The simplest ring oscillator is made from three strongly nonlinear elements repressing each other unidirectionally, resulting in the emergence of a limit cycle. A popular implementation of this scheme uses repressor genes in bacteria, creating the synthetic genetic oscillator known as the Repressilator. We consider the main collective modes produced when two identical Repressilators are mean-field-coupled via the quorum-sensing mechanism. In-phase and anti-phase oscillations of the coupled oscillators emerge from two Andronov–Hopf bifurcations of the homogeneous steady state. Using the rate of the repressor's production and the value of coupling strength as the bifurcation parameters, we performed one-parameter continuations of limit cycles and two-parameter continuations of their bifurcations to show how bifurcations of the in-phase and anti-phase oscillations influence the dynamical behaviors for this system. Pitchfork bifurcation of the unstable in-phase cycle leads to the creation of novel inhomogeneous limit cycles with very different amplitudes, in contrast to the well-known asymmetrical limit cycles arising from oscillation death. The Neimark–Sacker bifurcation of the anti-phase cycle determines the border of an island in two-parameter space containing almost all the interesting regimes including the set of resonant limit cycles, the area with stable inhomogeneous cycle, and very large areas with chaotic regimes resulting from torus destruction and period doubling of resonant cycles and inhomogeneous cycles. We discuss the structure of the chaos skeleton to show the role of inhomogeneous cycles in its formation. Many regions of multistability and transitions between regimes are presented. These results provide new insights into the coupling-dependent mechanisms of multistability and collective regime symmetry breaking in populations of identical multidimensional oscillators.

1.
J.
Peña Ramirez
 et al, “
The sympathy of two pendulum clocks: Beyond Huygens’ observations
,”
Sci. Rep.
6
(
1
),
23580
(
2016
).
2.
B.
van der Pol
and
J.
van der Mark
, “
The heartbeat considered as a relaxation oscillation, and an electrical model of the heart
,”
London Edinburgh Dublin Philos. Mag. J. Sci.
6
(
38
),
763
775
(
1928
).
3.
Y.
Kuramoto
and
D.
Battogtokh
, “
Coexistence of coherence and incoherence in nonlocally coupled phase oscillators
,”
Nonlinear Phenom. Complex Syst.
5
(
4
),
380
385
(
2002
).
4.
O. E.
Omel'chenko
 et al, “
Stationary patterns of coherence and incoherence in two-dimensional arrays of non-locally-coupled phase oscillators
,”
Phys. Rev. E
85
(
3
),
036210
(
2012
).
5.
M. R.
Tinsley
,
S.
Nkomo
, and
K.
Showalter
, “
Chimera and phase-cluster states in populations of coupled chemical oscillators
,”
Nat. Phys.
8
(
9
),
662
665
(
2012
).
6.
E. A.
Martens
 et al, “
Chimera states in mechanical oscillator networks
,”
Proc. Natl. Acad. Sci. U.S.A.
110
(
26
),
10563
10567
(
2013
).
7.
M. J.
Panaggio
and
D. M.
Abrams
, “
Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators
,”
Nonlinearity
28
(
3
),
R67
(
2015
).
8.
V. K.
Vanag
and
I. R.
Epstein
, “
Patterns of nanodroplets: The Belousov-Zhabotinsky-aerosol OT-microemulsion system
,” in
Self-Organized Morphology in Nanostructured Materials
, edited by
K.
Al-Shamery
and
J.
Parisi
(
Springer
,
Berlin
,
2008
), pp.
89
113
.
9.
R.
Karnatak
,
R.
Ramaswamy
, and
A.
Prasad
, “
Synchronization regimes in conjugate coupled chaotic oscillators
,”
Chaos
19
(
3
),
033143
(
2009
).
10.
M.
Dasgupta
,
M.
Rivera
, and
P.
Parmananda
, “
Suppression and generation of rhythms in conjugately coupled nonlinear systems
,”
Chaos
20
(
2
),
023126
(
2010
).
11.
W.
Han
 et al, “
Amplitude death, oscillation death, wave, and multistability in identical Stuart–Landau oscillators with conjugate coupling
,”
Commun. Nonlinear Sci. Numer. Simul.
39
,
73
80
(
2016
).
12.
T.
Stankovski
 et al, “
Coupling functions: Universal insights into dynamical interaction mechanisms
,”
Rev. Mod. Phys.
89
(
4
),
045001
(
2017
).
13.
C. M.
Waters
and
B. L.
Bassler
, “
QUORUM SENSING: Cell-to-cell communication in bacteria
,”
Annu. Rev. Cell Dev. Biol.
21
(
1
),
319
346
(
2005
).
14.
J.
Wolf
 et al, “
Transduction of intracellular and intercellular dynamics in yeast glycolytic oscillations
,”
Biophys. J.
78
(
3
),
1145
1153
(
2000
).
15.
D.
Gonze
,
N.
Markadieu
, and
A.
Goldbeter
, “
Selection of in-phase or out-of-phase synchronization in a model based on global coupling of cells undergoing metabolic oscillations
,”
Chaos
18
(
3
),
037127
(
2008
).
16.
S.
De Monte
 et al, “
Dynamical quorum sensing: Population density encoded in cellular dynamics
,”
Proc. Natl. Acad. Sci. U.S.A.
104
(
47
),
18377
18381
(
2007
).
17.
A. F.
Taylor
 et al, “
Dynamical quorum sensing and synchronization in large populations of chemical oscillators
,”
Science
323
(
5914
),
614
617
(
2009
).
18.
B.-W.
Li
 et al, “
Synchronization and quorum sensing in an ensemble of indirectly coupled chaotic oscillators
,”
Phys. Rev. E
86
(
4
),
046207
(
2012
).
19.
K.
Ponrasu
 et al, “
Symmetry breaking dynamics induced by mean-field density and low-pass filter
,”
Chaos
30
(
5
),
053120
(
2020
).
20.
F. K.
Balagadde
 et al, “
A synthetic Escherichia coli predator-prey ecosystem
,”
Mol. Syst. Biol.
4
,
187
(
2008
).
21.
L.
You
 et al, “
Programmed population control by cell-cell communication and regulated killing
,”
Nature
428
(
6985
),
868
871
(
2004
).
22.
S.
Hennig
,
G.
Rödel
, and
K.
Ostermann
, “
Artificial cell-cell communication as an emerging tool in synthetic biology applications
,”
J. Biol. Eng.
9
,
13
(
2015
).
23.
M. B.
Elowitz
and
S.
Leibler
, “
A synthetic oscillatory network of transcriptional regulators
,”
Nature
403
(
6767
),
335
338
(
2000
).
24.
J.
Garcia-Ojalvo
,
M. B.
Elowitz
, and
S. H.
Strogatz
, “
Modeling a synthetic multicellular clock: Repressilators coupled by quorum sensing
,”
Proc. Natl. Acad. Sci. U.S.A.
101
(
30
),
10955
10960
(
2004
).
25.
E.
Ullner
 et al, “
Multistability and clustering in a population of synthetic genetic oscillators via phase-repulsive cell-to-cell communication
,”
Phys. Rev. Lett.
99
(
14
),
148103
(
2007
).
26.
E.
Ullner
 et al, “
Multistability of synthetic genetic networks with repressive cell-to-cell communication
,”
Phys. Rev. E
78
(
3
),
031904
(
2008
).
27.
A.
Koseska
 et al, “
Cooperative differentiation through clustering in multicellular populations
,”
J. Theor. Biol.
263
(
2
),
189
202
(
2010
).
28.
H.
Niederholtmeyer
 et al, “
Rapid cell-free forward engineering of novel genetic ring oscillators
,”
eLife
4
,
e09771
(
2015
).
29.
L.
Potvin-Trottier
 et al, “
Synchronous long-term oscillations in a synthetic gene circuit
,”
Nature
538
(
7626
),
514
517
(
2016
).
30.
X. J.
Gao
and
M. B.
Elowitz
, “
Synthetic biology: Precision timing in a cell
,”
Nature
538
(
7626
),
462
463
(
2016
).
31.
E. H.
Hellen
 et al, “
An electronic analog of synthetic genetic networks
,”
PLoS ONE
6
(
8
),
e23286
(
2011
).
32.
E. H.
Hellen
,
J.
Kurths
, and
S. K.
Dana
, “
Electronic circuit analog of synthetic genetic networks: Revisited
,”
Eur. Phys. J. Spec. Top.
226
(
9
),
1811
1828
(
2017
).
33.
O.
Buse
,
R.
Pérez
, and
A.
Kuznetsov
, “
Dynamical properties of the repressilator model
,”
Phys. Rev. E
81
(
6
),
066206
(
2010
).
34.
O.
Buşe
,
A.
Kuznetsov
, and
R. A.
Pérez
, “
Existence of limit cycles in the repressilator equations
,”
Int. J. Bifurcation Chaos
19
(
12
),
4097
4106
(
2009
).
35.
K. M.
Page
and
R.
Perez-Carrasco
, “
Degradation rate uniformity determines success of oscillations in repressive feedback regulatory networks
,”
J. R. Soc. Interface
15
(
142
),
20180157
(
2018
).
36.
E. H.
Hellen
and
E.
Volkov
, “
Flexible dynamics of two quorum-sensing coupled repressilators
,”
Phys. Rev. E
95
(
2
),
022408
(
2017
).
37.
E. H.
Hellen
and
E.
Volkov
, “
How to couple identical ring oscillators to get quasiperiodicity, extended chaos, multistability, and the loss of symmetry
,”
Commun. Nonlinear Sci. Numer. Simul.
62
,
462
479
(
2018
).
38.
E. H.
Hellen
 et al, “
Electronic implementation of a repressilator with quorum sensing feedback
,”
PLoS ONE
8
(
5
),
e62997
(
2013
).
39.
B.
Ermentrout
,
Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students
(
SIAM
,
Philadelphia
,
PA
,
2002
).
40.
E. J.
Doedel
 et al, AUTO-07P: Continuation and bifurcation software for ordinary differential equations, Concordia University, 2012, see http://indy.cs.concordia.ca/auto.
41.
A.
Wolf
 et al, “
Determining Lyapunov exponents from a time series
,”
Physica D
16
(
3
),
285
317
(
1985
).
42.
A. N.
Pisarchik
and
U.
Feudel
, “
Control of multistability
,”
Phys. Rep.
540
(
4
),
167
218
(
2014
).
43.
R.
Li
and
B.
Bowerman
, “
Symmetry breaking in biology
,”
Cold Spring Harbor Perspect. Biol.
2
(
3
),
a003475
a003475
(
2010
).
44.
K.
Sathiyadevi
 et al, “
Spontaneous symmetry breaking due to the trade-off between attractive and repulsive couplings
,”
Phys. Rev. E
95
(
4
),
042301
(
2017
).
45.
T.
Banerjee
 et al, “
Transition from homogeneous to inhomogeneous limit cycles: Effect of local filtering in coupled oscillators
,”
Phys. Rev. E
97
(
4
),
042218
(
2018
).
46.
A.
Koseska
,
E.
Volkov
, and
J.
Kurths
, “
Oscillation quenching mechanisms: Amplitude vs. oscillation death
,”
Phys. Rep.
531
(
4
),
173
199
(
2013
).
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