Toroidal Manifold Research Articles

Numerical detection of suppression of quasi‐periodic solutions

AbstractDynamical systems can show so‐called quasi‐periodic solutions, which are composed of two or more so‐called incommensurable frequencies. Solving these systems in the time‐domain is not favorable, due to the fact that quasi‐periodic solutions have no finite periodicity, thus it is unclear how long simulations must be carried out in order to capture all relevant information. The time‐solution however fills a toroidal manifold densely. To calculate the invariant manifold directly one can use the hyper‐time parametrization. One can easily extract maximum values from the hyper‐time solution. The trade‐off however is, that the parametrization loses its validity if the quasi‐periodic solution synchronizes. When continuing quasi‐periodic solution branches it is essential to detect an approach to such synchronization points. We present an algorithm, with which one can detect synchronization by suppression of quasi‐periodic solutions, on the basis of a solution in the hyper‐time parametrization.

Complicated Dynamical Behaviors of a Geometrical Oscillator with a Mass Parameter

In this paper, we consider a special kind of geometrical nonlinear oscillator with a mass parameter admitting two different dynamical states leading to a double-valued potential energy. A cylindrical manifold is introduced to formulate the equation of motion to describe the distinguished dynamical behaviors. With the help of Hamiltonian, complex bifurcations are demonstrated with varying parameters including periodic solutions, the steady states and the blowing up phenomenon near [Formula: see text] to infinity. A toroidal manifold is introduced to map the infinities into [Formula: see text] on the torus exhibiting saddle-node-like behavior, where the uniqueness of solution is lost, for which a special “collision” parameter is introduced to define the possible motion leaving from infinities. Numerical calculation is carried out to generate bifurcation diagrams using Poincaré sections for the perturbed system to exhibit complex dynamics including the coexistence of periodic solutions, chaos from the coexisting periodic doubling and also instant chaos from the coexisting periodic solutions. The results demonstrated herein this paper provide a brand new insight into the understanding of enriched nonlinear dynamics and an essential explanation about “collision” of mechanical system with both geometrical and mass parameters.

CPT-even electrodynamics in a multidimensional torus: Casimir effect at finite temperature

The energy–momentum tensor for the electromagnetism with Lorentz breaking even term of the Standard Model Extended (SME) photon sector confined in a hyper torus is determined. A generalized partition function method is used, following in parallel the thermofield dynamics formalism written in N-dimensional toroidal manifold. After considering general aspects of the SME photon sector in a toroidal manifold, the influence of the isotropic CPT-even electromagnetic sector of the SME is analysed. The approach is then applied to the Casimir effect at finite temperature, corresponding to a topology (S^{1})^{r}times R^{N-r}, where N is the dimension of the Minkowski space-time, and r is the number of compactified dimensions.influence of the isotropic CPT-even electromagnetic sector of the SME is analysed.

A design strategy for water-based noise suppression systems in liquid rocket engines firing tests

This paper presents a streamlined design procedure for water-based noise suppression systems that are applicable to multiple classes of rocket engines. A newly adapted steady quasi-one-dimensional two-phase model is employed to predict the evolution of the exhaust gases interacting with water droplets. Such a model is embedded into a two-step optimization procedure with the objective of finding the most efficient combination of the suppressor operative parameters. This information is then used to design the hardware of the system, which consists in a set of injectors, with the task of producing atomized water jets directed towards the exhaust gases, and a toroidal manifold, with the task of delivering water to the injectors at uniform conditions of pressure and velocity. Finally, the proposed design procedure is applied to a 15 kN thrust class oxygen/methane liquid rocket engine. Technical specifications of the resulting water-based noise suppression system are provided along with a detailed three-dimensional CAD model.

Flow models of perforated manifolds and plates for the design of a large thermal storage tank for district heating with minimal maldistribution and thermocline growth

Large water tanks are used as thermal energy storage components in district heating systems to store sensible heat produced by intermittent energy sources and to decouple the production of thermal energy from its demand. Good thermal stratification is crucial for energy storage efficiency, thus flow maldistribution and mixing of water layers at different temperatures should be minimized. This paper proposes an innovative internal flow distribution configuration for a large-size thermal energy storage, and develops new simplified analytical models for the choice of its design parameters. In the novel configuration, water is injected into (and collected from) the cap volumes of the tank by flowing radially inward (outward) through several small orifices of a peripheral toroidal manifold. Two horizontal perforated plates cover the full cross sections downstream of the manifolds and rectify the vertical flow, thus reducing mixing. Uniform perforation pitch was analytically demonstrated to be the most reasonable solution both for the toroidal distributors and for the rectifying plates. A 1D model was developed to predict the time evolution of the vertical temperature distribution in the tank. The turbulence-related parameters that could not be inferred from the existing fluid-mechanics literature were initially estimated with CFD simulations. The results of CFD-calibrated model were then compared to experimental data obtained from a full-scale large water-tank facility recently built in Brescia according to the proposed design. After a re-calibration of the exponent defining the decay of homogeneous turbulence downstream of the perforated plates, good agreement was found between measured and predicted vertical temperatures. With the novel inlet design, a thermocline of about 0.5 m is established immediately downstream of the perforated plate, and remains practically constant along time. The model is important to minimize and control the thermocline thickness so as to maximize the recoverable thermal energy, not only at the tank design stage but also to identify optimal loading and unloading protocols.

Toroidal topology of population activity in grid cells

The medial entorhinal cortex is part of a neural system for mapping the position of an individual within a physical environment1. Grid cells, a key component of this system, fire in a characteristic hexagonal pattern of locations2, and are organized in modules3 that collectively form a population code for the animal’s allocentric position1. The invariance of the correlation structure of this population code across environments4,5 and behavioural states6,7, independent of specific sensory inputs, has pointed to intrinsic, recurrently connected continuous attractor networks (CANs) as a possible substrate of the grid pattern1,8–11. However, whether grid cell networks show continuous attractor dynamics, and how they interface with inputs from the environment, has remained unclear owing to the small samples of cells obtained so far. Here, using simultaneous recordings from many hundreds of grid cells and subsequent topological data analysis, we show that the joint activity of grid cells from an individual module resides on a toroidal manifold, as expected in a two-dimensional CAN. Positions on the torus correspond to positions of the moving animal in the environment. Individual cells are preferentially active at singular positions on the torus. Their positions are maintained between environments and from wakefulness to sleep, as predicted by CAN models for grid cells but not by alternative feedforward models12. This demonstration of network dynamics on a toroidal manifold provides a population-level visualization of CAN dynamics in grid cells.

Stability of cluster formations in adaptive Kuramoto networks

This paper studies stability properties of multi-cluster formations in Kuramoto networks with adaptive coupling. Sufficient conditions for the local asymptotic stability of the corresponding synchronization invariant toroidal manifold are derived and formulated in terms of the intra-cluster interconnection topology and plasticity parameters of the adaptive couplings. The proposed sufficient stability conditions qualitatively mimic certain counterpart results for Kuramoto networks with static coupling which require sufficiently strong and dense intra-cluster connections and sufficiently weak and sparse inter-cluster ones. Remarkably, the existence of cluster formations depends on the interconnection structure between nodes belonging to different clusters and does not require any coupling links between nodes that form a cluster. On the other hand, the stability properties of clusters depend on the interconnection structure inside the clusters. This dependence constitutes the main contribution of the paper. Also, a numerical example is provided to validate the proposed theoretical findings.

Synchronization and Multicluster Capabilities of Oscillatory Networks With Adaptive Coupling

We prove the existence of a multidimensional nontrivial invariant toroidal manifold for the Kuramoto network with adaptive coupling. The constructed invariant manifold corresponds to the multicluster behavior of the oscillators phases. Contrary to the static coupling, the adaptive coupling strengths exhibit quasiperiodic oscillations preserving zero phase-difference within clusters. The derived sufficient conditions for the existence of the invariant manifold provide a tradeoff between the natural frequencies of the oscillators, coupling plasticity parameters, and the interconnection structure of the network. Furthermore, we study the robustness of the invariant manifold with respect to the perturbations of the interconnection topology, and establish structural, and quantitative constraints on the perturbation adjacency matrix preserving the invariant manifold. Additionally, we demonstrate the application of the new results to the problem of interconnection topology design, which consists in endowing the desired multicluster behavior to the network by controlling its interconnection structure.

On the universality of the frequency spectrum and band-gap optimization of quasicrystalline-generated structured rods.

The dynamical properties of periodic two-component phononic rods, whose elementary cells are generated adopting the Fibonacci substitution rules, are studied through the recently introduced method of the toroidal manifold. The method allows all band gaps and pass bands featuring the frequency spectrum to be represented in a compact form with a frequency-dependent flow line on the surface describing their ordered sequence. The flow lines on the torus can be either closed or open: in the former case, (i) the frequency spectrum is periodic and the elementary cell corresponds to a canonical configuration, (ii) the band gap density depends on the lengths of the two phases; in the latter, the flow lines cover ergodically the torus and the band gap density is independent of those lengths. It is then shown how the proposed compact description of the spectrum can be exploited (i) to find the widest band gap for a given configuration and (ii) to optimize the layout of the elementary cell in order to maximize the low-frequency band gap. The scaling property of the frequency spectrum, that is a distinctive feature of quasicrystalline-generated phononic media, is also confirmed by inspecting band-gap/pass-band regions on the torus for the elementary cells of different Fibonacci orders. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 2)'.

Стiйкiсть iнварiантного многовиду нелiнiйної системи диференцiальних рiвнянь

The theory of extensions of the dynamical equations on the torus is an important section of the theory of ordinary differential equations that is intensively evolving and has an important applied application to various tasks of science and technology. This theory describes processes that have oscillatory character. One of the important questions of mathematical theory of multifrequency oscillations is the problem of the existence and stability of invariant toroidal manifolds, the problem of the structural stability of the invariant manifold, its preservation under small perturbations for the systems of differential equations that are defined in the direct product of a torus and Euclidean space. Such manifolds serve as carriers of multifrequency oscillations in the system. The basics of this theory have been developed by A. M. Samoilenko.In this paper the class of differential equations, defined in the direct product of m-dimensional torus T m and n-dimensional Euclidean space R n for which there exist conditions for the existence of an asymptotically stable invariant toroidal manifold, is investigated. We formulate and prove sufficient conditions for the existence and asymptotic stability of invariant toroidal class of nonlinear extensions of dynamical systems on torus, that has specific properties in ω-limit set Ω of the trajectories ϕ_t (ϕ). Two theorems, that define the conditions for the existence of asymptotically stable invariant sets for linear expansion of the dynamical system on torus and corresponding perturbed system, are given. We search an invariant manifold of a nonlinear system by the iterative method, proposed by A. M. Samoilenko.

Quantum Computing, Seifert Surfaces, and Singular Fibers

The fundamental group π 1 ( L ) of a knot or link L may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In this paper, one defines braids whose closure is the L of such a quantum computer model and computes their braid-induced Seifert surfaces and the corresponding Alexander polynomial. In particular, some d-fold coverings of the trefoil knot, with d = 3 , 4, 6, or 12, define appropriate links L, and the latter two cases connect to the Dynkin diagrams of E 6 and D 4 , respectively. In this new context, one finds that this correspondence continues with Kodaira’s classification of elliptic singular fibers. The Seifert fibered toroidal manifold Σ ′ , at the boundary of the singular fiber E 8 ˜ , allows possible models of quantum computing.

On the Stability of Toroidal Manifold for One Class of Dynamical Systems

We study the problem of stability of an invariant toroidal manifold for one class of linear extensions of a dynamical system on a torus. The obtained result is used to investigate the problem of existence of invariant manifold for a nonlinear system of differential equations.

Existence of an Invariant Torus for a Degenerate Linear Extension of Dynamical Systems

Under the assumptions that a degenerate system defined on the direct product of a torus and a Euclidean space can be reduced to a central canonical form and that the corresponding homogeneous nondegenerate system is exponentially dichotomous on the semiaxes, we establish a necessary and sufficient condition for the existence of a unique invariant torus of the degenerate linear system. We also establish conditions for the preservation of an asymptotically stable invariant toroidal manifold for a degenerate linear extension of the dynamical system on a torus under small perturbations in the set of nonwandering points.

On preservation of an exponentially stable invariant torus

Abstract New conditions of the preservation of an exponentially stable invariant toroidal manifold of linear extension of one-dimensional dynamical system on torus under small perturbations in ω-limit set are established. This approach is applied to the investigation of the qualitative behaviour of solutions of linear extensions of dynamical systems with simple structure of limit sets.

On Preservation of the Invariant torus for Multifrequency Systems

We establish new conditions for the preservation of an asymptotically stable invariant toroidal manifold of the linear extension of a dynamical system on a torus under small perturbations in a set of nonwandering points. The proposed approach is applied to the investigation of the existence and stability of the invariant tori of linear extensions of the dynamical systems with simple structures of limit sets and recurrent trajectories.

Graph manifolds, left-orderability and amalgamation

We show that every irreducible toroidal integer homology sphere graph manifold has a left-orderable fundamental group. This is established by way of a specialization of a result due to Bludov and Glass for the almagamated products that arise, and in this setting work of Boyer, Rolfsen and Wiest may be applied. Our result then depends on input from 3-manifold topology and Heegaard Floer homology.

Exceptional surgeries on [formula omitted]-pretzel knots

We give a complete description of exceptional surgeries on pretzel knots of type ( − 2 , p , p ) with p ⩾ 5 . It is known that such a knot admits a unique toroidal surgery yielding a toroidal manifold with a unique incompressible torus. By cutting along the torus, we obtain two connected components, one of which is a twisted I-bundle over the Klein bottle. We show that the other is homeomorphic to the one obtained by certain Dehn filling on the magic manifold. On the other hand, we show that all such pretzel knots admit no Seifert fibered surgeries.

Exceptional Dehn surgery on large arborescent knots

A Dehn surgery on a knot K in S is exceptional if it produces a reducible, toroidal or Seifert fibred manifold. It is known that a large arborescent knot admits no such surgery unless it is a type II arborescent knot. The main theorem of this paper shows that up to isotopy there are exactly three large arborescent knots admitting exceptional surgery, each of which admits exactly one exceptional surgery, producing a toroidal manifold.

Non-invertible knots having toroidal Dehn surgery of hitting number four

We show that there exist infinitely many non-invertible, hyperbolic knots that admit toroidal Dehn surgery of hitting number four. The resulting toroidal manifold contains a unique incompressible torus meeting the core of the attached solid torus in four points, but no incompressible torus meeting it less than four points. For a hyperbolic knot in the 3-sphere S 3 , at most finitely many Dehn surgeries yield non-hyperbolic 3-manifolds by Thurston's hyperbolic Dehn surgery theorem. Such Dehn surgeries are called exceptional Dehn surgeries. A typical one is Dehn surgery creating an incompressible torus, called toroidal Dehn surgery. By Gordon-Luecke (7, 8), the surgery slope of toroidal Dehn surgery is integral or half-integral, and the latter happens only for Eudave- Munoz knots (3). Thus the study of integral toroidal Dehn surgery is the next challenging task. We now introduce the notion of hitting number for toroidal Dehn surgery. Let K be a hyperbolic knot in S 3 . Suppose that the resulting 3-manifold by r-Dehn surgery, denoted by KðrÞ, is toroidal. For an incompressible torus T contained in KðrÞ, let jKV Tj denote the number of points in KV T, where Kis the core of the attached solid torus of KðrÞ. For a pair ðK; rÞ, we call minfjKV Tj : T is an incompressible torus in KðrÞg the hitting number of ðK; rÞ. Since K is hyperbolic, a hitting number is positive. This is a natural measure of the complexity of toroidal Dehn surgery. We should mention that the only possible odd hitting number is one.

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.