Chaotic behavior near a periodicity hub is characterized in five different three-dimensional systems, namely, the paradigmatic Rössler system, the Rosenzweig–MacArthur predator–prey model, a semiconductor laser model, the Gaspard–Nicolis chemical oscillator, and the Nishio–Inaba electronic circuit. Return maps of local maxima for a selected dynamical variable in each system were extracted from numerical solutions. By rescaling the data and assuming full ergodicity in the unit interval, we show that excellent fits to the ubiquitously U-shaped invariant densities are obtained with weighted combinations of the beta and Kumaraswamy distributions.
REFERENCES
1.
L. P.
Shilnikov
, “A case of the existence of a countable number of periodic motions
,” Sov. Math. Dokl.
6
, 163
(1965
).2.
L. P.
Shilnikov
and A.
Shilnikov
, “Shilnikov bifurcation
,” Scholarpedia
2
(8
), 1891
(2007
). 3.
C.
Silva
, “Shil’nikov’s theorem—A tutorial
,” IEEE Trans. Circuits Syst. I
40
, 675
(1993
). 4.
P.
Gaspard
, “Generation of a countable set of homoclinic flows through bifurcation
,” Phys. Lett.
97
, 1
(1983
). 5.
P.
Gaspard
, R.
Kapral
, and G.
Nicolis
, “Bifurcation phenomena near homoclinic systems: A two-parameter analysis
,” J. Stat. Phys.
35
, 697
(1984
). 6.
P.
Glendinning
and C.
Sparrow
, “Local and global behavior near homoclinic orbits
,” J. Stat. Phys.
35
, 645
(1984
). 7.
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
). 8.
J. A. C.
Gallas
, “The structure of infinite periodic and chaotic hub cascades in phase diagrams of simple autonomous flows
,” Int. J. Bifurc. Chaos
20
, 197
(2010
). 9.
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
). 10.
J. G.
Freire
and J. A. C.
Gallas
, “Non-Shilnikov cascades of spikes and hubs in a semiconductor laser with optoelectronic feedback
,” Phys. Rev. E
82
, 037202
(2010
). 11.
R.
Vitolo
, P.
Glendinning
, and J. A. C.
Gallas
, “Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows
,” Phys. Rev. E
84
, 015216
(2011
). 12.
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(R)
(2011
). 13.
M. A.
Nascimento
, H.
Varela
, and J. A. C.
Gallas
, “Periodicity hubs and spirals in an electrochemical oscillator
,” J. Sol. State Eletrochem.
19
, 3287
(2015
). 14.
S.
Malykh
, Y.
Bakhanova
, A.
Kazakov
, K.
Pusuluri
, and A.
Shilnikov
, “Homoclinic chaos in the Rössler model
,” Chaos
30
, 113126
(2020
). 15.
P.
Kumaraswamy
, “A generalized probability density function for double-bounded random processes
,” J. Hydrol.
46
, 79
(1980
). 16.
M. C.
Jones
, “Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages
,” Stat. Meth.
6
, 70
(2009
). 17.
O. E.
Rössler
, “An equation for continuous chaos
,” Phys. Lett. A
57
, 397
(1976
).18.
R. B.
do Carmo
, see https://repositorio.ufpe.br/handle/123456789/34289 (2019) for “Digital library of theses, UFPE.”19.
M.
Rosenzweig
and R.
MacArthur
, “Graphical representation and stability conditions of predator–prey interactions
,” Am. Naturalist
97
, 209
(1963
). 20.
Y. A.
Kuznetsov
, O.
de Feo
, and S.
Rinaldi
, “Belyakov homoclinic bifurcations in a tritrophic food chain model
,” SIAM J. Appl. Math.
62
, 462
(2001
). 21.
K.
Al-Naimee
, F.
Marino
, M.
Ciszak
, R.
Meucci
, and F. T.
Arecchi
, “Chaotic spiking and incomplete homoclinic scenarios in semiconductor lasers with optoelectronic feedback
,” New J. Phys.
11
, 073022
(2009
). 22.
P.
Gaspard
and G.
Nicolis
, “What can we learn from homoclinic orbits in chaotic dynamics?
” J. Stat. Phys.
31
, 499
(1983
). 23.
Y.
Nishio
, N.
Inaba
, S.
Mori
, and T.
Saito
, “Rigorous analyses of windows in a symmetric circuit
,” IEEE Trans. Circ. Syst.
37
, 473
(1990
). © 2025 Author(s). Published under an exclusive license by AIP Publishing.
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