Three-Dimensional Torus Breakdown and Chaos With Two Zero Lyapunov Exponents in Coupled Radio-Physical Generators

Stankevich, Nataliya; Shchegoleva, Natal’ya; Сатаев, И. Р.; Кузнецов, А. П.

doi:10.1115/1.4048025

Cited by 17 publications

(8 citation statements)

References 40 publications

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“…Such transition to chaotic attractors (including Shilnikov ones), is observed quite often (see e.g. [38,39,40,41,42,43,44,45,46,47]) and can lead to the birth of "flow-like" chaotic attractors possessing one positive and one very close to zero Lyapunov exponent (for flow systems chaos with additional close to zero Lyapunov exponent in the spectrum is observed in this case). In Sec.…”

Section: (A)mentioning

confidence: 93%

Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

Shykhmamedov¹,

Karatetskaia²,

Казаков³

et al. 2020

Preprint

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We study bifurcation mechanisms of the appearance of hyperchaotic attractors in three-dimensional maps. We consider, in some sense, the simplest cases when such attractors are homoclinic, i.e. they contain only one saddle fixed point and entirely its unstable manifold. We assume that this manifold is two-dimensional, which gives, formally, a possibility to obtain two positive Lyapunov exponents for typical orbits on the attractor (hyperchaos). For realization of this possibility, we propose several bifurcation scenarios of the onset of homoclinic hyperchaos that include cascades of both supercritical period-doubling bifurcations with saddle periodic orbits and supercritical Neimark-Sacker bifurcations with stable periodic orbits, as well as various combinations of these cascades. In the paper, these scenarios are illustrated by an example of three-dimensional Mirá map.

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Section: (A)mentioning

confidence: 93%

Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

Shykhmamedov¹,

Karatetskaia²,

Казаков³

et al. 2020

Preprint

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“…The first two taxonomic ranks are provided by the integers m and p, respectively. It is thus possible to discriminate some behaviors as follows [Klein & Baier, 1991;Letellier & Rössler, 2020;Stankevich et al, 2020].…”

Section: Essential Properties For Defining Taxamentioning

confidence: 99%

“…Governing equations (a) [Stankevich et al, 2020] ẋ = y [Rössler, 1976] [Sprott, 1997] [Ueda, 1993] [Klein & Baier, 1991] [Hénon & Heiles, 1964] [Charó et al, 2019] [Klein & Baier, 1991] [Rössler, 1979b] [Anishchenko & Nikolaev, 2005]…”

Section: Referencementioning

confidence: 99%

“…For similar reason, it is not possible to get conjugated toroidal chaos CT 2 c in a three-dimensional space without self-intersection. Conjugated toroidal chaos C 1 T 2 c ⊂ R 4 is thus structurally different from the toroidal chaos C 1 T 2 d or the toroidal chaos C 1 T 2 -produced by the oscillator investigated by [Stankevich et al, 2020] and by the driven van der Pol system [Ueda, 1993], respectively -which result from a Curry-Yorke scenario and can be embedded within a three-dimensional space.…”

Section: Referencementioning

confidence: 99%

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Dynamical Taxonomy: some taxonomic ranks to systematically classify every chaotic attractor

Letellier,

Stankevich,

Rössler

2021

Preprint

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Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine what are the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labelling. Addressing these problems correspond to the development of a dynamical taxonomy, exhibiting the key properties discriminating the variety of chaotic behaviors discussed in the abundant literature. Starting from the hierarchy of chaos initially proposed by one of us, we systematized the description of chaotic regimes observed in three-and four-dimensional spaces, covering a large variety of known (and less known) examples of chaotic systems. Starting with the spectrum of Lyapunov exponents as the first taxonomic ranks, we extended the description to higher ranks with some concepts inherited from topology (bounding torus, surface of section, first-return map...). By treating extensively the Rössler and the Lorenz attractors, we extended the description of branched manifold -the highest taxonomic rank for classifying chaotic attractor -by a linking matrix (or linker) to multi-component attractors (bounded by a torus whose genus g ≥ 3).

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“…Refs. [15][16][17][18][19][20][21] , however reasons for the appearance of such chaotic attractors in many cases are not clear. As far as we know, the corresponding explanation is given only in the following two cases: (i) when a system is subjected to an external quasiperiodic or skew forcing, see e.g.…”

Section: Introductionmentioning

confidence: 99%

On the origin of chaotic attractors with two zero Lyapunov exponents in a system of five biharmonically coupled phase oscillators

Grines,

Kazakov,

Sataev

2022

Preprint

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We study chaotic dynamics in a system of four differential equations describing the dynamics of five identical globally coupled phase oscillators with biharmonic coupling.We show that this system exhibits strange spiral attractors (Shilnikov attractors) with two zero (indistinguishable from zero in numerics) Lyapunov exponents in a wide region of the parameter space. We explain this phenomenon by means of bifurcation analysis of the three-dimensional Poincaré map for the system under consideration.We show that the chaotic dynamics develop here near a codimension three bifurcation, when a periodic orbit (fixed point in the Poincaré map) has the triplet (1, 1, 1) of multipliers. As it is known, the asymptotic flow normal form for this bifurcation coincides with the three-dimensional Arneodo-Coullet-Spiegel-Tresser (ACST) system in which spiral attractors exist. Based on this, we conclude that the additional near-zero Lyapunov exponent for orbits in the observed attractors appear due to the fact that the corresponding three-dimensional Poincaré map is close to the time-shift map of the three-dimensional ACST-system.

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Three-Dimensional Torus Breakdown and Chaos With Two Zero Lyapunov Exponents in Coupled Radio-Physical Generators

Cited by 17 publications

References 40 publications

Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

Dynamical Taxonomy: some taxonomic ranks to systematically classify every chaotic attractor

On the origin of chaotic attractors with two zero Lyapunov exponents in a system of five biharmonically coupled phase oscillators

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