Manifolds Of Points Research Articles

Partial Preservation of Frequencies and Floquet Exponents of Invariant Tori in the Reversible KAM Context 2

We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman’s method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus).

Deep neural networks for texture classification—A theoretical analysis

We investigate the use of Deep Neural Networks for the classification of image datasets where texture features are important for generating class-conditional discriminative representations. To this end, we first derive the size of the feature space for some standard textural features extracted from the input dataset and then use the theory of Vapnik–Chervonenkis dimension to show that hand-crafted feature extraction creates low-dimensional representations which help in reducing the overall excess error rate. As a corollary to this analysis, we derive for the first time upper bounds on the VC dimension of Convolutional Neural Network as well as Dropout and Dropconnect networks and the relation between excess error rate of Dropout and Dropconnect networks. The concept of intrinsic dimension is used to validate the intuition that texture-based datasets are inherently higher dimensional as compared to handwritten digits or other object recognition datasets and hence more difficult to be shattered by neural networks. We then derive the mean distance from the centroid to the nearest and farthest sampling points in an n-dimensional manifold and show that the Relative Contrast of the sample data vanishes as dimensionality of the underlying vector space tends to infinity.

The Health Network Tennengau – The first population oriented integrated chronic care programme for persons with multiple chronic conditions in Austria

We report of our work on a “thick description” of a population oriented integrated care programme in Austria, the Health Network Tennengau (HNT). The HNT is a bottom-up network comprised of social and health service providers and voluntary organisations in the Tennengau region, which is a rural area in the state of Salzburg. The HNT has its origin in a pilot project for medical home nursing care introduced in 1995 and has since gradually evolved into a comprehensive network, targeting all inhabitants of the Tennengau region. However, the activities are especially geared towards senior citizens with multiple medical and social requirements. Document analysis and interviews conducted during the Horizon 2020 project SELFIE show that the HTN puts the patient at the centre of the care process and aims to facilitate integrated care around the patient, namely to provide information, counselling, coordination and interface management through case and care management and improved communication. The interconnectedness with non-medical service providers is a central concern for all stakeholder of the HNT. Thus, the HNT is at its core not a medical programme, but a network between doctors, hospitals, care facilities and diverse social institutions in the region. The HNT has been successful for more than 20 years due to voluntary work and a culture of respect between all stakeholders. However, the sustainability has always been at risk as despite its success, continuous financing has been hard to secure, resulting in limited compensation for the involved partners, a lack of resources for public relations work, as well as a lack of funding especially at the start of pilot projects. As a consequence, the HNT still relies to a high degree on voluntary work. However, this also constitutes a weakness of the HNT, as it is not easily transferable. In other regions, not only the culture, but also voluntary stakeholders are lacking to build up integrated care without proper payment. Thus, the HNT offers manifold points for development that can only be spread by securing a sustainable financing. However, the fact that the HNT has persevered despite a lot of such adverse factors makes it an interesting case to learn from.

Semi-supervised local multi-manifold Isomap by linear embedding for feature extraction

In this paper, we mainly propose a semi-supervised local multi-manifold Isomap learning framework by linear embedding, termed SSMM-Isomap, that can apply the labeled and unlabeled training samples to perform the joint learning of neighborhood preserving local nonlinear manifold features and a linear feature extractor. The formulation of SSMM-Isomap aims at minimizing pairwise distances of intra-class points in the same manifold and maximizing the distances over different manifolds. To enhance the performance of nonlinear manifold feature learning, we also incorporate the neighborhood reconstruction error to preserve local topology structures between both labeled and unlabeled samples. To enable our SSMM-Isomap to extract local manifold features from the outside new data, we also add a feature approximation error that correlates manifold features with embedded features by the jointly learnt feature extractor. Thus, the learnt linear extractor can extract the local manifold features from the new data efficiently by direct embedding. To optimize the proposed objective function, two effective schemes are presented, i.e., Scaling by MAjorizing a Complicated Function and Eigen-decomposition. Notice that the comparison of the proposed two solvers is also described. We mainly evaluate SSMM-Isomap for manifold feature learning, data clustering and classification. Extensive simulation results verify the effectiveness of our SSMM-Isomap algorithm, compared with other related feature learning techniques.

A study on validating non-linear dimensionality reduction using persistent homology

During the process of non-linear dimensionality reduction, manifolds represented by point clouds are at risk of changing their topology. We review techniques for quality assessment of manifold learning and propose to use persistent homology to evaluate the topological impact of manifold learning by comparing the Betti numbers of test manifolds before and after dimensionality reduction. We propose a benchmark suite of test manifolds based on the Swiss roll dataset with added geometrical and topological complexity. The experiments demonstrate the effectivity of the approach by analysing examples of test manifolds where the embedding failed. Betti numbers based on persistent homology are also used to select suitable sampling rates for the manifold point clouds and to determine optimal values for the nearest neighbour parameter k of selected manifold learning methods. The results indicate that the more complex the manifold is the more sample points and larger values for k are required.

On the topology of the Lorenz system.

We present a new paradigm for three-dimensional chaos, and specifically for the Lorenz equations. The main difficulty in these equations and for a generic flow in dimension 3 is the existence of singularities. We show how to use knot theory as a way to remove the singularities. Specifically, we claim: (i)for certain parameters, the Lorenz system has an invariant one-dimensional curve, which is a trefoil knot. The knot is a union of invariant manifolds of the singular points. (ii)The flow is topologically equivalent to an Anosov flow on the complement of this curve, and moreover to a geodesic flow. (iii)When varying the parameters, the system exhibits topological phase transitions, i.e. for special parameter values, it will be topologically equivalent to an Anosov flow on a knot complement. Different knots appear for different parameter values and each knot controls the dynamics at nearby parameters.

Saddle-node equilibrium points on the stability boundary of nonlinear autonomous dynamical systems

ABSTRACTA complete characterization of the stability boundary of an asymptotically stable equilibrium point in the presence of type-k saddle-node non-hyperbolic equilibrium points, with k ≥ 0, on the stability boundary is developed in this paper. Under the transversality condition, it is shown that the stability boundary is composed of the stable manifolds of the hyperbolic equilibrium points on the stability boundary, the stable manifolds of type-0 saddle-node equilibrium points on the stability boundary and the stable centre and centre manifolds of the type-r saddle-node equilibrium points with r ≥ 1 on the stability boundary. This characterization is the first step to understanding the behaviour of stability regions and stability boundaries in the occurrence of saddle-node bifurcations on the stability boundary.

Killing symmetry on the Finsler manifold

Symmetry and conservation law are discussed on the Finsler manifold M. We adopt the point Finsler approach, where we consider the geometry on a point manifold M not on TM. Generalized vector fields are defined on oriented curves on M, and Finsler non-linear connections are considered on M, not on the tangent space TM. Killing vector fields K are defined as generalized vector fields as , and the Killing symmetry is also reformulated simply as by using the Killing 1-form and the spray operator defined by using the non-linear connection. is related to the generalization of Killing tensors on the Finsler manifold, and our ansatz of and give an analytical method of finding higher derivative conserved quantities, which may be called hidden conserved quantities. We show two examples: the Carter constant on Kerr spacetime and the Runge–Lentz vectors in Newtonian gravity.

A directed batch growing approach to enhance the topology preservation of self-organizing map

The growing self-organizing map (GSOM) possesses effective capability to generate feature maps and visualizing high-dimensional data without pre-determining their size. Most of the proposed growing SOM algorithms use an incremental learning strategy. The conventional growing approach of GSOM is based on filling all available position around the candidate neuron which can decrease the topology preservation quality of the map due to the misconfiguration and twisting of the map which could be a consequence of unexpected network growth and improper neuron addition and weight initialization. To overcome this problem, in this paper we introduce a batch learning strategy for growing self-organizing maps called DBGSOM which direct the growing process based on the accumulative error around the candidate boundary neuron. In the proposed growing approach, just one new neuron is added around each candidate boundary neuron. The DBGSOM offers suitable mechanisms to find a proper growing positions and allocating initial weight vectors for the new neurons.The potential of the DBGSOM was investigated with one synthetic dataset and six real-world benchmark datasets in terms of topology preservation and mapping quality. Experimental results showed that the proposed growing strategy provides an enhanced topology preserved map and reduces the susceptibility of twisting compared to GSOM. Furthermore, the proposed method has a better clustering ability than GSOM and SOM. According to the lower number of neurons generated by DBGSOM, it needs less time to learn the manifold of the data points compared to GSOM.

A Theoretical Framework for Lagrangian Descriptors

This paper provides a theoretical background for Lagrangian Descriptors (LDs). The goal of achieving rigorous proofs that justify the ability of LDs to detect invariant manifolds is simplified by introducing an alternative definition for LDs. The definition is stated for [Formula: see text]-dimensional systems with general time dependence, however we rigorously prove that this method reveals the stable and unstable manifolds of hyperbolic points in four particular 2D cases: a hyperbolic saddle point for linear autonomous systems, a hyperbolic saddle point for nonlinear autonomous systems, a hyperbolic saddle point for linear nonautonomous systems and a hyperbolic saddle point for nonlinear nonautonomous systems. We also discuss further rigorous results which show the ability of LDs to highlight additional invariants sets, such as [Formula: see text]-tori. These results are just a simple extension of the ergodic partition theory which we illustrate by applying this methodology to well-known examples, such as the planar field of the harmonic oscillator and the 3D ABC flow. Finally, we provide a thorough discussion on the requirement of the objectivity (frame-invariance) property for tools designed to reveal phase space structures and their implications for Lagrangian descriptors.

Exact Solutions in the Invariant Manifolds of the Generalized Integrable Hénon–Heiles System and Exact Traveling Wave Solutions of Klein–Gordon–Schrödinger Equations

In this paper, we consider the exact explicit solutions for the famous generalized Hénon–Heiles (H–H) system. Corresponding to the three integrable cases, on the basis of the investigation of the dynamical behavior and level curves of the planar dynamical systems, we find all possible explicit exact parametric representations of solutions in the invariant manifolds of equilibrium points in the four-dimensional phase space. These solutions contain quasi-periodic solutions, homoclinic solutions, periodic solutions as well as blow-up solutions. Therefore, we answer the question: what are the flows in the center manifolds and homoclinic manifolds of the generalized Hénon–Heiles (H–H) system. As an application of the above results, we consider the traveling wave solutions for the coupled [Formula: see text]-dimensional Klein–Gordon–Schrödinger Equations with quadratic power nonlinearity.

Flow bench testing of prototype intake manifolds for ultralight aircraft engine

The article describes the research on the intake manifold for ultralight aircraft engine Vaxell 100i. It presents the actual and new redesigned manifold construction and points out the design requirements. The results of previously made numerical simulation of air flow inside the manifold are discussed. Computer analysis confirmed the appropriateness of internal guide vanes usage to improve the uniform air distribution between cylinders. For verification, a flow bench test stand was made for multicylinder intake manifold testing. A prototype manifold was built with the possibility of guide vane adjustment. The best variant had almost 5 times better uniformity of air distribution comparing to variant without the guide vane. Flow bench results confirmed the conclusions from numerical simulations.

Herman's Approach to Quasi-Periodic Perturbations in the Reversible KAM Context 2

We revisit non-autonomous systems depending quasi-periodically in time within the reversible context 2 of KAM theory and obtain Whitney smooth families of invariant tori in such systems via Herman's method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question.

The distinct stochastic and deterministic dynamics between period-adding and period-doubling bifurcations of neural bursting patterns

Period-adding bifurcations and period-doubling bifurcations of neural firing patterns, which were both observed in the biological experiment on a neural pacemaker and simulated in a theoretical model (Chay model), manifested different stochastic dynamics near the bifurcation points. For period-adding bifurcations, a noise-induced stochastic bursting, whose behavior is stochastic transition between period-k and period-( $$k+1$$ ) bursts, lying between period-k and period-( $$k+1$$ ) burstings ( $$k=1,2,3$$ ). For period-doubling bifurcations, period-1 bursting is changed to period-2 bursting firstly and then to period-4 bursting. No stochastic firing patterns similar to those lying in the period-adding bifurcation were detected. Using the method of the fast–slow variables dissection, the deterministic burstings in both period-adding bifurcation and period-doubling bifurcation are classified into “fold/homoclinic” bursting with a saddle-node point and a saddle-homoclinic point, which behaves as a critical phase sensitive to noisy disturbance. For the bursting pattern near the period-doubling bifurcation point, the trajectory of burstings is far from the saddle-homoclinic point. Near the period-adding bifurcation points from period-k to period-( $$k+1$$ ) burstings, the trajectories of the bursting patterns pass through the neighborhood of the saddle-homoclinic point and exhibit a platform. The platform appears after the k-th spike for the period-k bursting and between the k-th spike and the ( $$k+1$$ )-th spike for the period-( $$k+1$$ ) bursting. For some bursts of period-k bursting, noise can induce a novel spike near the platform to form a burst with $$k+1$$ spikes, and for some bursts of period-( $$k+1$$ ) bursting, the last spike can be terminated by noise to form a burst with k spikes. It is the cause that the stochastic bursting whose behavior is stochastic transition between period-k burst and period-( $$k+1$$ ) burst is induced by noise, and the stochastic transition happens within or near the platform, i.e., the neighborhood of the saddle-homoclinic point. More detailed transition dynamics can be explained by the stable/unstable manifold of the saddle point. The underlying deterministic dynamics between period-adding and period-doubling bifurcation points that can lead to distinct stochastic dynamics are identified, which are helpful for understanding the roles of noise and provide critical phase to apply control strategy to modulate the firing patterns.

Whitney smooth families of invariant tori within the reversible context 2 of KAM theory

We prove a general theorem on the persistence of Whitney infinitely smooth families of invariant tori in the reversible context 2 of KAM theory. This context refers to the situation where dim Fix G < (codim T)/2 where Fix G is the fixed point manifold of the reversing involution G and T is the invariant torus in question. Our result is obtained as a corollary of the theorem by H.W.Broer, M.-C.Ciocci, H.Hanssmann, and A.Vanderbauwhede of 2009 concerning quasi-periodic stability of invariant tori with singular "normal" matrices in reversible systems.

Holonomicity of relative characters and applications to multiplicity bounds for spherical pairs

We prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic geometry and thus implies also analogous results in the p-adic case. As an application, we give a short proof of some results of Kobayashi-Oshima and Kroetz-Schlichtkrull on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. To deduce this application we prove the relative and quantitative analogs of the Bernstein–Kashiwara theorem, which states that the space of solutions of a holonomic system of differential equations in the space of distributions is finite-dimensional. We also deduce that, for every algebraic group \({{G}}\) defined over \(\mathbb {R}\), the space of \({{G(\mathbb {R})}}\)-equivariant distributions on the manifold of real points of any algebraic \({{G}}\)-manifold \({{X}}\) is finite-dimensional if \({{G}}\) has finitely many orbits on \({{X}}\).

Analysis of medium-energy transfers to the Moon

This study analyzes a recently discovered class of exterior transfers to the Moon. These transfers terminate in retrograde ballistic capture orbits, i.e., orbits with negative Keplerian energy and angular momentum with respect to the Moon. Yet, their Jacobi constant is relatively low, for which no forbidden regions exist, and the trajectories do not appear to mimic the dynamics of the invariant manifolds of the Lagrange points. This paper shows that these orbits shadow instead lunar collision orbits. We investigate the dynamics of singular, lunar collision orbits in the Earth–Moon planar circular restricted three-body problem, and reveal their rich phase space structure in the medium-energy regime, where invariant manifolds of the Lagrange point orbits break up. We show that lunar retrograde ballistic capture trajectories lie inside the tube structure of collision orbits. We also develop a method to compute medium-energy transfers by patching together orbits inside the collision tube and those whose apogees are located in the appropriate quadrant in the Sun–Earth system. The method yields the novel family of transfers as well as those ending in direct capture orbits, under particular energetic and geometrical conditions.

Subcritical Hopf Equilibrium Points in Boundary of the Stability Region

A complete characterization of the boundary of the stability region of a class of nonlinear autonomous dynamical systems is developed admitting the existence of Subcritical Hopf nonhyperbolic equilibrium points on the boundary of the stability region. The characterization of the stability region developed in this paper is an extension of the characterization already developed in the literature, which considers only hyperbolic equilibrium point. Under the transversality condition, it is shown the boundary of the stability region is comprised of the stable manifolds of all equilibrium points on the boundary of the stability region, including the stable manifolds of the subcritical Hopf equilibrium points of type k, with 0&lt;=k&lt;=(n-2), which belong to the boundary of the stability region.

Simple indicators for Lorentzian causets

Several classes of directed acyclic graphs have been investigated in the last two decades, in the context of the causal set program, in search for good discrete models of spacetime. We introduce some statistical indicators that can be used for comparing these graphs and for assessing their closeness to the ideal Lorentzian causal sets (‘causets’)—those obtained by sprinkling points in a Lorentzian manifold. In particular, with the reversed triangular inequality of Special Relativity in mind, we introduce ‘longest/shortest path plots’, an easily implemented tool to visually detect the extent to which a generic causet matches the wide range of path lengths between events of Lorentzian causets. This tool can attribute some degree of ‘Lorentzianity’—in particular ‘non-locality’—also to causets that are not (directly) embeddable and that, due to some regularity in their structure, would not pass the key test for Lorentz invariance: the absence of preferred reference frames. We compare the discussed indicators and use them for assessing causets both of stochastic and of deterministic, algorithmic origin, finding examples of the latter that behave optimally w.r.t. our longest/shortest path plots.

The dynamics of a fold-and-twist map

Inspired by a discrete-time predator–prey model we introduce a planar, noninvertible map composed of a rigid rotation over an angle φ and a quadratic map depending on a parameter a. We study the dynamics of this map with a particular emphasis on the transitions from orderly to chaotic dynamics. For a=3 a stable fixed point bifurcates through a Hopf–Neĭmark–Sacker bifurcation which gives rise to the alternation of periodic and quasi-periodic dynamics organized by Arnold tongues in the (φ,a)-plane. Inside a tongue a periodic attractor typically either undergoes a period doubling cascade, which leads to chaotic dynamics, or a Hopf–Neĭmark–Sacker bifurcation, which leads in turn to a new family of Arnold tongues. Numerical evidence suggests the existence of strange attractors with both one and two positive Lyapunov exponents. The former attractors are conjectured to be Hénon-like, i.e., they are formed by the closure of the unstable manifold of a periodic point of saddle type. The folded nature of such attractors is the novel feature of this paper.