power requirements than attempts to simulate neural networks on traditional hardware. The dynamics of a system of coupled oscillators can exhibit attractive limit cycles that represent synchronized states.
  • Math Morphing Proximate and Evolutionary Mechanisms

    Math Morphing Proximate and Evolutionary Mechanisms

    Curriculum Units by Fellows of the Yale-New Haven Teachers Institute 2009 Volume V: Evolutionary Medicine Math Morphing Proximate and Evolutionary Mechanisms Curriculum Unit 09.05.09 by Kenneth William Spinka Introduction Background Essential Questions Lesson Plans Website Student Resources Glossary Of Terms Bibliography Appendix Introduction An important theoretical development was Nikolaas Tinbergen's distinction made originally in ethology between evolutionary and proximate mechanisms; Randolph M. Nesse and George C. Williams summarize its relevance to medicine: All biological traits need two kinds of explanation: proximate and evolutionary. The proximate explanation for a disease describes what is wrong in the bodily mechanism of individuals affected Curriculum Unit 09.05.09 1 of 27 by it. An evolutionary explanation is completely different. Instead of explaining why people are different, it explains why we are all the same in ways that leave us vulnerable to disease. Why do we all have wisdom teeth, an appendix, and cells that if triggered can rampantly multiply out of control? [1] A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Beno?t Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. http://www.kwsi.com/ynhti2009/image01.html A fractal often has the following features: 1. It has a fine structure at arbitrarily small scales.
  • Thesis Is the final Work of My Ph.D

    Thesis Is the final Work of My Ph.D

    UvA-DARE (Digital Academic Repository) Bifurcation of random maps Zmarrou, H. Publication date 2008 Document Version Final published version Link to publication Citation for published version (APA): Zmarrou, H. (2008). Bifurcation of random maps. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:06 Oct 2021 Bifurcation of random maps Hicham Zmarrou Bifurcation of random maps Academisch proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. D.C. van den Boom ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op vrijdag 28 maart 2008, te 14:00 uur door Hicham Zmarrou geboren te Oued Amlil, Marokko Promotiecommissie: Promotor: Prof.