Double-scroll attractors are one of the pillars of modern chaos theory. However, rigorous computer-free analysis of their existence and global structure is often elusive. Here, we address this fundamental problem by constructing an analytically tractable piecewise-smooth system with a double-scroll attractor. We derive a Poincaré return map to prove the existence of the double-scroll attractor and explicitly characterize its global dynamical properties. In particular, we reveal a hidden set of countably many saddle orbits associated with infinite-period Smale horseshoes. These complex hyperbolic sets emerge from an ordered iterative process that yields sequential intersections between different horseshoes and their preimages. This novel distinctive feature differs from the classical Smale horseshoes, directly intersecting with their own preimages. Our global analysis suggests that the structure of the classical Chua attractor and other figure-eight attractors might be more complex than previously thought.
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April 2023
Research Article|
April 10 2023
The hidden complexity of a double-scroll attractor: Analytic proofs from a piecewise-smooth system
Special Collection:
Non-smooth Dynamics
Vladimir N. Belykh
;
Vladimir N. Belykh
a)
(Conceptualization, Formal analysis, Investigation, Writing – original draft, Writing – review & editing)
1
Department of Mathematics, Volga State University of Water Transport
, 5A, Nesterov str., Nizhny Novgorod 603950, Russia
2
Department of Control Theory, Lobachevsky State University of Nizhny Novgorod
, 23, Gagarin Ave., 603950 Nizhny Novgorod, Russia
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Nikita V. Barabash
;
Nikita V. Barabash
b)
(Conceptualization, Investigation, Writing – review & editing (equal))
1
Department of Mathematics, Volga State University of Water Transport
, 5A, Nesterov str., Nizhny Novgorod 603950, Russia
2
Department of Control Theory, Lobachevsky State University of Nizhny Novgorod
, 23, Gagarin Ave., 603950 Nizhny Novgorod, Russia
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Igor Belykh
Igor Belykh
c)
(Conceptualization, Investigation, Writing – original draft (equal), Writing – review & editing (equal))
3
Department of Mathematics and Statistics, Georgia State University
, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
c)Author to whom correspondence should be addressed: ibelykh@gsu.edu
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c)Author to whom correspondence should be addressed: ibelykh@gsu.edu
a)
Electronic mail: belykh@unn.ru
b)
Electronic mail: barabash@itmm.unn.ru
Note: This article is part of the Focus Issue, Non-smooth Dynamics.
Chaos 33, 043119 (2023)
Article history
Received:
December 17 2022
Accepted:
March 22 2023
Citation
Vladimir N. Belykh, Nikita V. Barabash, Igor Belykh; The hidden complexity of a double-scroll attractor: Analytic proofs from a piecewise-smooth system. Chaos 1 April 2023; 33 (4): 043119. https://doi.org/10.1063/5.0139064
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