We performed an exhaustive numerical analysis of the two-dimensional Chialvo map by obtaining the parameter planes based on the computation of periodicities and Lyapunov exponents. Our results allowed us to determine the different regions of dynamical behavior, identify regularities in the distribution of periodicities in regions indicating regular behavior, find some pseudofractal structures, identify regions such as the “eyes of chaos” similar to those obtained in parameter planes of continuous systems, and, finally, characterize the statistical properties of chaotic attractors leading to possible hyperchaotic behavior.
REFERENCES
1.
J. L.
Hindmarsh
, R. M.
Rose
, and A. F.
Huxley
, “A model of neuronal bursting using three coupled first order differential equations
,” Proc. R. Soc. Lond. B
221
, 87
–102
(1984
).2.
A. L.
Hodgkin
and A. F.
Huxley
, “A quantitative description of membrane current and its application to conduction and excitation in nerve
,” J. Physiol.
117
(4
), 500
–544
(1952
). 3.
R.
FitzHugh
, “Mathematical models of threshold phenomena in the nerve membrane
,” Bull. Math. Biophys.
17
, 257
–278
(1955
). 4.
S.
Rakshit
, A.
Ray
, B.
Bera
, and D.
Ghosh
, “Synchronization and firing patterns of coupled Rulkov neuronal map
,” Nonlinear Dyn.
1
, 1
–21
(2018
).5.
N. F.
Rulkov
, “Regularization of synchronized chaotic bursts
,” Phys. Rev. Lett.
86
, 183
–186
(2001
). 6.
G. I.
Strelkova
, S. A.
Bogomolov
, E. V.
Rybalova
, and V. S.
Anishchenko
, “Spatiotemporal structures in an ensemble of nonlocally coupled nekorkin maps
,” Izv. Sarat. Univ. Phys.
19
, 86
–94
(2019
).7.
E. M.
Izhikevich
, “Simple model of spiking neurons
,” IEEE Trans. Neural Netw.
14
, 1569
–1572
(2003
). 8.
S. S.
Muni
, “Mode-locked orbits, doubling of invariant curves in discrete Hindmarsh-Rose neuron model
,” Phys. Scr.
98
, 085205
(2023
). 9.
D. R.
Chialvo
, “Generic excitable dynamics on a two-dimensional map
,” Chaos Solit. Fractals
5
, 461
–479
(1995
). 10.
S. S.
Muni
, H. O.
Fatoyinbo
, and I.
Ghosh
, “Dynamical effects of electromagnetic flux on Chialvo neuron map: Nodal and network behaviors
,” Int. J. Bifurcat. Chaos
32
, 2230020
(2022
). 11.
P.
Pilarczyk
, J.
Signerska-Rynkowska
, and G.
Graff
, “Topological-numerical analysis of a two-dimensional discrete neuron model
,” Chaos
33
, 043110
(2023
). 12.
R.
Smidtaite
and M.
Ragulskis
, “Finite-time divergence in chialvo hyperneuron model of nilpotent matrices
,” Chaos Solit. Fractals
179
, 114482
(2024
). 13.
F. L.
Trujillo
, J.
Signerska-Rynkowska
, and P.
Bartłomiejczyk
, “Periodic and chaotic dynamics in a map-based neuron model
,” Math. Methods Appl. Sci.
46
, 11906
–11931
(2023
). 14.
M.
Courbage
and V. I.
Nekorkin
, “Map based models in neurodynamics
,” Int. J. Bifurcat. Chaos
20
, 1631
–1651
(2010
). 15.
B.
Ibarz
, J.
Casado
, and M.
Sanjuán
, “Map-based models in neuronal dynamics
,” Phys. Rep.
501
, 1
–74
(2011
). 16.
S. S.
Muni
, “Ergodic and resonant torus doubling bifurcation in a three-dimensional quadratic map
,” Nonlinear Dyn.
112
, 1
(2024
). 17.
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
). 18.
J. A. C.
Gallas
, “Dissecting shrimps: Results for some one-dimensional physical models
,” Physica A
202
, 196
–223
(1994
). 19.
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
).20.
F.
Wang
and H.
Cao
, “Mode locking and quasiperiodicity in a discrete-time Chialvo neuron model
,” Commun. Nonlinear Sci. Numer. Simul.
56
, 481
–489
(2018
). 21.
R. E.
Ecke
, J. D.
Farmer
, and D. K.
Umberger
, “Scaling of the Arnold tongues
,” Nonlinearity
2
, 175
–196
(1989
). 22.
C.
Rosa
, M. J.
Correia
, and P. C.
Rech
, “Arnold tongues and quasiperiodicity in a prey–predator model
,” Chaos Solit. Fractals
40
, 2041
–2046
(2009
). 23.
J. K.
Jang
, X.
Ji
, C.
Joshi
, Y.
Okawachi
, M.
Lipson
, and A. L.
Gaeta
, “Observation of Arnold tongues in coupled soliton Kerr frequency combs
,” Phys. Rev. Lett.
123
, 153901
(2019
). 24.
E.
Behta
, G. H.
Goldsztein
, and L. Q.
English
, “Phase-locking dynamics for electronic relaxation oscillators via threshold pulse-modulation: Comparing experimental and analytical Arnold tongues
,” Physica D
454
, 133849
(2023
). 25.
L.
Glass
and R.
Perez
, “Fine structure of phase locking
,” Phys. Rev. Lett.
48
, 1772
–1775
(1982
). 26.
P. B.
Main
, P. J.
Mosley
, and A. V.
Gorbach
, “Revealing Arnold’s tongues in photon-pair generation and quantum frequency conversion,” in OSA Advanced Photonics Congress (AP) 2020 (IPR, NP, NOMA, Networks, PVLED, PSC, SPPCom, SOF) (Optica Publishing Group, 2020), p. JM2E.1.27.
S. W.
Shaw
, “Arnold tongues and subharmonics in the forced oscillations of a mechanical clock,” in 1985 24th IEEE Conference on Decision and Control (IEEE, 1985), pp. 976–981.28.
I.
Bashkirtseva
, L.
Ryashko
, J.
Used
, J. M.
Seoane
, and M. A.
Sanjuán
, “Noise-induced complex dynamics and synchronization in the map-based Chialvo neuron model
,” Commun. Nonlinear Sci. Numer. Simul.
116
, 106867
(2023
). 29.
H.
Cao
, Y.
Wang
, S.
Banerjee
, Y.
Cao
, and J.
Mou
, “A discrete Chialvo–Rulkov neuron network coupled with a novel memristor model: Design, dynamical analysis, DSP implementation and its application
,” Chaos Solit. Fractals
179
, 114466
(2024
). 30.
G.
Vivekanandhan
, H.
Natiq
, Y.
Merrikhi
, K.
Rajagopal
, and S.
Jafari
, “Dynamical analysis and synchronization of a new memristive Chialvo neuron model
,” Electronics
12
, 545
(2023
). 31.
S.
Coombes
and P. C.
Bressloff
, “Mode locking and Arnold tongues in integrate-and-fire neural oscillators
,” Phys. Rev. E
60
, 2086
–2096
(1999
). 32.
A. L.
Lin
, A.
Hagberg
, E.
Meron
, and H. L.
Swinney
, “Resonance tongues and patterns in periodically forced reaction-diffusion systems
,” Phys. Rev. E
69
, 066217
(2004
). 33.
G. M.
Ramírez-Ávila
, S.
Depickère
, I. M.
Jánosi
, and J. A. C.
Gallas
, “Distribution of spiking and bursting in Rulkov’s neuron model
,” Eur. Phys. J. Spec. Top.
231
, 319
–328
(2022
).34.
Y.
Wang
, X.
Zhang
, and S.
Liang
, “New phenomena in Rulkov map based on Poincaré cross section
,” Nonlinear Dyn.
111
, 19447
–19458
(2023
). 35.
M.
Sekikawa
, T.
Kousaka
, T.
Tsubone
, N.
Inaba
, and H.
Okazaki
, “Bifurcation analysis of mixed-mode oscillations and Farey trees in an extended Bonhoeffer–van der Pol oscillator
,” Physica D
433
, 133178
(2022
). 36.
J. G.
Freire
and J. A.
Gallas
, “Stern–Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer–van der Pol and the FitzHugh–Nagumo models of excitable systems
,” Phys. Lett. A
375
, 1097
–1103
(2011
). 37.
C. T.
Annand
, S. M.
Fleming
, and J. G.
Holden
, “Farey trees explain sequential effects in choice response time
,” Front. Physiol.
12
, 611145
(2021
). 38.
O.
Rossler
, “An equation for hyperchaos
,” Phys. Lett. A
71
, 155
–157
(1979
). 39.
F. D.
Tappert
, G. J.
Goni
, and M. J.
Brown
, “Chaos and hyperchaos in shallow water acoustics
,” J. Acoust. Soc. Am.
84
, S152
(2005
). 40.
L.
Munteanu
, C.
Brißan
, and V.
Chiroiu
, “Chaos–hyperchaos transition in a class of models governed by Sommerfeld effect
,” Nonlinear Dyn.
78
, 1877
–1889
(2014
). 41.
A. S.
Elwakil
and M. P.
Kennedy
, “Inductorless hyperchaos generator
,” Microelectron. J.
30
, 739
–743
(1999
). 42.
K.
Stefański
, “Modelling chaos and hyperchaos with 3-D maps
,” Chaos Solit. Fractals
9
, 83
–93
(1998
).43.
Q.
Xu
, L.
Huang
, N.
Wang
, H.
Bao
, H.
Wu
, and M.
Chen
, “Initial-offset-boosted coexisting hyperchaos in a 2D memristive Chialvo neuron map and its application in image encryption
,” Nonlinear Dyn.
111
, 20447
–20463
(2023
). 44.
J. C.
Sprott
, Chaos and Time-Series Analysis
(Oxford University Press
, New York
, 2003
).45.
B.
Mandelbrot
, “How long is the coast of britain? Statistical self-similarity and fractional dimension
,” Science
156
, 636
–638
(1967
). 46.
H. M.
Hastings
, Fractals: A User’s Guide for the Natural Sciences
(Oxford University Press
, 1994
).47.
G. M.
Ramírez-Ávila
and J. A. C.
Gallas
, “How similar is the performance of the cubic and the piecewise-linear circuits of Chua?
,” Phys. Lett. A
375
, 143
–148
(2010
).48.
G. M.
Ramírez-Ávila
and J. A. C.
Gallas
, “Self-similarities in the parameter space of Chua’s circuit with discrete and continuous nonlinearities (in Spanish)
,” Revista Boliviana de Física
18
, 1
–6
(2011
), http://www.scielo.org.bo/scielo.php?pid=S1562-38232011000200001&script=sci_arttext49.
J. A. C.
Gallas
, “Periodic oscillations of the forced Brusselator
,” Mod. Phys. Lett. B
29
, 1530018
(2015
). 50.
J. A. C.
Gallas
, “Non-quantum chirality in a driven Brusselator
,” J. Phys.-Condens. Mat.
34
, 144002
(2022
). 51.
X.-B.
Rao
, X.-P.
Zhao
, J.-S.
Gao
, and J.-G.
Zhang
, “Self-organizations with fast-slow time scale in a memristor-based Shinriki’s circuit
,” Commun. Nonlinear Sci. Numer. Simul.
94
, 105569
(2021
). 52.
L.
Xu
, Y.-D.
Chu
, and Q.
Yang
, “Novel dynamical scenario of the two-stage Colpitts oscillator
,” Chaos Solit. Fractals
138
, 109998
(2020
). 53.
G. M.
Ramírez-Ávila
, T.
Kapitaniak
, and D.
Gonze
, “Dynamical analysis of a periodically forced chaotic chemical oscillator
,” Chaos
34
, 073154
(2024
). 54.
A.
Pikovsky
and A.
Politi
, Lyapunov Exponents: A Tool to Explore Complex Dynamics
(Cambridge University Press
, 2016
).55.
R. C.
Hilborn
, “Quantifying chaos,” in Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers (Oxford University Press, 2000).56.
S. L.
Kingston
, A.
Mishra
, M.
Balcerzak
, T.
Kapitaniak
, and S. K.
Dana
, “Instabilities in quasiperiodic motion lead to intermittent large-intensity events in Zeeman laser
,” Phys. Rev. E
104
, 034215
(2021
). 57.
V.
Nekorkin
and L.
Vdovin
, “Map-based model of the neural activity
,” Izv. VUZ. Appl. Nonlinear Dyn.
15
, 36
(2007
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
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