We construct an autonomous low-dimensional system of differential equations by replacement of real-valued variables with complex-valued variables in a self-oscillating system with homoclinic loops of a saddle. We provide analytical and numerical indications and argue that the emerging chaotic attractor is a uniformly hyperbolic chaotic attractor of Smale–Williams type. The four-dimensional phase space of the flow consists of two parts: a vicinity of a saddle equilibrium with two pairs of equal eigenvalues where the angular variable undergoes a Bernoulli map, and a region which ensures that the trajectories return to the origin without angular variable changing. The trajectories of the flow approach and leave the vicinity of the saddle equilibrium with the arguments of complex variables undergoing a Bernoulli map on each return. This makes possible the formation of the attractor of a Smale–Williams type in Poincaré cross section. In essence, our model resembles complex amplitude equations governing the dynamics of wave envelops or spatial Fourier modes. We discuss the roughness and generality of our scheme.
REFERENCES
1.
L.
Shil’nikov
and D.
Turaev
, “Simple bifurcations leading to hyperbolic attractors
,” Comput. Math. Appl.
34
, 173
–193
(1997
). 2.
D.
Turaev
and L.
Shil’nikov
, “Blue sky catastrophes,” in Doklady Mathematics c/c of Doklady-Akademiia Nauk (Maik Nauka, 1995), Vol. 51, pp. 404–407.3.
D. V.
Anosov
, “Dynamical systems in the 1960s: The hyperbolic revolution,” in Mathematical Events of the Twentieth Century (Springer, 2006), pp. 1–17.4.
S.
Smale
, “Differentiable dynamical systems
,” Bull. Am. Math. Soc.
73
, 747
–817
(1967
). 5.
D.
Anosov
, G.
Gould
, S.
Aranson
, V.
Grines
, R.
Plykin
, A.
Safonov
, E.
Sataev
, S.
Shlyachkov
, V.
Solodov
, A.
Starkov
et al., Dynamical Systems IX: Dynamical Systems with Hyperbolic Behaviour
(Springer
, 1995
).6.
A.
Katok
and B.
Hasselblatt
, Introduction to the Modern Theory of Dynamical Systems
(Cambridge University Press
, 1997
), Vol. 54.7.
L. P.
Shil’nikov
, “Mathematical problems of nonlinear dynamics: A tutorial
,” Int. J. Bifurcat. Chaos
7
, 1953
–2001
(1997
). 8.
S. P.
Kuznetsov
, “Dynamical chaos and uniformly hyperbolic attractors: From mathematics to physics
,” Phys. Usp.
54
, 119
(2011
). 9.
S. P.
Kuznetsov
, Hyperbolic Chaos: A Physicist’s View
(Higher Education Press, Springer
, Beijing
, 2012
).10.
V. S.
Afraimovich
, “Strange attractors and quasi-attractors,” in Nonlinear Dynamics and Turbulence, edited by G. I. Barenblatt, G. Iooss, and D. D. Joseph (Pitman, New York, 1983), pp. 1–28.11.
A. S.
Gonchenko
, S. V.
Gonchenko
, A. O.
Kazakov
, and A.
Kozlov
, “Elements of contemporary theory of dynamical chaos: A tutorial. Part I. Pseudohyperbolic attractors
,” Int. J. Bifurcat. Chaos
28
, 1830036
(2018
). 12.
D. V.
Turaev
and L. P.
Shil’nikov
, “An example of a wild strange attractor
,” Sbornik Math.
189
, 291
(1998
). 13.
V. S.
Afraimovich
, V.
Bykov
, and L. P.
Shil’nikov
, “On the origin and structure of the Lorenz attractor
,” DoSSR
234
, 336
–339
(1977
).14.
J.
Guckenheimer
and R. F.
Williams
, “Structural stability of Lorenz attractors
,” Publ. Math. Inst. Hautes Études Sci.
50
, 59
–72
(1979
). 15.
L.
Shilnikov
, “Bifurcations and strange attractors,” arXiv:math/0304457 (2003).16.
R.
Barrio
, A.
Shilnikov
, and L.
Shilnikov
, “Kneadings, symbolic dynamics and painting Lorenz chaos
,” Int. J. Bifurcat. Chaos
22
, 1230016
(2012
). 17.
T.
Xing
, R.
Barrio
, and A.
Shilnikov
, “Symbolic quest into homoclinic chaos
,” Int. J. Bifurcat. Chaos
24
, 1440004
(2014
). 18.
E. N.
Lorenz
, “Deterministic nonperiodic flow
,” J. Atmos. Sci.
20
, 130
–141
(1963
). 19.
R. F.
Williams
, “Expanding attractors
,” Publ. Math. Inst. Hautes Études Sci.
43
, 169
–203
(1974
). 20.
S. P.
Kuznetsov
, “Example of a physical system with a hyperbolic attractor of the Smale–Williams type
,” Phys. Rev. Lett.
95
, 144101
(2005
). 21.
S. P.
Kuznetsov
and A.
Pikovsky
, “Autonomous coupled oscillators with hyperbolic strange attractors
,” Physica D
232
, 87
–102
(2007
). 22.
V.
Kruglov
and S.
Kuznetsov
, “An autonomous system with attractor of Smale–Williams type with resonance transfer of excitation in a ring array of van der Pol oscillators
,” Commun. Nonlinear Sci. Numer. Simul.
16
, 3219
–3223
(2011
). 23.
V. P.
Kruglov
, S. P.
Kuznetsov
, and A.
Pikovsky
, “Attractor of Smale-Williams type in an autonomous distributed system
,” Regul. Chaotic Dyn.
19
, 483
–494
(2014
). 24.
S. P.
Kuznetsov
and I. R.
Sataev
, “Hyperbolic attractor in a system of coupled non-autonomous van der Pol oscillators: Numerical test for expanding and contracting cones
,” Phys. Lett. A
365
, 97
–104
(2007
). 25.
D.
Wilczak
, “Uniformly hyperbolic attractor of the Smale–Williams type for a Poincaré map in the Kuznetsov system
,” SIAM J. Appl. Dyn. Syst.
9
, 1263
–1283
(2010
). 26.
S.
Kuznetsov
and E.
Seleznev
, “A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system
,” J. Exp. Theor. Phys.
102
, 355
–364
(2006
). 27.
L. P.
Shil’nikov
, “On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type
,” Mat. Sbornik
119
, 461
–472
(1968
).28.
The eigenvectors that span the plane tangent to the stable manifold are and at . The initial conditions close to the stable manifold were picked , . There were overall 4096 trajectories with initial angles evenly distributed. The numerical integration of the particular trajectory stopped when exceeded 1.
29.
I. S.
Aranson
and L.
Kramer
, “The world of the complex Ginzburg–Landau equation
,” Rev. Mod. Phys.
74
, 99
(2002
). 30.
V. P.
Kruglov
and L.-B.
Khadzhieva
, “Uniformly hyperbolic attractor in a system based on coupled oscillators with “figure-eight” separatrix
,” Izv. VUZ. Appl. Nonlinear Dyn.
24
, 54
–64
(2016
). 31.
S. P.
Kuznetsov
and Y. V.
Sedova
, “Hyperbolic chaos in the Bonhoeffer–van der Pol oscillator with additional delayed feedback and periodically modulated excitation parameter
,” Izv. VUZ. Appl. Nonlinear Dyn.
27
, 77
–95
(2019
). 32.
S. P.
Kuznetsov
and Y. V.
Sedova
, “Robust hyperbolic chaos in Froude pendulum with delayed feedback and periodic braking
,” Int. J. Bifurcat. Chaos
29
, 1930035
(2019
). 33.
S.
Kuznetsov
, V.
Kruglov
, and Y.
Sedova
, “Mechanical systems with hyperbolic chaotic attractors based on Froude pendulums
,” Russ. J. Nonlinear Dyn.
16
, 51
–58
(2020
).34.
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
). 35.
I.
Shimada
and T.
Nagashima
, “A numerical approach to ergodic problem of dissipative dynamical systems
,” Prog. Theor. Phys.
61
, 1605
–1616
(1979
). 36.
A.
Pikovsky
and A.
Politi
, Lyapunov Exponents: A Tool to Explore Complex Dynamics
(Cambridge University Press
, 2016
).37.
The orthogonalizations were performed with DGEQRF and DORGQR routines from LAPACK library. LEs were calculated for randomly selected trajectories, and the provided results are average values of LEs and standard deviations. Values were rounded to the first significant digits of deviations.
38.
Y.-C.
Lai
, C.
Grebogi
, J.
Yorke
, and I.
Kan
, “How often are chaotic saddles nonhyperbolic?
,” Nonlinearity
6
, 779
(1993
). 39.
V. S.
Anishchenko
, A.
Kopeikin
, J.
Kurths
, T.
Vadivasova
, and G.
Strelkova
, “Studying hyperbolicity in chaotic systems
,” Phys. Lett. A
270
, 301
–307
(2000
). 40.
P. V.
Kuptsov
, “Fast numerical test of hyperbolic chaos
,” Phys. Rev. E
85
, 015203
(2012
). 41.
Adjacent equations are such that the Jacobi matrix of the evolution operator is conjugate transposed, and all of its elements are multiplied by . The reason for using the conjugate transposed matrix is that its eigenvectors are orthogonal to eigenvectors (with eigenvalues not conjugate transpose of each other) of the original matrix. One can also calculate the Lyapunov exponent corresponding to the perturbation vector in forward time and the Lyapunov exponent of the adjacent perturbation vector in backward time and see that they converge to the same value.
42.
M. C.
Cross
and P. C.
Hohenberg
, “Pattern formation outside of equilibrium
,” Rev. Mod. Phys.
65
, 851
(1993
). 43.
L.
Turukina
and A.
Pikovsky
, “Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators
,” Phys. Lett. A
375
, 1407
–1411
(2011
). © 2021 Author(s).
2021
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