Abstract

Using an example a system of two coupled generators of quasi-periodic oscillations, we study the occurrence of chaotic dynamics with one positive, two zero, and several negative Lyapunov exponents. It is shown that such dynamic arises as a result of a sequence of bifurcations of two-frequency torus doubling and involves saddle tori occurring at their doublings. This transition is associated with typical structure of parameter plane, like cross-road area and shrimp-shaped structures, based on the two-frequency quasi-periodic dynamics. Using double Poincaré section, we have shown destruction of three-frequency torus.

References

References
1.
Pikovsky
,
A.
,
Kurths
,
J.
,
Rosenblum
,
M.
, and
Kurths
,
J.
,
2003
,
Synchronization: A Universal Concept in Nonlinear Sciences
, Vol.
12
,
Cambridge University Press
, Cambridge, UK.
2.
Anishchenko
,
V. S.
,
Vadivasova
,
T. E.
, and
Strelkova
,
G. I.
,
2014
, “
Synchronization of Periodic Self-Sustained Oscillations
,”
Deterministic Nonlinear Systems
,
Springer
, Cham, Switzerland, pp.
217
243
.10.1007/978-3-319-06871-8
3.
Shilnikov
,
A.
,
Shilnikov
,
L.
, and
Turaev
,
D.
,
2004
, “
On Some Mathematical Topics in Classical Synchronization: A Tutorial
,”
Int. J. Bifurcation Chaos
,
14
(
07
), pp.
2143
2160
.10.1142/S0218127404010539
4.
Belykh
,
V. N.
,
Belykh
,
I. V.
,
Hasler
,
M.
, and
Nevidin
,
K. V.
,
2003
, “
Cluster Synchronization in Three-Dimensional Lattices of Diffusively Coupled Oscillators
,”
Int. J. Bifurcation Chaos
,
13
(
04
), pp.
755
779
.10.1142/S0218127403006923
5.
McGraw
,
P. N.
, and
Menzinger
,
M.
,
2005
, “
Clustering and the Synchronization of Oscillator Networks
,”
Phys. Rev. E
,
72
(
1
), p.
015101
.10.1103/PhysRevE.72.015101
6.
Czolczynski
,
K.
,
Perlikowski
,
P.
,
Stefanski
,
A.
, and
Kapitaniak
,
T.
,
2009
, “
Clustering and Synchronization of n Huygens' Clocks
,”
Phys. A: Stat. Mech. Appl.
,
388
(
24
), pp.
5013
5023
.10.1016/j.physa.2009.08.033
7.
Abrams
,
D. M.
, and
Strogatz
,
S. H.
,
2006
, “
Chimera States in a Ring of Nonlocally Coupled Oscillators
,”
Int. J. Bifurcation Chaos
,
16
(
01
), pp.
21
37
.10.1142/S0218127406014551
8.
Bordyugov
,
G.
,
Pikovsky
,
A.
, and
Rosenblum
,
M.
,
2010
, “
Self-Emerging and Turbulent Chimeras in Oscillator Chains
,”
Phys. Rev. E
,
82
(
3
), p.
035205
.10.1103/PhysRevE.82.035205
9.
Omelchenko
,
I.
,
Maistrenko
,
Y.
,
Hövel
,
P.
, and
Schöll
,
E.
,
2011
, “
Loss of Coherence in Dynamical Networks: Spatial Chaos and Chimera States
,”
Phys. Rev. Lett.
,
106
(
23
), p.
234102
.10.1103/PhysRevLett.106.234102
10.
Ashwin
,
P.
, and
Burylko
,
O.
,
2015
, “
Weak Chimeras in Minimal Networks of Coupled Phase Oscillators
,”
Chaos: An Interdiscip. J. Nonlinear Sci.
,
25
(
1
), p.
013106
.10.1063/1.4905197
11.
Nayfeh
,
A. H.
, and
Balachandran
,
B.
,
1989
, “
Modal Interactions in Dynamical and Structural Systems
,”
ASME Appl. Mech. Rev.
,
42
(
11
), pp.
175
201
.10.1115/1.3152389
12.
Balachandran
,
B.
, and
Nayfeh
,
A.
,
1992
, “
Cyclic Motions Near a Hopf Bifurcation of a Four-Dimensional System
,”
Nonlinear Dyn.
,
3
(
1
), pp.
19
39
.10.1007/BF00045469
13.
Emelianova
,
Y. P.
,
Kuznetsov
,
A.
,
Sataev
,
I.
, and
Turukina
,
L.
,
2013
, “
Synchronization and Multi-Frequency Oscillations in the Low-Dimensional Chain of the Self-Oscillators
,”
Phys. D: Nonlinear Phenom.
,
244
(
1
), pp.
36
49
.10.1016/j.physd.2012.10.012
14.
Emelianova
,
Y. P.
,
Emelyanov
,
V. V.
, and
Ryskin
,
N. M.
,
2014
, “
Synchronization of Two Coupled Multimode Oscillators With Time-Delayed Feedback
,”
Commun. Nonlinear Sci. Numer. Simul.
,
19
(
10
), pp.
3778
3791
.10.1016/j.cnsns.2014.03.031
15.
Kaneko
,
K.
,
1983
, “
Doubling of Torus
,”
Prog. Theor. Phys.
,
69
(
6
), pp.
1806
1810
.10.1143/PTP.69.1806
16.
Kaneko
,
K.
,
1984
, “
Oscillation and Doubling of Torus
,”
Prog. Theor. Phys.
,
72
(
2
), pp.
202
215
.10.1143/PTP.72.202
17.
Anishchenko
,
V.
,
Safonova
,
M.
,
Feudel
,
U.
, and
Kurths
,
J.
,
1994
, “
Bifurcations and Transition to Chaos Through Three-Dimensional Tori
,”
Int. J. Bifurcation Chaos
,
04
(
03
), pp.
595
607
.10.1142/S0218127494000423
18.
Alaggio
,
R.
, and
Rega
,
G.
,
2000
, “
Characterizing Bifurcations and Classes of Motion in the Transition to Chaos Through 3D-Tori of a Continuous Experimental System in Solid Mechanics
,”
Phys. D: Nonlinear Phenom.
,
137
(
1–2
), pp.
70
93
.10.1016/S0167-2789(99)00169-4
19.
Pazó
,
D.
,
Sánchez
,
E.
, and
Matias
,
M. A.
,
2001
, “
Transition to High-Dimensional Chaos Through Quasiperiodic Motion
,”
Int. J. Bifurcation Chaos
,
11
(
10
), pp.
2683
2688
.10.1142/S0218127401003747
20.
Anishchenko
,
V.
, and
Nikolaev
,
S.
,
2005
, “
Generator of Quasi-Periodic Oscillations Featuring Two-Dimensional Torus Doubling Bifurcations
,”
Tech. Phys. Lett.
,
31
(
10
), pp.
853
855
.10.1134/1.2121837
21.
Landau
,
L. D.
,
1944
, “
On the Problem of Turbulence
,”
Dokl. Akad. Nauk USSR
,
44
, p.
311
.
22.
Hopf
,
E.
,
1948
, “
A Mathematical Example Displaying Features of Turbulence
,”
Commun. Pure Appl. Math.
,
1
(
4
), pp.
303
322
.10.1002/cpa.3160010401
23.
Kuznetsov
,
A. P.
,
Kuznetsov
,
S. P.
,
Sataev
,
I. R.
, and
Turukina
,
L. V.
,
2013
, “
About Landau–Hopf Scenario in a System of Coupled Self-Oscillators
,”
Phys. Lett. A
,
377
(
45–48
), pp.
3291
3295
.10.1016/j.physleta.2013.10.013
24.
Stankevich
,
N. V.
,
Dvorak
,
A.
,
Astakhov
,
V.
,
Jaros
,
P.
,
Kapitaniak
,
M.
,
Perlikowski
,
P.
, and
Kapitaniak
,
T.
,
2018
, “
Chaos and Hyperchaos in Coupled Antiphase Driven Toda Oscillators
,”
Regular Chaotic Dyn.
,
23
(
1
), pp.
120
126
.10.1134/S1560354718010094
25.
Garashchuk
,
I. R.
,
Sinelshchikov
,
D. I.
,
Kazakov
,
A. O.
, and
Kudryashov
,
N. A.
,
2019
, “
Hyperchaos and Multistability in the Model of Two Interacting Microbubble Contrast Agents
,”
Chaos: An Interdiscip. J. Nonlinear Sci.
,
29
(
6
), p.
063131
.10.1063/1.5098329
26.
Stankevich
,
N.
,
Kuznetsov
,
A.
,
Popova
,
E.
, and
Seleznev
,
E.
,
2019
, “
Chaos and Hyperchaos Via Secondary Neimark–Sacker Bifurcation in a Model of Radiophysical Generator
,”
Nonlinear Dyn.
,
97
(
4
), pp.
2355
2370
.10.1007/s11071-019-05132-0
27.
Pikovsky
,
A.
, and
Politi
,
A.
,
2016
,
Lyapunov Exponents: A Tool to Explore Complex Dynamics
,
Cambridge University Press
, Cambridge, UK.
28.
Stankevich
,
N.
,
Astakhov
,
O.
,
Kuznetsov
,
A.
, and
Seleznev
,
E.
,
2018
, “
Exciting Chaotic and Quasi-Periodic Oscillations in a Multicircuit Oscillator With a Common Control Scheme
,”
Tech. Phys. Lett.
,
44
(
5
), pp.
428
431
.10.1134/S1063785018050267
29.
Kuznetsov
,
A.
,
Kuznetsov
,
S.
,
Shchegoleva
,
N.
, and
Stankevich
,
N.
,
2019
, “
Dynamics of Coupled Generators of Quasiperiodic Oscillations: Different Types of Synchronization and Other Phenomena
,”
Phys. D: Nonlinear Phenom.
,
398
, pp.
1
12
.10.1016/j.physd.2019.05.014
30.
Kuznetsov
,
A. P.
, and
Stankevich
,
N. V.
,
2018
, “
Dynamics of Coupled Generators of Quasi-Periodic Oscillations With Equilibrium State
,”
Izvestiya VUZ. Appl. Nonlinear Dyn.
,
26
(
2
), pp.
41
58
.10.18500/0869-6632-2018-26-2-41-58
31.
Kuznetsov
,
A.
,
Kuznetsov
,
S.
,
Mosekilde
,
E.
, and
Stankevich
,
N.
,
2013
, “
Generators of Quasiperiodic Oscillations With Three-Dimensional Phase Space
,”
Eur. Phys. J. Spec. Top.
,
222
(
10
), pp.
2391
2398
.10.1140/epjst/e2013-02023-x
32.
Benettin
,
G.
,
Galgani
,
L.
,
Giorgilli
,
A.
, and
Strelcyn
,
J.-M.
,
1980
, “
Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems; A Method for Computing All of Them—Part 1: Theory
,”
Meccanica
,
15
(
1
), pp.
9
20
.10.1007/BF02128236
33.
Ermentrout
,
B.
,
2002
,
Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students
, Vol.
14
,
Society for Industrial and Applied Mathematics, Philadelphia
, PA.
34.
Froeschlé
,
C.
,
Guzzo
,
M.
, and
Lega
,
E.
,
2000
, “
Graphical Evolution of the Arnold Web: From Order to Chaos
,”
Science
,
289
(
5487
), pp.
2108
2110
.10.1126/science.289.5487.2108
35.
Broer
,
H.
,
Simó
,
C.
, and
Vitolo
,
R.
,
2008
, “
The Hopf-Saddle-Node Bifurcation for Fixed Points of 3D-Diffeomorphisms: The Arnol'd Resonance Web
,”
Bull. Belgian Math. Soc. Simon Stevin
,
15
(
5
), pp.
769
787
.10.36045/bbms/1228486406
36.
Sekikawa
,
M.
,
Inaba
,
N.
,
Kamiyama
,
K.
, and
Aihara
,
K.
,
2014
, “
Three-Dimensional Tori and Arnold Tongues
,”
Chaos: Interdiscip. J. Nonlinear Sci.
,
24
(
1
), p.
013137
.10.1063/1.4869303
37.
Vitolo
,
R.
,
Broer
,
H.
, and
Simó
,
C.
,
2011
, “
Quasi-Periodic Bifurcations of Invariant Circles in Low-Dimensional Dissipative Dynamical Systems
,”
Regular Chaotic Dyn.
,
16
(
1–2
), pp.
154
184
.10.1134/S1560354711010060
38.
Zhusubaliyev
,
Z. T.
, and
Mosekilde
,
E.
,
2009
, “
Novel Routes to Chaos Through Torus Breakdown in Non-Invertible Maps
,”
Phys. D: Nonlinear Phenom.
,
238
(
5
), pp.
589
602
.10.1016/j.physd.2008.12.012
39.
Zhusubaliyev
,
Z. T.
,
Laugesen
,
J. L.
, and
Mosekilde
,
E.
,
2010
, “
From Multi-Layered Resonance Tori to Period-Doubled Ergodic Tori
,”
Phys. Lett. A
,
374
(
25
), pp.
2534
2538
.10.1016/j.physleta.2010.04.022
40.
Arnol'd
,
V. I.
,
2012
,
Geometrical Methods in the Theory of Ordinary Differential Equations
, Vol.
250
,
Springer Science & Business Media
, Berlin.
41.
Carcasses
,
J.
,
Mira
,
C.
,
Bosch
,
M.
,
Simó
,
C.
, and
Tatjer
,
J.
,
1991
, “
‘Crossroad Area—Spring Area’ Transition (i) Parameter Plane Presentation
,”
Int. J. Bifurcation Chaos
,
01
(
01
), pp.
183
196
.10.1142/S0218127491000117
42.
Gallas
,
J. A.
,
1993
, “
Structure of the Parameter Space of the Hénon Map
,”
Phys. Rev. Lett.
,
70
(
18
), pp.
2714
2717
.10.1103/PhysRevLett.70.2714
43.
Moon
,
F.
, and
Holmes
,
W.
,
1985
, “
Double Poincaré Sections of a Quasi-Periodically Forced, Chaotic Attractor
,”
Phys. Lett. A
,
111
(
4
), pp.
157
160
.10.1016/0375-9601(85)90565-1
44.
Stankevich
,
N.
,
Kuznetsov
,
A.
,
Popova
,
E.
, and
Seleznev
,
E.
,
2017
, “
Experimental Diagnostics of Multi-Frequency Quasiperiodic Oscillations
,”
Commun. Nonlinear Sci. Numer. Simul.
,
43
, pp.
200
210
.10.1016/j.cnsns.2016.07.007
45.
Stoop
,
R.
,
Benner
,
P.
, and
Uwate
,
Y.
,
2010
, “
Real-World Existence and Origins of the Spiral Organization of Shrimp-Shaped Domains
,”
Phys. Rev. Lett.
,
105
(
7
), p.
074102
.10.1103/PhysRevLett.105.074102
46.
Barrio
,
R.
,
Blesa
,
F.
,
Serrano
,
S.
, and
Shilnikov
,
A.
,
2011
, “
Global Organization of Spiral Structures in Biparameter Space of Dissipative Systems With Shilnikov Saddle-Foci
,”
Phys. Rev. E
,
84
(
3
), p.
035201
.10.1103/PhysRevE.84.035201
47.
Simó
,
C.
,
1979
, “
On the Hénon-Pomeau Attractor
,”
J. Stat. Phys.
,
21
(
4
), pp.
465
494
.10.1007/BF01009612
48.
Rössler
,
O.
,
Stewart
,
H.
, and
Wiesenfeld
,
K.
,
1990
, “
Unfolding a Chaotic Bifurcation
,”
Proc. R. Soc. London. Ser. A
,
431
(
1882
), pp.
371
383
.10.1098/rspa.1990.0137
49.
Vitolo
,
R.
,
Broer
,
H.
, and
Simó
,
C.
,
2010
, “
Routes to Chaos in the Hopf-Saddle-Node Bifurcation for Fixed Points of 3D-Diffeomorphisms
,”
Nonlinearity
,
23
(
8
), pp.
1919
1947
.10.1088/0951-7715/23/8/007
You do not currently have access to this content.