We present a systematic approach to identify the similarities and differences between a chaotic system with delayed feedback and two mutually delay-coupled systems. We consider the general case in which the coupled systems are either unsynchronized or in a generally synchronized state, in contrast to the mostly studied case of identical synchronization. We construct a new time-series for each of the two coupling schemes, respectively, and present analytic evidence and numerical confirmation that these two constructed time-series are statistically equivalent. From the construction, it then follows that the distribution of time-series segments that are small compared to the overall delay in the system is independent of the value of the delay and of the coupling scheme. By focusing on numerical simulations of delay-coupled chaotic lasers, we present a practical example of our findings.

1.
F.
Takens
, “
Detecting strange attractors in turbulence
,” in
Dynamical Systems and Turbulence
, Lecture Notes in Mathematics Vol. 898, edited by
D. A.
Rand
and
L.-S.
Young
(
Springer-Verlag
,
1981
), pp.
366
381
.
2.
P.
Grassberger
and
I.
Procaccia
, “
Measuring the strangeness of strange attractors
,”
Physica D
9
,
189
208
(
1983
).
3.
J.-P.
Eckmann
and
D.
Ruelle
, “
Ergodic theory of chaos and strange attractors
,”
Rev. Mod. Phys.
57
,
617
(
1985
).
4.
A.
Wolf
,
J. B.
Swift
,
H. L.
Swinney
, and
J. A.
Vastano
, “
Determining Lyapunov exponents from a time series
,”
Physica D
16
,
285
317
(
1985
).
5.
M. C.
Mackey
and
L.
Glass
, “
Oscillation and chaos in physiological control systems
,”
Science
197
,
287
289
(
1977
).
6.
J. D.
Farmer
, “
Chaotic attractors of an infinite-dimensional dynamical system
,”
Physica D
4
,
366
393
(
1982
).
7.
K.
Ikeda
and
K.
Matsumoto
, “
High-dimensional chaotic behavior in systems with time-delayed feedback
,”
Physica D
29
,
223
(
1987
).
8.
T.
Erneux
, “
Applied delay differential equations
,” in
Surveys and Tutorials
, Applied Mathematical Sciences Vol. 3 (
Springer
,
2009)
.
9.
G.
Van der Sande
,
M. C.
Soriano
,
I.
Fischer
, and
C. R.
Mirasso
, “
Dynamics, correlation scaling, and synchronization behavior in rings of delay-coupled oscillators
,”
Phys. Rev. E
77
,
055202
R
(
2008
).
10.
M. C.
Soriano
,
J.
García-Ojalvo
,
C. R.
Mirasso
, and
I.
Fischer
, “
Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers
,”
Rev. Mod. Phys.
85
,
421
470
(
2013
).
11.
D.
Lenstra
,
B. H.
Verbeek
, and
A. J.
den Boef
, “
Coherence collapse in single-mode semiconductor lasers due to optical feedback
,”
IEEE J. Quantum Electron.
21
,
674
679
(
1985
).
12.
I.
Fischer
,
G. H. M.
van Tartwijk
,
A. M.
Levine
,
W.
Elsäßer
,
E.
Göbel
, and
D.
Lenstra
, “
Fast pulsing and chaotic itinerancy with a drift in the coherence collapse of semiconductor lasers
,”
Phys. Rev. Lett.
76
,
220
223
(
1996
).
13.
T.
Heil
,
I.
Fischer
,
W.
Elsäßer
,
J.
Mulet
, and
C. R.
Mirasso
, “
Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers
,”
Phys. Rev. Lett.
86
,
795
798
(
2001
).
14.
T.
Heil
,
I.
Fischer
,
W.
Elsäßer
, and
A.
Gavrielides
, “
Dynamics of semiconductor lasers subject to delayed optical feedback: The short cavity regime
,”
Phys. Rev. Lett.
87
,
243901
(
2001
).
15.
H.
Fujino
and
J.
Ohtsubo
, “
Synchronization of chaotic oscillations in mutually coupled semiconductor lasers
,”
Opt. Rev.
8
,
351
357
(
2001
).
16.
M.
Nixon
,
M.
Friedman
,
E.
Ronen
,
A. A.
Friesem
,
N.
Davidson
, and
I.
Kanter
, “
Synchronized cluster formation in coupled laser networks
,”
Phys. Rev. Lett.
106
,
223901
(
2011
).
17.
J.
Tiana-Alsina
,
K.
Hicke
,
X.
Porte
,
M. C.
Soriano
,
M. C.
Torrent
,
J.
García-Ojalvo
, and
I.
Fischer
, “
Zero-lag synchronization and bubbling in delay-coupled lasers
,”
Phys. Rev. E
85
,
026209
(
2012
).
18.
G. D.
Vanwiggeren
and
R.
Roy
, “
Communication with chaotic lasers
,”
Science
279
,
1198
1200
(
1998
).
19.
A.
Argyris
,
D.
Syvridis
,
L.
Larger
,
V.
Annovazzi-Lodi
,
P.
Colet
,
I.
Fischer
,
J.
García-Ojalvo
,
C. R.
Mirasso
,
L.
Pesquera
, and
K. A.
Shore
, “
Chaos-based communications at high bit rates using commercial fiber-optic links
,”
Nature
438
,
343
(
2005
).
20.
L.
Larger
and
J. P.
Goedgebuer
, “
Cryptography using optical chaos
,”
C. R. Phys.
5
,
609
(
2004
).
21.
A.
Uchida
,
K.
Amano
,
M.
Inoue
,
K.
Hirano
,
S.
Naito
,
H.
Someya
,
I.
Oowada
,
T.
Kurashige
,
M.
Shiki
,
S.
Yoshimori
,
K.
Yoshimura
, and
P.
Davis
, “
Fast physical random bit generation with chaotic semiconductor lasers
,”
Nature Photon.
2
,
728
732
(
2008
).
22.
C.-W.
Shih
, “
Influence of boundary conditions on pattern formation and spatial chaos in lattice systems
,”
SIAM J. Appl. Math.
61
,
335
368
(
2000
).
23.
G.
Giacomelli
and
A.
Politi
, “
Relationship between delayed and spatially extended dynamical systems
,”
Phys. Rev. Lett.
76
,
2686
(
1996
).
24.
O.
D'Huys
,
I.
Fischer
,
J.
Danckaert
, and
R.
Vicente
, “
Spectral and correlation properties of rings of delay-coupled elements: Comparing linear and nonlinear systems
,”
Phys. Rev. E
85
,
056209
(
2012
).
25.
R.
Vicente
,
J.
Daudén
,
P.
Colet
, and
R.
Toral
, “
Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop
,”
IEEE J. Quantum Electron.
41
,
541
(
2005
).
26.
K.
Kanno
and
A.
Uchida
, “
Consistency and complexity in coupled semiconductor lasers with time-delayed optical feedback
,”
Phys. Rev. E
86
,
066202
(
2012
).
27.
S.
Heiligenthal
,
T.
Dahms
,
S.
Yanchuk
,
T.
Jüngling
,
V.
Flunkert
,
I.
Kanter
,
E.
Schöll
, and
W.
Kinzel
, “
Strong and weak chaos in nonlinear networks with time-delayed couplings
,”
Phys. Rev. Lett.
107
,
234102
(
2011
).
28.
S.
Heiligenthal
,
T.
Jüngling
,
O.
D'Huys
,
D. A.
Arroyo-Almanza
,
M. C.
Soriano
,
I.
Fischer
,
I.
Kanter
, and
W.
Kinzel
, “
Strong and weak chaos in networks of semiconductor lasers with time-delayed couplings
,”
Phys. Rev. E
88
,
012902
(
2013
).
You do not currently have access to this content.