Birhythmicity is evident in many nonlinear systems, which include physical and biological systems. In some living systems, birhythmicity is necessary for response to the varying environment while unnecessary in some physical systems as it limits their efficiency. Therefore, its control is an important area of research. This paper proposes a space-dependent intermittent control scheme capable of controlling birhythmicity in various dynamical systems. We apply the proposed control scheme in five nonlinear systems from diverse branches of natural science and demonstrate that the scheme is efficient enough to control the birhythmic oscillations in all the systems. We derive the analytical condition for controlling birhythmicity by applying harmonic decomposition and energy balance methods in a birhythmic van der Pol oscillator. Further, the efficacy of the control scheme is investigated through numerical and bifurcation analyses in a wide parameter space. Since the proposed control scheme is general and efficient, it may be employed to control birhythmicity in several dynamical systems.

1.
R.
Lozi
and
S.
Ushiki
, “
Coexisting chaotic attractors in Chua’s circuit
,”
Int. J. Bifurcat. Chaos
1
,
923
(
1991
).
2.
A. N.
Pisarchik
,
R.
Jaimes-Reátegui
,
J. R.
Villalobos-Salazar
,
J. H.
García-López
, and
S.
Boccaletti
, “
Synchronization of chaotic systems with coexisting attractors
,”
Phys. Rev. Lett.
96
,
244102
(
2006
).
3.
A. N.
Pisarchik
and
U.
Feudel
, “
Control of multistability
,”
Phys. Rep.
540
,
167
(
2014
).
4.
C. A. K.
Kwuimy
and
C.
Nataraj
, “Recurrence and joint recurrence analysis of multiple attractors energy harvesting system,” in Structural Nonlinear Dynamics and Diagnosis, edited by M. Belhaq (Springer International Publishing, Switzerland, 2015), pp. 97–123.
5.
D.
Biswas
and
T.
Banerjee
,
Time-Delayed Chaotic Dynamical Systems
(
Springer International Publishing
,
2018
).
6.
S.
Kar
and
D. S.
Ray
, “
Large fluctuations and nonlinear dynamics of birhythmicity
,”
EPL
67
,
137
(
2004
).
7.
A.
Goldbeter
,
Biochemical Oscillations and Cellular Rythms. The Molecular Basis of Periodic and Chaotic Behavior
(
Cambridge University Press
,
1996
).
8.
A.
Goldbeter
, “
Computational approaches to cellular rhythms
,”
Nature
420
,
238
245
(
2002
).
9.
R.
Arumugam
,
T.
Banerjee
, and
P. S.
Dutta
, “
Rhythmogenesis, birhythmicity and chaos in a metapopulation model
,”
Eur. Phys. J. ST
4226
,
2145
(
2017
).
10.
R.
Yamapi
and
G.
Filatrella
, “
Noise effects on a birhythmic Josephson junction coupled to a resonator
,”
Phys. Rev. E
89
,
052905
(
2014
).
11.
L.
Weicker
,
T.
Erneux
,
D.
Rosin
, and
D.
Gauthier
, “
Multirhythmicity in an optoelectronic oscillator with large delay
,”
Phys. Rev. E
91
,
012910
(
2015
).
12.
F.
Arecchi
,
R.
Meucci
,
G.
Puccioni
, and
J.
Tredicce
, “
Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser
,”
Phys. Rev. Lett.
49
,
1217
1220
(
1982
).
13.
M.
Alamgir
and
I.
Epstein
, “
Birythmicity and compound oscillation in coupled chemical oscillators: Chlorite-bromate-iodide system
,”
J. Am. Chem. Soc.
105
,
2500
2501
(
1983
).
14.
D.
Cohen
and
J.
Keener
, “
Multiplicity and stability of oscillatory states in a continuous stirred tank reactor with exothermic consecutive reaction a b c
,”
Chem. Eng. Sci.
31
,
115
122
(
1976
).
15.
M.
Morita
,
K.
Iwamoto
, and
M.
Senō
, “
A new permanganate-nitrite-formic acid-methanol oscillator
,”
Bull. Chem. Soc. Jpn.
61
,
3467
3470
(
1988
).
16.
B.
Johnson
,
J.
Griffiths
, and
S.
Scott
, “
Characterisation of oscillations in the H 2 + O 2 reaction in a continuous-flow reactor
,”
J. Chem. Soc. Faraday Trans.
87
,
523
533
(
1991
).
17.
T.
Nagy
,
E.
Verner
,
V.
Gáspár
,
H.
Kori
, and
I.
Kiss
, “
Delayed feedback induced multirhythmicity in the oscillatory electrodissolution of copper
,”
Chaos
25
,
064608
(
2015
).
18.
F.
Morán
and
A.
Goldbeter
, “
Onset of birhythmicity in a regulated biochemical system
,”
Biophys. Chem.
20
,
149
(
1984
).
19.
A.
Goldbeter
and
J.
Yan
, “
Multi-synchronization and other patterns of multi-rhythmicity in oscillatory biological systems
,”
Interface Focus
12
,
20210089
(
2022
).
20.
S.
Kar
and
D. S.
Ray
, “
Large fluctuations and nonlinear dynamics of birhythmicity
,”
Phys. Rev. Lett.
90
,
238102
(
2003
).
21.
J. C.
Leloup
and
A.
Goldbeter
, “
Chaos and birhythmicity in a model for circadian oscillations of the PER and TIM proteins in Drosophila
,”
J. Theor. Biol.
198
,
445
(
1999
).
22.
A.
Goldbeter
and
J. L.
Martiel
, “
Birhythmicity in a model for the cyclic AMP signalling system of the slime mold Dictyostelium discoideum
,”
FEBS Lett.
191
,
149
(
1985
).
23.
J. L.
Martiel
and
A.
Goldbeter
, “
A model based on receptor desensitization for cyclic AMP signaling in Dictyostelium cells
,”
Biophys. J.
52
,
807
(
1987
).
24.
P.
Ghosh
,
S.
Sen
,
S. S.
Riaz
, and
D. S.
Ray
, “
Controlling birhythmicity in a self-sustained oscillator by time-delayed feedback
,”
Phys. Rev. E
83
,
036205
(
2011
).
25.
R.
Sevilla-Escoboza
,
A. N.
Pisarchik
,
R.
Jaimes-Reátegui
, and
G.
Huerta-Cuellar
, “
Selective monostability in multi-stable systems
,”
Proc. R. Soc. London A
471
,
2015005
(
2015
).
26.
D.
Biswas
,
T.
Banerjee
, and
J.
Kurths
, “
Control of birhythmicity through conjugate self-feedback: Theory and experiment
,”
Phys. Rev. E
94
,
042226
(
2016
).
27.
D.
Biswas
,
T.
Banerjee
, and
J.
Kurths
, “
Control of birhythmicity: A self-feedback approach
,”
Chaos
27
,
063110
(
2017
).
28.
D.
Biswas
,
T.
Banerjee
, and
J.
Kurths
, “
Effect of filtered feedback on birhythmicity: Suppression of birhythmic oscillation
,”
Phys. Rev. E
99
,
062210
(
2019
).
29.
D.
Biswas
,
T.
Banerjee
, and
J.
Kurths
, “
Impulsive feedback control of birhythmicity: Theory and experiment
,”
Chaos
32
,
053125
(
2022
).
30.
A. N.
Pisarchik
and
B. K.
Goswami
, “
Annihilation of one of the coexisting attractors in a bistable system
,”
Phys. Rev. Lett.
84
,
1423
(
2000
).
31.
A. N.
Pisarchik
,
Y. O.
Barmenkov
, and
A. V.
Kiryanov
, “
Experimental demonstration of attractor annihilation in a multistable fiber laser
,”
Phys. Rev. E
68
,
066211
(
2003
).
32.
B. E.
Martinez-Zerega
,
A. N.
Pisarchik
, and
L. S.
Tsimring
, “
Using periodic modulation to control coexisting attractors induced by delayed feedback
,”
Phys. Lett. A
318
,
102
(
2003
).
33.
J.
Foss
,
F.
Moss
, and
J.
Milton
, “
Noise, multistability, and delayed recurrent loops
,”
Phys. Rev. E
55
,
4536
(
1997
).
34.
S.
Gadaleta
and
G.
Dangelmayr
, “
Learning to control a complex multistable system
,”
Phys. Rev. E
63
,
036217
(
2001
).
35.
B. K.
Goswami
,
S.
Euzzor
,
K. A.
Naimee
,
A.
Geltrude
,
R.
Meucci
, and
F. T.
Arecchi
, “
Control of stochastic multistable systems: Experimental demonstration
,”
Phys. Rev. E
80
,
016211
(
2009
).
36.
E.
Shajan
and
M. D.
Shrimali
, “
Controlling multistability with intermittent noise
,”
Chaos, Solitons Fractals
160
,
112187
(
2022
).
37.
P. R.
Sharma
,
A.
Sharma
,
M. D.
Shrimali
, and
A.
Prasad
, “
Targeting fixed-point solutions in nonlinear oscillators through linear augmentation
,”
Phys. Rev. E
83
,
067201
(
2011
).
38.
P. R.
Sharma
,
M. D.
Shrimali
,
A.
Prasad
, and
U.
Feudel
, “
Control of multistability in hidden attractors
,”
Phys. Lett. A
377
,
2329
(
2013
).
39.
K.
Yadav
,
N.
Kamal
, and
M. D.
Shrimali
, “
Intermittent feedback induces attractor selection
,”
Phys. Rev. E
95
,
042215
(
2017
).
40.
P.
Gawthrop
,
I.
Loram
,
H.
Gollee
, and
M.
Lakie
, “
Intermittent control models of human standing: Similarities and differences
,”
Biol. Cybern.
108
,
159
168
(
2014
).
41.
Q.
Wang
,
Y.
He
,
G.
Tan
, and
M.
Wu
, “
State-dependent intermittent control of nonlinear systems
,”
IET Control Theory Appl.
11
,
1884
1893
(
2017
).
42.
T.
Yang
and
L.
Chua
, “
Impulsive stabilization for control and synchronization of chaotic systems: Theory and application to secure communication
,”
IEEE Trans. Cir. Syst. I: Fund. Theory Appl.
44
,
976
(
1997
).
43.
Y.-W.
Wang
,
Z.-H.
Guan
, and
J.-W.
Xiao
, “
Impulsive control for synchronization of a class of continuous systems
,”
Chaos
14
,
199
(
2004
).
44.
C.
Li
,
L.
Chen
, and
K.
Aihara
, “
Impulsive control of stochastic systems with applications in chaos control, chaos synchronization, and neural networks
,”
Chaos
18
,
023132
(
2008
).
45.
M.
Schröder
,
M.
Mannattil
,
D.
Dutta
,
S.
Chakraborty
, and
M.
Timme
, “
Transient uncoupling induces synchronization
,”
Phys. Rev. Lett.
115
,
054101
(
2015
).
46.
S.
Ghosh
and
D.
Ray
, “
Chemical oscillator as a generalized Rayleigh oscillator
,”
J. Chem. Phys.
139
,
164112
(
2013
).
47.
S.
Saha
,
G.
Gangopadhyay
, and
D.
Ray
, “
Systematic designing of bi-rhythmic and tri-rhythmic models in families of Van der Pol and Rayleigh oscillators
,”
Commun.Nonlinear Sci. Numer. Simul.
85
,
105234
(
2020
).
48.
W.
Abou-Jaoudé
,
M.
Chaves
, and
J.-L.
Gouzé
, “
A theoretical exploration of birhythmicity in the p53-Mdm2 network
,”
PLoS One
6
,
e17075
(
2011
).
49.
F.
Kaiser
and
C.
Eichwald
, “
Bifurcation structure of a driven, multi-limit-cycle van der pol oscillator (I): The superharmonic resonance structure
,”
Int. J. Bifurcat. Chaos Appl. Sci. Eng.
1
,
485
(
1991
).
50.
C.
Eichwald
and
F.
Kaiser
, “
Bifurcation structure of a driven, multi-limit-cycle van der pol oscillator (I): Symmetry-breaking crisis and intermittency
,”
Int. J. Bifurcat. Chaos Appl. Sci. Eng.
1
,
711
(
1991
).
51.
H. G. E.
Kadji
,
J. B. C.
Orou
,
R.
Yamapi
, and
P.
Woafo
, “
Nonlinear dynamics and strange attractors in the biological system
,”
Chaos Solitons Fractals
32
,
862
(
2007
).
52.
D. W.
Jordan
and
P.
Smith
,
Nonlinear Ordinary Differential Equations
(
Oxford University Press
,
New York
,
1999
).
53.
R.
Yamapi
,
B. R. N.
Nbendjo
, and
H. G. E.
Kadji
, “
Dynamics and active control of motion of a driven multi-limit-cycle Van der Pol oscillator
,”
Int. J. Bifurcat. Chaos
17
,
1343
1354
(
2007
).
54.
B.
Ermentrout
, Simulating, analyzing, and animating dynamical systems: A guide to XPPAUT for researchers and students (software, environments, tools) (SIAM,
2002
).
55.
W.
Abou-Jaoudé
,
D.
Ouattara
, and
M.
Kaufman
, “
From structure to dynamics: Frequency tuning in the p53–Mdm2 network: I. Logical approach
,”
J. Theor. Biol.
258
,
561
(
2009
).
56.
A.
Liënard
, “Etude des oscillations entretenues,” Revue Gënërale de l’ëlectricitë 23, 901–912 and 946–954 (1928).
57.
N.
Levinson
and
O. K.
Smith
, “
A general equation for relaxation oscillations
,”
Duke Math. J.
9
,
382
403
(
1942
).
58.
N.
Levinson
, “
Transformation theory of non-linear differential equations of the second order
,”
Ann. Math.
45
,
723
737
(
1944
).
59.
A.
Ciliberto
,
B.
Novak
, and
J.
Tyson
, “
Steady states and oscillations in the p53/Mdm2 network
,”
Cell Cycle
4
,
488
(
2005
).
60.
P.
Horowitz
and
W.
Hill
,
The Art of Electronics
(
Cambridge University Press
,
2015
).
61.
E. S.
Medeiros
,
I. L.
Caldas
,
M. S.
Baptista
, and
U.
Feudel
, “
Trapping phenomenon attenuates the consequences of tipping points for limit cycles
,”
Sci. Rep.
7
,
42351
(
2017
).
You do not currently have access to this content.