When a chaotic attractor is produced by a three-dimensional strongly dissipative system, its ultimate characterization is reached when a branched manifold—a template—can be used to describe the relative organization of the unstable periodic orbits around which it is structured. If topological characterization was completed for many chaotic attractors, the case of toroidal chaos—a chaotic regime based on a toroidal structure—is still challenging. We here investigate the topology of toroidal chaos, first by using an inductive approach, starting from the branched manifold for the Rössler attractor. The driven van der Pol system—in Robert Shaw’s form—is used as a realization of that branched manifold. Then, using a deductive approach, the branched manifold for the chaotic attractor produced by the Deng toroidal system is extracted from data.
REFERENCES
1.
E. N.
Lorenz
, “Deterministic nonperiodic flow
,” J. Atmos. Sci.
20
, 130
–141
(1963
). 2.
3.
A.-T.
Vandermonde
, “Remarques sur les Problèmes de Situation
,” Mémoires de l’Académie Royale des Sciences (Paris) (1771
), pp.566
–574
.4.
C. F.
Gauss
, “Note dated 22 January 1833,” in Werke, edited by C. Schäfer (Königliche Gesellschaft der Wissenschaften zu Göttingen, Leipzig, 1867), Vol. 5, p. 605.5.
R. L.
Ricca
and B.
Nipoti
, “Gauss’ linking number revisited
,” J. Knot Theor. Ramif.
20
, 1325
–1343
(2011
). 6.
7.
P. G.
Tait
, “On knots i
,” Trans. R. Soc. Edinb.
28
, 145
–190
(1877
). 8.
P. G.
Tait
, “On knots ii
,” Trans. R. S. Edinb.
32
, 327
–342
(1884
). 9.
P. G.
Tait
, “On knots iii
,” Trans. R. Soc. Edinb.
32
, 493
–506
(1884
). 10.
11.
H.
Poincaré
, “Sur le problème des trois corps et les équations de la dynamique
,” Acta Math.
13
, 1
–270
(1890
).12.
H.
Poincaré
, Les méthodes nouvelles de la mécanique céleste iii
(Gauthier-Vilard
, Paris
, 1899
).13.
R. F.
Williams
, “Expanding attractors
,” Publ. Math. Inst. Hautes Études Sci.
43
, 169
–203
(1974
). 14.
O. E.
Rössler
, “Chaotic behavior in simple reaction system
,” Z. Naturforsch. A
31
, 259
–264
(1976
). 15.
O. E.
Rössler
, “An equation for continuous chaos
,” Phys. Lett. A
57
, 397
–398
(1976
). 16.
O. E.
Rössler
, “Chaos in abstract kinetics: Two prototypes
,” Bull. Math. Biol.
39
, 275
–289
(1977
). 17.
J.
Birman
and R. F.
Williams
, “Knotted periodic orbits in dynamical systems I. Lorenz’s equations
,” Topology
22
, 47
–82
(1983
). 18.
J.
Birman
and R. F.
Williams
, “Knotted periodic orbits in dynamical systems II. Knot holders for fibred knots
,” Contemp. Math.
20
, 1
–60
(1983
). 19.
R.
Gilmore
, “Topological analysis of chaotic dynamical systems
,” Rev. Mod. Phys.
70
, 1455
–1529
(1998
). 20.
G. B.
Mindlin
, H. G.
Solari
, M. A.
Natiello
, R.
Gilmore
, and X.-J.
Hou
, “Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction
,” J. Nonlinear Sci.
1
, 147
–173
(1991
). 21.
G. M.
Mindlin
and R.
Gilmore
, “Topological analysis and synthesis of chaotic time series
,” Physica D
58
, 229
–242
(1992
). 22.
N. B.
Tufillaro
, T.
Abbott
, and J.
Reilly
, An Experimental Approach to Nonlinear Dynamics and Chaos
(Addison-Wesley
, Redwood City, CA
, 1992
).23.
M.
Lefranc
and P.
Glorieux
, “Topological analysis of chaotic signals from a CO laser with modulated losses
,” Int. J. Bifurc. Chaos
3
, 643
–650
(1993
). 24.
L.
Kocarev
, Z.
Tasev
, and D.
Dimovski
, “Topological description of a chaotic attractor with spiral structure
,” Phys. Lett. A
190
, 399
–402
(1994
). 25.
C.
Letellier
, P.
Dutertre
, and G.
Gouesbet
, “Characterization of the Lc system, taking into account the equivariance of the vector field
,” Phys. Rev. E
49
, 3492
–3495
(1994
). 26.
T.
Braun
, R. R. B.
Correia
, and N.
Altmann
, “Topological model of homoclinic chaos in a glow discharge
,” Phys. Rev. E
51
, 4165
–4168
(1995
). 27.
C.
Letellier
, P.
Dutertre
, and B.
Maheu
, “Unstable periodic orbits and templates of the Rössler system: Toward a systematic topological characterization
,” Chaos
5
, 271
–282
(1995
). 28.
R. W.
Ghrist
and P. J.
Holmes
, “An ODE whose solutions contain all knots and links
,” Int. J. Bifurc. Chaos
6
, 779
–800
(1996
). 29.
C.
Letellier
, P.
Dutertre
, J.
Reizner
, and G.
Gouesbet
, “Evolution of multimodal map induced by an equivariant vector field
,” J. Phys. A
29
, 5359
–5373
(1996
). 30.
G.
Boulant
, M.
Lefranc
, S.
Bielawski
, and D.
Derozier
, “A nonhorseshoe template in a chaotic laser model
,” Int. J. Bifurc. Chaos
08
, 965
–975
(1998
). 31.
C.
Letellier
, E.
Roulin
, and O. E.
Rössler
, “Inequivalent topologies of chaos in simple equations
,” Chaos Solitons Fractals
28
, 337
–360
(2006
). 32.
C.
Letellier
and R.
Gilmore
, “The universal template is a subtemplate of the double-scroll template
,” J. Phys. A
46
, 065102
(2013
). 33.
R.
Gilmore
, C.
Letellier
, and M.
Lefranc
, “Chaos topology
,” Scholarpedia
3
, 4592
(2008
). 34.
W. F.
Langford
, “Numerical studies of torus bifurcations,” in Numerical Methods for Bifurcation Problems, International Series of Numerical Mathematics Vol. 70, edited by T. Küpper, H. D. Mittelmann and H. Weber (Birkhäuser, Basel, 1984), pp. 285–295.35.
B.
Deng
, “Constructing homoclinic orbits and chaotic attractors
,” Int. J. Bifurc. Chaos
4
, 823
–841
(1994
). 36.
D.
Li
, “A three-scroll chaotic attractor
,” Phys. Lett. A
372
, 387
–393
(2008
). 37.
O. E.
Rössler
, “Continuous chaos: Four prototype equations
,” Ann. N. Y. Acad. Sci.
316
, 376
–392
(1979
). 38.
C.
Letellier
and O. E.
Rössler
, “An updated hierarchy of chaos,” in Chaos: The World of Nonperiodic Oscillations (Springer, Cham, Switzerland, 2020), pp. 181–203.39.
G.
Mindlin
and H.
Solari
, “Tori and Klein bottles in four-dimensional chaotic flows
,” Physica D
102
, 177
–186
(1997
). 40.
J.
Curry
and J. A.
Yorke
, “A transition from Hopf bifurcation to chaos: Computer experiments with maps on
,” Lecture Notes Math.
668
, 48
–66
(1978
). 41.
T. D.
Tsankov
and R.
Gilmore
, “Topological aspects of the structure of chaotic attractors in
,” Phys. Rev. E
69
, 056206
(2004
). 42.
M.
Rosalie
and C.
Letellier
, “Systematic template extraction from chaotic attractors: i. Genus-one attractors with an inversion symmetry
,” J. Phys. A Math. Theor.
46
, 375101
(2013
). 43.
L.
Le Sceller
, C.
Letellier
, and G.
Gouesbet
, “Algebraic evaluation of linking numbers of unstable periodic orbits in chaotic attractors
,” Phys. Rev. E
49
, 4693
–4695
(1994
). 44.
C.
Letellier
, “Caractérisation topologique et reconstruction des attracteurs étranges,” Ph.D. thesis (University of Paris VII, Paris, France, 1994).45.
H.
Bai-Lin
, Elementary Symbolic Dynamics and Chaos in Dissipative Systems
(World Scientific Publishing
, Singapore
, 1989
).46.
R.
Shaw
, “Strange attractor, chaotic behavior and information flow
,” Z. Naturforsch. A
36
, 80
–112
(1981
). 47.
O.
Ménard
, C.
Letellier
, J.
Maquet
, L. L.
Sceller
, and G.
Gouesbet
, “Analysis of a non synchronized sinusoidally driven dynamical system
,” Int. J. Bifurc. Chaos
10
, 1759
–1772
(2000
). 48.
C.
Letellier
and G.
Gouesbet
, “Topological characterization of reconstructed attractors modding out symmetries
,” J. Phys. ii
6
, 1615
–1638
(1996
). 49.
R.
Miranda
and E.
Stone
, “The proto-Lorenz system
,” Phys. Lett. A
178
, 105
–113
(1993
). 50.
C.
Letellier
and R.
Gilmore
, “Covering dynamical systems: Two-fold covers
,” Phys. Rev. E
63
, 016206
(2001
). 51.
52.
S.
Mangiarotti
, M.
Peyre
, and M.
Huc
, “A chaotic model for the epidemic of Ebola virus disease in West Africa (2013-2016)
,” Chaos
26
, 113112
(2016
). © 2021 Author(s).
2021
Author(s)
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