The generalized four-dimensional Rössler system is studied. Main bifurcation scenarios leading to a hyperchaos are described phenomenologically and their implementation in the model is demonstrated. In particular, we show that the formation of hyperchaotic invariant sets is related mainly to cascades (finite or infinite) of nondegenerate bifurcations of two types: period-doubling bifurcations of saddle cycles with a one-dimensional unstable invariant manifold and Neimark-Sacker bifurcations of stable cycles. The onset of the discrete hyperchaotic Shilnikov attractors containing a saddle-focus cycle with a two-dimensional unstable invariant manifold is confirmed numerically in a Poincaré map of the model. A new phenomenon, “jump of hyperchaoticity,” when the attractor under consideration becomes hyperchaotic due to the boundary crisis of some other attractor, is discovered.
REFERENCES
1.
O.
Rossler
, “An equation for hyperchaos
,” Phys. Lett. A
71
, 155
–157
(1979
). 2.
O. E.
Rössler
and C.
Letellier
, “Hyperchaos,” in Chaos (Springer, 2020), pp. 55–62.3.
G.
Baier
and M.
Klein
, “Maximum hyperchaos in generalized Hénon maps
,” Phys. Lett. A
151
, 281
–284
(1990
). 4.
M.
Klein
, G.
Baier
, and O.
Rössler
, “From N-tori to hyperchaos
,” Chaos Solitons Fractals
1
, 105
–118
(1991
). 5.
T.
Tsubone
and T.
Saito
, “Hyperchaos from a 4-D manifold piecewise-linear system
,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl.
45
, 889
–894
(1998
). 6.
S.
Nikolov
and S.
Clodong
, “Occurrence of regular, chaotic and hyperchaotic behavior in a family of modified Rossler hyperchaotic systems
,” Chaos Solitons Fractals
22
, 407
–431
(2004
). 7.
S.
Nikolov
and S.
Clodong
, “Hyperchaos–chaos–hyperchaos transition in modified Rössler systems
,” Chaos Solitons Fractals
28
, 252
–263
(2006
). 8.
T.
Matsumoto
, L.
Chua
, and K.
Kobayashi
, “Hyper chaos: Laboratory experiment and numerical confirmation
,” IEEE Trans. Circuits Syst.
33
, 1143
–1147
(1986
). 9.
R.
Stoop
, J.
Peinke
, J.
Parisi
, B.
Röhricht
, and R.
Huebener
, “A p-GE semiconductor experiment showing chaos and hyperchaos
,” Physica D
35
, 425
–435
(1989
). 10.
T.
Kapitaniak
, L.
Chua
, and G.-Q.
Zhong
, “Experimental hyperchaos in coupled Chua’s circuits
,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl.
41
, 499
–503
(1994
). 11.
A.
Tamasevicuius
, A.
Cenys
, G.
Mykolaitis
, A.
Namajunas
, and E.
Lindberg
, “Hyperchaotic oscillator with gyrators
,” Electron. Lett.
33
, 542
–544
(1997
). 12.
T. F.
Fonzin
, J.
Kengne
, and F.
Pelap
, “Dynamical analysis and multistability in autonomous hyperchaotic oscillator with experimental verification
,” Nonlinear Dyn.
93
, 653
–669
(2018
). 13.
P.
Colet
, R.
Roy
, and K.
Wiesenfeld
, “Controlling hyperchaos in a multimode laser model
,” Phys. Rev. E
50
, 3453
(1994
). 14.
J.-Y.
Hsieh
, C.-C.
Hwang
, A.-P.
Wang
, and W.-J.
Li
, “Controlling hyperchaos of the Rossler system
,” Int. J. Control
72
, 882
–886
(1999
). 15.
L.
Yang
, Z.
Liu
, and J.-M.
Mao
, “Controlling hyperchaos
,” Phys. Rev. Lett.
84
, 67
(2000
). 16.
H.
Zhang
, X.-K.
Ma
, M.
Li
, and J.-L.
Zou
, “Controlling and tracking hyperchaotic Rössler system via active backstepping design
,” Chaos Solitons Fractals
26
, 353
–361
(2005
). 17.
M.
Rafikov
and J. M.
Balthazar
, “On control and synchronization in chaotic and hyperchaotic systems via linear feedback control
,” Commun. Nonlinear Sci. Numer. Simul.
13
, 1246
–1255
(2008
). 18.
A.
Gonchenko
, S.
Gonchenko
, and L. P.
Shilnikov
, “Towards scenarios of chaos appearance in three-dimensional maps
,” Russ. J. Nonlinear Dyn.
8
, 3
–28
(2012
).19.
A.
Gonchenko
, S.
Gonchenko
, A.
Kazakov
, and D.
Turaev
, “Simple scenarios of onset of chaos in three-dimensional maps
,” Int. J. Bifurcat. Chaos
24
, 1440005
(2014
). 20.
A.
Gonchenko
and S.
Gonchenko
, “Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps
,” Physica D
337
, 43
–57
(2016
). 21.
G.
Benettin
, L.
Galgani
, A.
Giorgilli
, and J.-M.
Strelcyn
, “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory
,” Meccanica
15
, 9
–20
(1980
). 22.
R.
Vitolo
, H.
Broer
, and C.
Simó
, “Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems
,” Regul. Chaotic Dyn.
16
, 154
–184
(2011
). 23.
N. V.
Stankevich
, A.
Dvorak
, V.
Astakhov
, P.
Jaros
, M.
Kapitaniak
, P.
Perlikowski
, and T.
Kapitaniak
, “Chaos and hyperchaos in coupled antiphase driven Toda oscillators
,” Regul. Chaotic Dyn.
23
, 120
–126
(2018
). 24.
N.
Stankevich
, A.
Kuznetsov
, E.
Popova
, and E.
Seleznev
, “Chaos and hyperchaos via secondary Neimark–Sacker bifurcation in a model of radiophysical generator
,” Nonlinear Dyn.
97
, 2355
–2370
(2019
). 25.
I. R.
Garashchuk
, D. I.
Sinelshchikov
, A. O.
Kazakov
, and N. A.
Kudryashov
, “Hyperchaos and multistability in the model of two interacting microbubble contrast agents
,” Chaos
29
, 063131
(2019
). 26.
R.
Barrio
, M. A.
Martínez
, S.
Serrano
, and D.
Wilczak
, “When chaos meets hyperchaos: 4D Rössler model
,” Phys. Lett. A
379
, 2300
–2305
(2015
). 27.
I. R.
Garashchuk
, A. O.
Kazakov
, and D. I.
Sinelshchikov
, “Synchronous ocsillations and symmetry breaking in a model of two interacting ultrasound contrast agents
,” Nonlinear Dyn.
101
, 1199
–1213
(2020
). 28.
S.
Yanchuk
and T.
Kapitaniak
, “Chaos–hyperchaos transition in coupled Rössler systems
,” Phys. Lett. A
290
, 139
–144
(2001
). 29.
S.
Yanchuk
and T.
Kapitaniak
, “Symmetry-increasing bifurcation as a predictor of a chaos-hyperchaos transition in coupled systems
,” Phys. Rev. E
64
, 056235
(2001
). 30.
P.
So
, E.
Ott
, T.
Sauer
, B. J.
Gluckman
, C.
Grebogi
, and S. J.
Schiff
, “Extracting unstable periodic orbits from chaotic time series data
,” Phys. Rev. E
55
, 5398
(1997
). 31.
B.
Ermentrout
, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students
(SIAM
, 2002
), Vol. 14
.32.
Y. A.
Kuznetsov
, H. G.
Meijer
, and L.
van Veen
, “The fold-flip bifurcation
,” Int. J. Bifurcat. Chaos
14
, 2253
–2282
(2004
). 33.
J.
Gheiner
, “Codimension-two reflection and non-hyperbolic invariant lines
,” Nonlinearity
7
, 109
(1994
). 34.
L. P.
Shilnikov
, “The theory of bifurcations and turbulence. I
,” in Methods of Qualitative Theory of Differential Equations
, edited by E. A. Leontovich (Gorki University Press
, 1986
), pp. 150
–163
[English translation in Selecta Math. Sov.
10
, 43
–53
(1991
)].35.
S. V.
Gonchenko
and D. V.
Turaev
, “On three types of dynamics and the notion of attractor
,” Proc. Steklov Inst. Math.
297
, 116
–137
(2017
). 36.
S. V.
Gonchenko
, A. S.
Gonchenko
, and A. O.
Kazakov
, “Three types of attractors and mixed dynamics of nonholonomic models of rigid body motion
,” Proc. Steklov Inst. Math.
308
, 125
–140
(2020
). © 2021 Author(s).
2021
Author(s)
You do not currently have access to this content.
