Numerical test for hyperbolicity in chaotic systems with multiple time delays

Kuptsov, Pavel V.; Кузнецов, С. П.

doi:10.1016/j.cnsns.2017.08.016

Cited by 12 publications

(4 citation statements)

References 42 publications

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“…For example in Refs. [19,20] a special form of the inner product is introduced for analysis of hyperbolicity of chaos in time delay systems. In our analysis however, it is enough to consider the simplest standard dot product.…”

Section: Covariant Lyapunov Vectors and Angles Between Tangent Subspacesmentioning

confidence: 99%

“…Notice that the action of the adjoint propagator as well as the action of the inverted one corresponds to steps backward in time [22]. The form of the adjoint propagator depends on the chosen inner product [19,20], and the standard dot product produces its simplest version: the adjoint propagator is obtained from the original one simply by transposition as F T (t 1 , t 2 ). The steps are performed again with K vectors that are QR-decomposed after each action of the propagator F T (t 1 , t 2 ):…”

Section: Covariant Lyapunov Vectors and Angles Between Tangent Subspacesmentioning

confidence: 99%

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Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction

Kuptsov

Кузнецов

2018

Regul. Chaot. Dyn.

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Pseudohyperbolic attractors are genuine strange chaotic attractors. They do not contain stable periodic orbits and are robust in a sense that such orbits do not appear under variations. The tangent space of these attractors is split into a direct sum of volume expanding and contracting subspaces and these subspaces never have tangencies with each other. Any contraction in the first subspace, if occur, is weaker than contractions in the second one. In this paper we analyze local structure of several chaotic attractors recently suggested in literature as pseudohyperbolic. The absence of tangencies and thus the presence of the pseudohyperbolicity is verified using the method of angles that includes computation of distributions of the angles between the corresponding tangent subspaces. Also we analyze how volume expansion in the first subspace and the contraction in the second one occurs locally. For this purpose we introduce a family of instant Lyapunov exponents. Unlike the well known finite time ones, the instant Lyapunov exponents show expansion or contraction on infinitesimal time intervals. Two types of instant Lyapunov exponents are defined. One is related to ordinary finite time Lyapunov exponents computed in the course of standard algorithm for Lyapunov exponents. Their sums reveal instant volume expanding properties. The second type of instant Lyapunov exponents shows how covariant Lyapunov vectors grow or decay on infinitesimal time. Using both instant and finite time Lyapunov exponents we demonstrate that specific to the pseudohyperbolicity average expanding and contracting properties are typically violated on infinitesimal time. Instantly volumes from the first subspace can sometimes be contacted, directions in the second subspace can sometimes be expanded, and the instant contraction in the first subspace can sometimes be stronger than the contraction in the second subspace.

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Section: Covariant Lyapunov Vectors and Angles Between Tangent Subspacesmentioning

confidence: 99%

Section: Covariant Lyapunov Vectors and Angles Between Tangent Subspacesmentioning

confidence: 99%

Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction

Kuptsov

Кузнецов

2018

Regul. Chaot. Dyn.

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“…The fast method of computation of the angles is developed in papers [36,40], and its implementation for time delay systems can be found in Refs. [34,35].…”

Section: The System and Methods Of Analysismentioning

confidence: 99%

Route to hyperbolic hyperchaos in a nonautonomous time-delay system

Kuptsov,

Kuznetsov

2019

Preprint

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We consider a time delay system whose excitation parameter is periodically modulated. Each new stage of excitation is seeded from the stage before the last, and due to the nonlinearity the seeding arrives with the doubled phase. As a result, the system operates as two coupled hyperbolic chaotic subsystems. Varying the relation between delay time and excitation period we affect the coupling strength between these subsystems as well as the intensity of phase doubling mechanism responsible for the hyperbolicity. Due to this the transition from non-hyperbolic to hyperbolic hyperchaos occurs. The following parts of transition scenario are revealed and analyzed: (a) intermittency as alternation of staying near a fixed point and chaotic bursts; (b) wandering between the fixed point and chaotic subset, which appears near it; (c) plain hyperchaos without hyperbolicity after termination of the visits to the fixed point; (d) transformation of hyperchaos to hyperbolic form.

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“…In the case of high-dimensional systems, to identify a contracting subspace it is more convenient to use not vectors belonging to it, but vectors defining its orthogonal complement, whose dimension is usually small [32]. The latter are obtained from solution of the adjoint system of linearized equations in variations [32,33,34,35].…”

Section: Angles Between Subspaces Of Perturbation Vectorsmentioning

confidence: 99%

Chaos and hyperchaos of geodesic flows on curved manifolds corresponding to mechanically coupled rotators: Examples and numerical study

Кузнецов¹

2018

Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki

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A system of N rotators is investigated with a constraint given by a condition of vanishing sum of the cosines of the rotation angles. Equations of the dynamics are formulated and results of numerical simulation for the cases N =3, 4, and 5 are presented relating to the geodesic flows on a two-dimensional, three-dimensional, and four-dimensional manifold, respectively, in a compact region (due to the periodicity of the configuration space in angular variables). It is shown that a system of three rotators demonstrates chaos, characterized by one positive Lyapunov exponent, and for systems of four and five elements there are, respectively, two and three positive exponents ("hyperchaos"). An algorithm has been implemented that allows calculating the sectional curvature of a manifold in the course of numerical simulation of the dynamics at points of a trajectory. In the case of N =3, curvature of the two-dimensional manifold is negative (except for a finite number of points where it is zero), and Anosov's geodesic flow is realized. For N =4 and 5, the computations show that the condition of negative sectional curvature is not fulfilled. Also the methodology is explained and applied for testing hyperbolicity based on numerical analysis of the angles between the subspaces of small perturbation vectors; in the case of N =3, the hyperbolicity is confirmed, and for N =4 and 5 the hyperbolicity does not take place.

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Numerical test for hyperbolicity in chaotic systems with multiple time delays

Cited by 12 publications

References 42 publications

Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction

Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction

Route to hyperbolic hyperchaos in a nonautonomous time-delay system

Chaos and hyperchaos of geodesic flows on curved manifolds corresponding to mechanically coupled rotators: Examples and numerical study

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