Toroidal Sets Research Articles

Universal bounds on the entropy of toroidal attractors

A toroidal set is a compactum K \subseteq \mathbb{R}^{3} which has a neighbourhood basis of solid tori. We study the topological entropy of toroidal attractors K , bounding it from below in terms of purely topological properties of K . In particular, we show that for a toroidal set K , either any smooth attracting dynamics on K has an entropy at least \log 2 , or (up to continuation) K admits smooth attracting dynamics which are stationary (hence with a zero entropy).

Knotted toroidal sets, attractors and incompressible surfaces

In this paper we give a complete characterization of those knotted toroidal sets that can be realized as attractors for discrete or continuous dynamical systems globally defined in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document}. We also see that the techniques used to solve this problem can be used to give sufficient conditions to ensure that a wide class of subcompacta of R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document} that are attractors for homeomorphisms must also be attractors for flows. In addition we study certain attractor-repeller decompositions of S3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {S}}^3$$\\end{document} which arise naturally when considering toroidal sets.

Howe duality in the toroidal setting

In this paper, we construct and study various dual pairs acting on the oscillator modules of the symplectic toroidal Lie algebras coordinated by irrational quantum tori. This extends the classical Howe dual pairs to the toroidal setup.

The geometric index and attractors of homeomorphisms of

AbstractIn this paper we focus on compacta$K \subseteq \mathbb {R}^3$which possess a neighbourhood basis that consists of nested solid tori$T_i$. We call these sets toroidal. Making use of the classical notion of the geometric index of a curve inside a torus, we introduce the self-geometric index of a toroidal setK, which roughly captures how each torus$T_{i+1}$winds inside the previous$T_i$as$i \rightarrow +\infty $. We then use this index to obtain some results about the realizability of toroidal sets as attractors for homeomorphisms of$\mathbb {R}^3$.

Dynamics of the mathematical model of phase-locked systems with delay

In the article, the conditions for the existence of limit cycles of the first kind are obtained for self-tuning systems with delay, which, in turn, determine the conditions for the occurrence of hidden synchronization modes in such systems. The principle of the proof is based on constructing a positively invariant toroidal set using two cylindrical surfaces, whose boundaries are determined by the limit cycles of a system of the second-order differential equations. Using the results obtained in the article for limit cycles, the possibility of using the curvature of the cycle for a comparative analysis of the proximity of the cycles of phase and non-phase systems, as well as for determining the mode of hidden synchronization, is shown.

Steklov spectral problems in a set with a thin toroidal hole

The paper concerns the Steklov spectral problem for the Laplace operator, and some variants in a 3-dimensional bounded domain, with a cavity Γ ε having the shape of a thin toroidal set, with a constant cross-section of diameter ε ≪ 1 . We construct the main terms of the asymptotic expansion of the eigenvalues in terms of real-analytic functions of the variable | ln ε | − 1 , and we prove that the relative asymptotic error is of much smaller order O ( ε | ln ε | ) as ε → 0 + . The asymptotic analysis involves eigenvalues and eigenfunctions of a certain integral operator on the smooth curve Γ , the axis of the cavity Γ ε .

Knots and Solenoids that Cannot Be Attractors of Self-homeomorphisms of ℝ3

AbstractAs a 1st step to understand how complicated attractors for dynamical systems can be, one may consider the following realizability problem: given a continuum $K \subseteq \mathbb{R}^3$, decide when $K$ can be realized as an attractor for a homeomorphism of $\mathbb{R}^3$. In this paper we introduce toroidal sets as those continua $K \subseteq \mathbb{R}^3$ that have a neighbourhood basis comprised of solid tori and, generalizing the classical notion of genus of a knot, give a natural definition of the genus of toroidal sets and study some of its properties. Using these tools we exhibit knots and solenoids for which the answer to the realizability problem stated above is negative.

On the Exponential Stability of a Trivial Torus for One Class of Nonlinear Impulsive Systems

We prove the exponential stability of a trivial torus for a class of nonlinear extensions of dynamical systems on a torus. The obtained results can be applied to the study of stability of toroidal sets for a certain class of impulsive dynamical systems.

Homomorphisms between different quantum toroidal and affine Yangian algebras

This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of sln, denoted by Uq1,q2,q3(n) and Yh1,h2,h3(n), respectively. Our motivation arises from the milestone work [11], where a similar relation between the quantum loop algebra Uq(Lg) and the Yangian Yh(g) has been established by constructing an isomorphism of C[[ħ]]-algebras Φ:Uˆexp⁡(ħ)(Lg)⟶∼Yˆħ(g) (with ˆ standing for the appropriate completions). These two completions model the behavior of the algebras in the formal neighborhood of h=0. The same construction can be applied to the toroidal setting with qi=exp⁡(ħi) for i=1,2,3 (see [11,22]). In the current paper, we are interested in the more general relation: q1=ωmneh1/m,q2=eh2/m,q3=ωmn−1eh3/m, where m,n≥1 and ωmn is an mn-th root of 1. Assuming ωmnm is a primitive n-th root of unity, we construct a homomorphism Φm,nωmn between the completions of the formal versions of Uq1,q2,q3(m) and Yh1/mn,h2/mn,h3/mn(mn).

Integrals of motion from quantum toroidal algebras**Dedicated to the memory of Petr Petrovich Kulish.

We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors.That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model.We also discuss the duality of XXZ models in quantum toroidal setting and the implications for the quantum KdV model. In particular, we conjecture that the spectrum of non-local integrals of motion of Bazhanov, Lukyanov, and Zamolodchikov is described by Gaudin Bethe ansatz equations associated to affine .

Perturbation Theorems for a Multifrequency System with Pulses

We consider a problem of preservation of a piecewise continuous invariant toroidal set for a class of multifrequency systems with pulses at nonfixed times under perturbations of the right-hand side. New theorems that impose constraints on perturbation terms not in the entire phase space but only in a nonwandering set of the dynamical system guarantee the existence of exponentially stable invariant toroidal set.

Invariant manifolds of one class of systems of impulsive differential equations

We consider general problems related to the existence of invariant toroidal sets for linear and weakly nonlinear systems of impulsive differential equations defined in the direct product of an m-dimensional torus and an n-dimensional Euclidean space. We investigate classes of problems for which the conditions for the existence of invariant toroidal manifolds are satisfied.

Existence of an invariant torus for one class of systems of differential equations

We consider the problem of the existence of an asymptotically stable toroidal set for a system of linear differential equations defined on an m-dimensional torus. We establish conditions under which a nonlinear system of differential equations has an invariant toroidal manifold.

QPS Transport Physics Flexibility Using Variable Coil Currents

An important goal for a stellarator design is to incorporate enough flexibility to experimentally test a range of physics issues. The proposed Quasi-Poloidal Stellarator device achieves this by allowing independently variable currents in the modular, vertical field, and toroidal coil sets. Numerical optimizations and modeling show that this can allow significant tests of neoclassical cross-field transport rates, reduced poloidal flow damping (relative to the tokamak), and magnetic island width control. This flexibility is achieved in a unique, very low aspect ratio (R0/<a> = 2.7) two-field period (racetrack-shaped) configuration that generates rotational transform from a combination of internal plasma currents and external shaping.

On One Method for the Introduction of Local Coordinates in a Neighborhood of an Invariant Toroidal Set

We consider one method for the introduction of local coordinates in a neighborhood of an m-dimensional invariant torus of a dynamical system of differential equations in the Euclidean space Rn in dimensions satisfying the inequalities m + 1 < n ≤ 2m.

Schur duality in the toroidal setting

The classical Frobenius-Schur duality gives a correspondence between finite dimensional representations of the symmetric and the linear groups. The goal of the present paper is to extend this construction to the quantum toroidal setup with only elementary (algebraic) methods. This work can be seen as a continuation of [J, D1 and C2] (see also [C-P and G-R-V]) where the cases of the quantum groups U q (sl(n)), Y(sl(n)) (the Yangian) and U q (sl(n)) are given. In the toroidal setting the two algebras involved are deformations of Cherednik's double affine Hecke algebra introduced in [C1] and of the quantum toroidal group as given in [G-K-V]. Indeed, one should keep in mind the geometrical construction in [G-R-V] and [G-K-V] in terms of equivariant K-theory of some flag manifolds. A similar K-theoretic construction of Cherednik's algebra has motivated the present work. At last, we would like to lay emphasis on the fact that, contrary to [J, D1 and C2], the representations involved in our duality are infinite dimensional. Of course, in the classical case, i.e.,q=1, a similar duality holds between the toroidal Lie algebra and the toroidal version of the symmetric group. The authors would like to thank V. Ginzburg for a useful remark on a preceding version of this paper.

The capacity of slender toroidal sets in ?N

The capacity of slender toroidal sets in ?N

Certain problems of existence of invariant toroidal sets of systems of difference equations

Certain problems of existence of invariant toroidal sets of systems of difference equations

Toroidal integral sets of systems with impulses

Toroidal integral sets of systems with impulses

Vibration and stability of a set of superconducting toroidal magnets

The vibration and elastic stability of a set of discrete superconducting magnets arranged to form a ’’bumpy’’ toroidal field is examined. The mutual destabilizing magnetic forces between magnet pairs are calculated using a numerical differential inductance technique. It is shown that the mutual attractive magnetic forces can produce elastic buckling of the entire toroidal set. Using the theory of periodic structures, the vibration modes of the set are found as functions of the coil current. The response of the set of magnets to an earthquake-type motion of the toroidal base is also calculated.