In this paper, we report the discovery of some novel dynamical scenarios for quasi-periodic shrimp-shaped structures embedded within chaotic phases in bi-parameter space of a discrete predator–prey system. By constructing high-resolution, two-dimensional stability diagrams based on Lyapunov exponents, we observe the abundance of both periodic and quasi-periodic shrimp-shaped organized domains in a certain parameter space of the system. A comprehensive comparative analysis is conducted to elucidate the similarities and differences between these two types of shrimps. Our analysis reveals that, unlike periodic shrimp, quasi-periodic shrimp induces (i) torus bubbling transition to chaos and (ii) multistability with multi-tori, torus-chaotic, and multi-chaotic coexisting attractors, resulting from the crossing of its two inner antennae. The basin sets of the coexisting attractors are analyzed, and we observe the presence of intriguing basin boundaries. We also verify that, akin to periodic shrimp structures, quasi-periodic shrimps also maintain the three-times self-similarity scaling. Furthermore, we encounter the occurrence of spiral organization for the self-distribution of quasi-periodic shrimps within a large chaotic domain. We believe that these novel findings will significantly enhance our understanding of shrimp-shaped structures and the intricate dynamics exhibited by their distribution in chaotic regimes.
REFERENCES
1.
O. V.
Popovych
, V.
Krachkovskyi
, and P. A.
Tass
, “Phase-locking swallows in coupled oscillators with delayed feedback
,” Phys. Rev. E
82
, 046203
(2010
). 2.
C.
Bonatto
and J. A. C.
Gallas
, “Periodicity hub and nested spirals in the phase diagram of a simple resistive circuit
,” Phys. Rev. Lett.
101
, 054101
(2008
). 3.
R.
Stoop
, P.
Benner
, and Y.
Uwate
, “Real-world existence and origins of the spiral organization of shrimp-shaped domains
,” Phys. Rev. Lett.
105
, 074102
(2010
). 4.
R.
Barrio
, F.
Blesa
, S.
Serrano
, and A.
Shilnikov
, “Global organization of spiral structures in biparameter space of dissipative systems with Shilnikov saddle-foci
,” Phys. Rev. E
84
, 035201
(2011
). 5.
J. A. C.
Gallas
, “Structure of the parameter space of the Hénon map
,” Phys. Rev. Lett.
70
, 2714
(1993
). 6.
G. C.
Layek
, An Introduction to Dynamical Systems and Chaos
, 2nd ed. (Springer
, 2024
).7.
R.
Costantino
, J.
Cushing
, B.
Dennis
, and R. A.
Desharnais
, “Experimentally induced transitions in the dynamic behaviour of insect populations
,” Nature
375
, 227
–230
(1995
). 8.
P.
Rohani
and O.
Miramontes
, “Chaos or quasiperiodicity in laboratory insect populations?
,” J. Anim. Ecol.
65
, 847
–849
(1996
). 9.
V. G.
Petrosyan
, V. V.
Golubkov
, N. A.
Zavyalov
, Z. I.
Goryainova
, N. N.
Dergunova
, A. V.
Omelchenko
, S. A.
Bessonov
, S. A.
Albov
, N. F.
Marchenko
, and L. A.
Khlyap
, “Patterns of population dynamics of Eurasian beaver (Castor fiber L.) after reintroduction into nature reserves of the European part of Russia
,” Russian J. Biological Invasions
7
, 355
–373
(2016
). 10.
S.
Cordero
and F. H.
Busse
, “Experiments on convection in rotating hemispherical shells: Transition to a quasi-periodic state
,” Geophys. Res. Lett.
19
, 733
–736
, https://doi.org/10.1029/92GL00574 (1992
). 11.
P.
Zhu
, L.
Duan
, and Q.
Kang
, “Transition to chaos in thermocapillary convection
,” Int. J. Heat Mass Transfer
57
, 457
–464
(2013
). 12.
G.
Broggi
, P. F.
Meier
, R.
Stoop
, and R.
Badii
, “Nonlinear dynamics of a model for parallel pumping in ferromagnets
,” Phys. Rev. A
35
, 365
(1987
). 13.
F.
Sanchez
, M.
LeFlohic
, G. M.
Stephan
, P.
LeBoudec
, and P. L.
Francois
, “Quasi-periodic route to chaos in erbium-doped fiber laser
,” IEEE J. Quantum Electron.
31
, 481
–488
(1995
). 14.
Q.
Xing
, L.
Chai
, W.
Zhang
, and C. Y.
Wang
, “Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:Sapphire laser
,” Opt. Commun.
162
, 71
–74
(1999
). 15.
J.
Peinke
, J.
Parisi
, R. P.
Huebener
, M.
Duong-Van
, and P.
Keller
, “Quasi-periodic behavior of dc-biased semiconductor electronic breakdown
,” Europhys. Lett.
12
, 13
–18
(1990
). 16.
M.
Chen
and Y. L.
Ju
, “Experimental study of quasi-periodic on-off phenomena in a small-scale traveling wave thermoacoustic heat engine
,” Cryogenics
85
, 23
–29
(2017
). 17.
B. C.
Bao
, P. Y.
Wu
, H.
Bao
, Q.
Xu
, and M.
Chen
, “Numerical and experimental confirmations of quasi-periodic behavior and chaotic bursting in third-order autonomous memristive oscillator
,” Chaos, Solitons Fractals
106
, 161
–170
(2018
). 18.
L.
Glass
and R.
Perez
, “Fine structure of phase locking
,” Phys. Rev. Lett.
48
, 1772
(1982
). 19.
S.
Fraser
and R.
Kapral
, “Analysis of flow hysteresis by a one-dimensional map
,” Phys. Rev. A
25
, 3223
(1982
). 20.
R.
Perez
and L.
Glass
, “Bistability, period doubling bifurcations and chaos in a periodically forced oscillator
,” Phys. Lett. A
90
, 441
–443
(1982
). 21.
P.
Gaspard
, R.
Kapral
, and G.
Nicolis
, “Bifurcation phenomena near homoclinic systems: A two-parameter analysis
,” J. Stat. Phys.
35
, 697
–727
(1984
). 22.
M.
Markus
and B.
Hess
, “Lyapunov exponents of the logistic map with periodic forcing
,” Comput. Graphics
13
, 553
–558
(1989
). 23.
J.
Milnor
, “Remarks on iterated cubic maps
,” Exp. Math.
1
, 5
–24
(1992
).24.
J. P.
Carcasses
, C.
Mira
, M.
Bosch
, C.
Simó
, and J. C.
Tatjer
, ““Crossroad area–spring area” transition (i) parameter plane representation
,” Int. J. Bifurcation Chaos
1
, 183
–196
(1991
). 25.
R.
Allam
and C.
Mira
, ““crossroad area-dissymmetrical spring area–symmetrical spring area”, and “double crossroad area–double spring area” transitions
,” Int. J. Bifurcation Chaos
3
, 429
–435
(1993
). 26.
E. N.
Lorenz
, “Compound windows of the Hénon-map
,” Physica D
237
, 1689
–1704
(2008
). 27.
C.
Bonatto
and J. A. C.
Gallas
, “Accumulation horizons and period adding in optically injected semiconductor lasers
,” Phys. Rev. E
75
, 055204
(2007
). 28.
J.
Slipantschuk
, E.
Ullner
, M. d. S.
Baptista
, M.
Zeineddine
, and M.
Thiel
, “Abundance of stable periodic behavior in a Red Grouse population model with delay: A consequence of homoclinicity
,” Chaos
20
, 045117
(2010
). 29.
P. C.
Rech
, “How to embed shrimps in parameter planes of the Lorenz system
,” Phys. Scr.
92
, 045201
(2017
). 30.
J. A.
Vélez
, J.
Bragard
, L. M.
Pérez
, A. M.
Cabanas
, O. J.
Suarez
, D.
Laroze
, and H. L.
Mancini
, “Periodicity characterization of the nonlinear magnetization dynamics
,” Chaos
30
, 093112
(2020
).31.
D. F. M.
Oliveira
, M.
Robnik
, and E. D.
Leonel
, “Shrimp-shape domains in a dissipative kicked rotator
,” Chaos
21
, 043122
(2011
). 32.
S. L. T.
de Souza
, A. A.
Lima
, I. L.
Caldas
, R. O.
Medrano-T
, and Z. O.
Guimarães-Filho
, “Self-similarities of periodic structures for a discrete model of a two-gene system
,” Phys. Lett. A
376
, 1290
–1294
(2012
). 33.
W.
Façanha
, B.
Oldeman
, and L.
Glass
, “Bifurcation structures in two-dimensional maps: The endoskeletons of shrimps
,” Phys. Lett. A
377
, 1264
–1268
(2013
).34.
J. A.
De Oliveira
, L. T.
Montero
, D. R.
Da Costa
, J. A.
Méndez-Bermúdez
, R. O.
Medrano-T
, and E. D.
Leonel
, “An investigation of the parameter space for a family of dissipative mappings
,” Chaos
29
, 053114
(2019
). 35.
E. R.
Viana
, R. M.
Rubinger
, H. A.
Albuquerque
, A. G.
de Oliveira
, and G. M.
Ribeiro
, “High-resolution parameter space of an experimental chaotic circuit
,” Chaos
20
, 023110
(2010
). 36.
F. F. G.
de Sousa
, R. M.
Rubinger
, J. C.
Sartorelli
, H. A.
Albuquerque
, and M. S.
Baptista
, “Parameter space of experimental chaotic circuits with high-precision control parameters
,” Chaos
26
, 083107
(2016
). 37.
F.
Prebianca
, D. W. C.
Marcondes
, H. A.
Albuquerque
, and M. W.
Beims
, “Exploring an experimental analog Chua’s circuit
,” Eur. Phys. J. B
92
, 1
–8
(2019
). 38.
R.
Stoop
, S.
Martignoli
, P.
Benner
, R. L.
Stoop
, and Y.
Uwate
, “Shrimps: Occurrence, scaling and relevance
,” Int. J. Bifurcation Chaos
22
, 1230032
(2012
). 39.
G. C.
Layek
and N. C.
Pati
, “Organized structures of two bidirectionally coupled logistic maps
,” Chaos
29
, 093104
(2019
). 40.
N. C.
Pati
, G. C.
Layek
, and N.
Pal
, “Bifurcations and organized structures in a predator-prey model with hunting cooperation
,” Chaos, Solitons Fractals
140
, 110184
(2020
). 41.
M.
Hossain
, N. C.
Pati
, S.
Pal
, S.
Rana
, N.
Pal
, and G. C.
Layek
, “Bifurcations and multistability in a food chain model with nanoparticles
,” Math. Comput. Simul.
190
, 808
–825
(2021
). 42.
J. A. C.
Gallas
, “Dissecting shrimps: Results for some one-dimensional physical models
,” Physica A
202
, 196
–223
(1994
). 43.
P. C.
Rech
, “Period-adding and spiral organization of the periodicity in a Hopfield neural network
,” Int. J. Mach. Learn. Cybern.
6
, 1
–6
(2015
). 44.
R. A.
da Silva
and P. C.
Rech
, “Spiral periodic structures in a parameter plane of an ecological model
,” Appl. Math. Comput.
254
, 9
–13
(2015
).45.
A.
Sack
, J. G.
Freire
, E.
Lindberg
, T.
Pöschel
, and J. A. C.
Gallas
, “Discontinuous spirals of stable periodic oscillations
,” Sci. Rep.
3
, 3350
(2013
). 46.
C. A. T.
Chávez
and S.
Curilef
, “Discontinuous spirals of stability in an optically injected semiconductor laser
,” Chaos
30
, 053107
(2020
).47.
J. G.
Freire
and J. A. C.
Gallas
, “Cyclic organization of stable periodic and chaotic pulsations in Hartley’s oscillator
,” Chaos, Solitons Fractals
59
, 129
–134
(2014
). 48.
G. M.
Ramírez-Ávila
, J.
Kurths
, and J. A. C.
Gallas
, “Ubiquity of ring structures in the control space of complex oscillators
,” Chaos
31
, 101102
(2021
).49.
A. A. C.
Recco
, J. C.
Sagás
, and P. C.
Rech
, “Multistability, period-adding, and fractality in a plasma oscillator
,” Phys. Plasmas
30
, 112301
(2023
). 50.
N. C.
Pati
and B.
Ghosh
, “Stability scenarios and period-doubling onset of chaos in a population model with delayed harvesting
,” Math. Methods Appl. Sci.
46
, 12930
–12945
(2023
). 51.
Z.
He
, X.
Lai
, and A.
Hou
, “Stability and Neimark–Sacker bifurcation of numerical discretization of delay differential equations
,” Chaos, Solitons Fractals
41
, 2010
–2017
(2009
). 52.
X.
Jiang
, X.
Chen
, M.
Chi
, and J.
Chen
, “On Hopf bifurcation and control for a delay systems
,” Appl. Math. Comput.
370
, 124906
(2020
).53.
B.
Ghosh
, S.
Sarda
, and S.
Sahu
, “Torus doubling route to chaos and chaos eradication in delayed discrete-time predator–prey models
,” Math. Methods Appl. Sci.
1
–20
(2022
), https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.8789.54.
F.
Gomez
, R. L.
Stoop
, and R.
Stoop
, “Universal dynamical properties preclude standard clustering in a large class of biochemical data
,” Bioinformatics
30
, 2486
–2493
(2014
). 55.
S.
Wiggins
, Introduction to Applied Nonlinear Dynamical Systems and Chaos
(Springer
, 2003
).56.
P. N.
Paraskevopoulos
, Modern Control Engineering
(CRC Press
, 2017
).57.
S.
Salivahanan
, R.
Rengaraj
, and G. R.
Venkatakrishnan
, Control Systems Engineering
(Pearson
, 2014
).58.
S. N.
Elaydi
, Discrete Chaos: With Applications in Science and Engineering
(CRC Press
, 2007
).59.
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
). 60.
J. C.
Sprott
, Chaos and Time-series Analysis
(Oxford University Press
, 2003
).61.
V. I.
Arnold
, “Small denominators, I: Mappings of the circumference into itself
,” AMS Trans. Series 2
46
, 213
(1965
).62.
C.
Rosa
, M. J.
Correia
, and P. C.
Rech
, “Arnold tongues and quasiperiodicity in a prey–predator model
,” Chaos, Solitons Fractals
40
, 2041
–2046
(2009
). 63.
F.
Montoya
, M.
Rivera
, J.
Escalona
, and P.
Parmananda
, “Construction of Arnold tongue structures for coupled periodic oscillators
,” Phys. Lett. A
377
, 3124
–3127
(2013
). 64.
C. C.
Felicio
and P. C.
Rech
, “Arnold tongues and the devil’s staircase in a discrete-time Hindmarsh–Rose neuron model
,” Phys. Lett. A
379
, 2845
–2847
(2015
). 65.
V.
Wiggers
and P. C.
Rech
, “Multistability and organization of periodicity in a van der Pol–Duffing oscillator
,” Chaos, Solitons Fractals
103
, 632
–637
(2017
). 66.
C.
Manchein
, L.
Santana
, R. M.
da Silva
, and M. W.
Beims
, “Noise-induced stabilization of the FitzHugh–Nagumo neuron dynamics: Multistability and transient chaos
,” Chaos
32
, 083102
(2022
). 67.
S. L. T.
de Souza
, A. M.
Batista
, M. S.
Baptista
, I. L.
Caldas
, and J. M.
Balthazar
, “Characterization in bi-parameter space of a non-ideal oscillator
,” Physica A
466
, 224
–231
(2017
). 68.
N. C.
Pati
, S.
Garai
, M.
Hossain
, G. C.
Layek
, and N.
Pal
, “Fear induced multistability in a predator-prey model
,” Int. J. Bifurcation Chaos
31
, 2150150
(2021
). 69.
M.
Mugnaine
, M. R.
Sales
, J. D.
Szezech Jr
, and R. L.
Viana
, “Dynamics, multistability, and crisis analysis of a sine-circle nontwist map
,” Phys. Rev. E
106
, 034203
(2022
). © 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
