In this paper, the dynamics of the paradigmatic Rössler system is investigated in a yet unexplored region of its three-dimensional parameter space. We prove a necessary condition in this space for which the Rössler system can be chaotic. By using standard numerical tools, like bifurcation diagrams, Poincaré sections, and first-return maps, we highlight both asymptotically stable limit cycles and chaotic attractors. Lyapunov exponents are used to verify the chaotic behavior while random numerical procedures and various plane cross sections of the basins of attraction of the coexisting attractors prove that both limit cycles and chaotic attractors are hidden. We thus obtain previously unknown examples of bistability in the Rössler system, where a point attractor coexists with either a hidden limit cycle attractor or a hidden chaotic attractor.
REFERENCES
1.
O. E.
Rössler
, “An equation for continuous chaos
,” Phys. Lett. A
57
, 397
–398
(1976
). 2.
O. E.
Rössler
, “Different types of chaos in two simple differential equations
,” Z. Naturforsch. A
31
, 1664
–1670
(1976
). 3.
E. N.
Lorenz
, “Deterministic nonperiodic flow
,” J. Atmos. Sci.
20
, 130
–141
(1963
). 4.
U.
Feudel
, “Complex dynamics in multistable systems
,” Int. J. Bifurcation Chaos
18
, 1607
–1626
(2007
). 5.
D. E.
Postnov
, T. E.
Vadivasova
, O. V.
Sosnovtseva
, A. G.
Balanov
, and V. S.
Anishchenko
, “Role of multistability in the transition to chaotic phase synchronization
,” Chaos
9
, 227
–232
(1999
). 6.
A. N.
Pisarchik
, R.
Jaimes-Reátegui
, and J. H.
García-López
, “Synchronization of coupled bistable chaotic systems: Experimental study
,” Philos. Trans. R. Soc. A
336
, 459
–473
(2007
). 7.
C. R.
Hens
, R.
Banerjee
, U.
Feudel
, and S. K.
Dana
, “How to obtain extreme multistability in coupled dynamical systems
,” Phys. Rev. E
85
, 035202(R)
(2012
). 8.
M. S.
Patel
, U.
Patel
, A.
Sen
, G. C.
Sethia
, C.
Hens
, S. K.
Dana
, U.
Feudel
, K.
Showalter
, C. N.
Ngonghala
, and R. E.
Amritkar
, “Experimental observation of extreme multistability in an electronic system of two coupled Rössler oscillators
,” Phys. Rev. E
89
, 022918
(2014
). 9.
P.
Jaros
, P.
Perlikowski
, and T.
Kapitaniak
, “Synchronization and multistability in the ring of modified Rössler oscillators
,” Eur. Phys. J. Spec. Top.
224
, 1541
–1552
(2015
). 10.
R.
Barrio
, F.
Blesa
, and S.
Serrano
, “Qualitative analysis of the Rössler equations: Bifurcations of limit cycles and chaotic attractors
,” Physica D
238
, 1087
–1100
(2009
). 11.
C.
Li
, W.
Hu
, J. C.
Sprott
, and X.
Wang
, “Multistability in symmetric chaotic systems
,” Eur. Phys. J. Spec. Top.
224
, 1493
–1506
(2015
). 12.
J. C.
Sprott
and C.
Li
, “Asymmetric bistability in the Rössler system
,” Acta Phys. Pol. B
48
, 97
–107
(2017
). 13.
G. A.
Leonov
, N. V.
Kuznetsov
, and V. I.
Vagaitsev
, “Localization of hidden Chua’s attractors
,” Phys. Lett. A
375
, 2230
–2233
(2011
). 14.
G. A.
Leonov
, N. V.
Kuznetsov
, and V. I.
Vagaitsev
, “Hidden attractor in smooth Chua systems
,” Physica D
241
, 1482
–1486
(2012
). 15.
G. A.
Leonov
and N. V.
Kuznetsov
, “Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits
,” Int. J. Bifurcation Chaos
23
, 1330002
(2013
). 16.
A. S.
Pikovski
, M. I.
Rabinovich
, and V. Y.
Trakhtengerts
, “Onset of stochasticity in decay confinement of parametric instability
,” J. Exp. Theor. Phys.
47
, 715
–719
(1978
), see http://www.stat.physik.uni-potsdam.de/∼pikovsky/pdffiles/1978/jetp.pdf.17.
N. V.
Kuznetsov
, G. A.
Leonov
, T. N.
Mokaev
, and S. M.
Seledzhi
, “Hidden attractor in the Rabinovich system, Chua circuits and PLL
,” AIP Conf. Proc.
1738
, 210008
(2016
). 18.
A. B.
Glukhovskii
and F. V.
Dolzhanskii
, “Three-component geostrophic model of convection in a rotating fluid
,” Acad. Sci. USSR Izv. Atmos. Ocean. Phys.
16
, 311
–318
(1980
), see https://ui.adsabs.harvard.edu/abs/1980FizAO..16..451G/abstract.19.
D.
Dudkowski
, S.
Jafari
, T.
Kapitaniak
, N. V.
Kuznetsov
, G. A.
Leonov
, and A.
Prasad
, “Hidden attractors in dynamical systems
,” Phys. Rep.
637
, 1
–50
(2016
). 20.
M. I.
Rabinovich
and A. L.
Fabrikant
, “Stochastic self-modulation of waves in nonequilibrium media
,” J. Exp. Theor. Phys.
50
, 311
–317
(1979
), see http://jetp.ac.ru/cgi-bin/dn/e_050_02_0311.pdf.21.
M.-F.
Danca
, N. V.
Kuznetsov
, and G.
Chen
, “Unusual dynamics and hidden attractors of the Rabinovich–Fabrikant system
,” Nonlinear Dyn.
88
, 791
–805
(2017
). 22.
Z.
Wei
, “Dynamical behaviors of a chaotic system with no equilibria
,” Phys. Lett. A
376
, 102
–108
(2011
). 23.
Z.
Wei
and Q.
Yang
, “Dynamical analysis of a new autonomous 3D chaotic system only with stable equilibria
,” Nonlinear Anal. Real World Appl.
12
, 106
–118
(2011
). 24.
Z.
Wei
and Q.
Yang
, “Dynamical analysis of the generalized Sprott C system with only two stable equilibria
,” Nonlinear Dyn.
68
, 543
–554
(2012
). 25.
X.
Wang
and G.
Chen
, “A chaotic system with only one stable equilibrium
,” Commun. Nonlinear Sci. Numer. Simul.
17
, 1264
–1272
(2012
). 26.
X.
Wang
and G.
Chen
, “Constructing a chaotic system with any number of equilibria
,” Nonlinear Dyn.
71
, 429
–436
(2013
). 27.
S.
Jafari
, J.
Sprott
, and S. M. R. H.
Golpayegani
, “Elementary quadratic chaotic flows with no equilibria
,” Phys. Lett. A
377
, 699
–702
(2013
). 28.
M.
Molaie
, S.
Jafari
, J. C.
Sprott
, and S. M. R. H.
Golpayegani
, “Simple chaotic flows with one stable equilibrium
,” Int. J. Bifurcation Chaos
23
, 1350188
(2013
). 29.
K.
Barati
, S.
Jafari
, J. C.
Sprott
, and V.-T.
Pham
, “Simple chaotic flows with a curve of equilibria
,” Int. J. Bifurcation Chaos
26
, 1630034
(2016
). 30.
T.
Gotthans
, J. C.
Sprott
, and J.
Petrzela
, “Simple chaotic flow with circle and square equilibrium
,” Int. J. Bifurcation Chaos
26
, 1650137
(2016
). 31.
S.
Jafari
, J. C.
Sprott
, and M.
Molaie
, “A simple chaotic flow with a plane of equilibria
,” Int. J. Bifurcation Chaos
26
, 1650098
(2016
). 32.
S.
Jafari
, J. C.
Sprott
, V.-T.
Pham
, C.
Volos
, and C.
Li
, “Simple chaotic 3D flows with surfaces of equilibria
,” Nonlinear Dyn.
86
, 1349
–1358
(2016
). 33.
B.
Munmuangsaen
and B.
Srisuchinwong
, “A hidden chaotic attractor in the classical Lorenz system
,” Chaos Soliton. Fract.
107
, 61
–66
(2018
). 34.
J. C.
Sprott
and B.
Munmuangsaen
, “Comment on a hidden chaotic attractor in the classical Lorenz system
,” Chaos Soliton. Fract.
113
, 261
–262
(2018
). 35.
C.
Letellier
, P.
Dutertre
, and B.
Maheu
, “Unstable periodic orbits and templates of the Rössler system: Toward a systematic topological characterization
,” Chaos
5
(1
), 271
–282
(1995
). 36.
H.
Poincaré
, “Sur les courbes définies par les équations différentielles (iii)
,” J. Math. Pures Appl. 4E Sér.
1
, 167
–244
(1885
), see https://link.springer.com/article/10.1007/BF02403068.37.
I.
Bendixson
, “Sur les courbes définies par des équations différentielles
,” Acta Math.
24
, 1
–88
(1901
). 38.
I.
Gardini
, “Hopf bifurcations and period-doubling transitions in Rössler model
,” Nuovo Cimento B
89
, 139
–160
(1985
). 39.
Y. A.
Kuznetsov
, “Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODE’s
,” SIAM J. Numer. Anal.
36
, 1104
–1124
(1999
). 40.
Y. A.
Kuznetsov
, Elements of Applied Bifurcation Theory
(Springer-Verlag
, New York
, 2004
).41.
M.
Hénon
, “On the numerical computation of Poincaré maps
,” Physica D
5
, 412
–414
(1982
). 42.
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
). 43.
H.
Haken
, “At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point
,” Phys. Lett. A
94
, 71
–72
(1983
). 44.
N.
Metropolis
, M.
Stein
, and P.
Stein
, “On finite limit sets for transformations on the unit interval
,” J. Comb. Theory Ser. A
15
, 25
–44
(1973
). 45.
R. M.
May
, “Simple mathematical models with very complicated dynamics
,” Nature
261
, 459
–467
(1976
). 46.
E.
Ott
, “Strange attractors and chaotic motions of dynamical systems
,” Rev. Mod. Phys.
53
, 655
–671
(1981
). 47.
48.
C.
Grebogi
, E.
Ott
, and J. A.
Yorke
, “Chaotic attractors in crisis
,” Phys. Rev. Lett.
48
, 1507
–1510
(1982
). 49.
C.
Grebogi
, E.
Ott
, and J. A.
Yorke
, “Crises, sudden changes in chaotic attractors, and transient chaos
,” Physica D
7
, 181
–200
(1983
). 50.
C.
Grebogi
, E.
Ott
, F.
Romeiras
, and J. A.
Yorke
, “Chaotic attractors in crisis
,” Phys. Rev. A
36
, 5365
–5380
(1987
). 51.
P.
Frederickson
, J. L.
Kaplan
, E. D.
Yorke
, and J. A.
Yorke
, “The Lyapunov dimension of strange attractors
,” J. Differ. Equ.
49
, 185
–207
(1983
). © 2020 Author(s).
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