Manifolds Of Points Research Articles

On topological classification of Morse–Smale diffeomorphisms on the sphere S n (n > 3)

We consider the class G(S n ) of orientation preserving Morse–Smale diffeomorphisms of the sphere S n of dimension n > 3, assuming that invariant manifolds of different saddle periodic points have no intersection. For any diffeomorphism f ∈ G(S n ), we define a coloured graph Γ f that describes a mutual arrangement of invariant manifolds of saddle periodic points of the diffeomorphism f. We enrich the graph Γ f by an automorphism P f induced by dynamics of f and define the isomorphism notion between two coloured graphs. The aim of the paper is to show that two diffeomorphisms f, f′ ∈ G(S n ) are topologically conjugated if and only if the graphs Γ f , Γ f ′ are isomorphic. Moreover, we establish the existence of a linear-time algorithm to distinguish coloured graphs of diffeomorphisms from the class G(S n ).

On a real caustic of type

We prove that the manifold of non-singular points of a stable real caustic germ of type and the manifolds of points of transversal intersection of its smooth branches consist only of contractible connected components. We also calculate the number of these components.

Research on the Lissajous Trajectory Transfer of Large Area Mass Ratio Solar Sail Spacecraft

Considering that the solar sail spacecraft working in the Lissajous orbit will be affected by the shadow of the earth or the solar wind, which sometime makes the spacecraft unable to work, we try to use the solar light pressure as the thrust to carry out the Lissajous orbit transfer in advance. Because the solar pressure parameters(cone angle, clock angle and solar pressure factor) will change the position of the libration points, we propose the idea of splicing the old and new libration point manifolds to transfer the escape spacecraft to the new Lissajous orbit to achieve orbit maintenance. The simulation results show that besides changing the cone angle and the solar pressure factor, the clock angle can also achieve orbit transfer, and the influence of the solar pressure factor on the periodic orbit is greater than the other two. This study provides a reference for the solar sail spacecraft in the aspects of periodic orbit transfer, orbit maintenance and avoidance of the earth's shadows.

A scenario of the homotopy type changing of the invariant saddle manifold closure

The paper deals with surface gradient-like diffeomorphisms. The closures of the invariant manifolds of saddle points of such systems contain nodal points in their closure. In the case when there is only one such point, the closure of the invariant manifold is a closed curve that is homeomorphic to the circle. In a general case the conjugating homeomorphism changes the homotopy type of the closed curve, while the diffeomorphisms themselves may remain in the same isotopic class. This means that in the space of diffeomorphisms two such systems are connected by an arc, but every such arc necessarily undergoes bifurcations. In this paper, we describe a scenario for changing the homotopy type of the closure of the invariant saddle manifold. Moreover, the constructed arc is stable in the space of diffeomorphisms.

On Realization of Topological Conjugacy Classes of Morse–Smale Cascades on the Sphere $$S^n$$

We consider the class $$G(S^n)$$ of orientation-preserving Morse–Smale diffeomorphisms defined on the sphere $$S^n$$ of dimension $$n\geq 4$$ under the assumption that the invariant manifolds of different saddle periodic points are disjoint. For diffeomorphisms in this class, we describe an algorithm for constructing representatives of all topological conjugacy classes.

Low-thrust trajectory design in low-energy regimes using variational equations

This paper proposes a novel description of the equations of motion for low-thrust trajectory design in the presence of a third-body perturbation. The framework is formulated using Gauss’ Variational Equations (GVE) with two distinct accelerations: the one produced by the electric engine and the disturbing term of the third-body effect, which is computed using the disturbing potential of the previously studied Keplerian Map. The presented GVE third-body (GVE-3B) framework allows for a simple and intuitive description of the low-thrust optimisation problem. It is accurate until very close to the sphere of influence of the perturbing body, and thus can be used to target trajectories in low-energy regimes. Together with the framework, this paper develops a methodology to generate low-energy first-guess solutions for low-thrust trajectories. Both the methodology and the framework are showcased in the design of two distinct missions: a rendezvous with asteroid 2017 SV19 during its next Earth encounter, after departing from the unstable invariant manifold of the L2 point in the Sun-Earth system, and the capture of asteroid 2018 AV2 to a stable invariant manifold of the same point.

Dynamic analysis of the nonlinear energy sink with local and global potentials: geometrically nonlinear damping

In this paper, the considered two-DOF system consists of a linear oscillator (LO) under external harmonic excitation and an attached lightweight nonlinear energy sink (NES) with local potential and geometrically nonlinear damping. With the application of complex-averaging method, the steady-state dynamical behavior of the system is investigated by the slow invariant manifold, folding singularities and equilibrium points. Different scenarios of strongly modulated responses are presented based on the geometry of SIM, and the numerical simulation results are in consistent with the analytical prediction. The incremental harmonic balance method is applied to detect the frequency response curves of the system around the fundamental resonance, and the accuracy of the theoretical analysis is fully verified by the numerical results obtained by direct integration of equations of motion of the system. It is demonstrated that the increase in external forcing amplitude, global nonlinear stiffness and local nonlinear stiffness can drive the frequency response curves move toward the right and widen the frequency bandwidth of the coexistence of multiple steady-state response regimes, while the increase in nonlinear damping the reverse. The numerical simulation results also show that the addition of geometrically nonlinear damping and local potential in the proposed NES can drastically enhance the capacity of the nonlinear vibration absorber to suppress the shock-induced response of the LO, and the proposed NES is effective for a comparatively broad range of applied impulsive energies, particularly for the high impulsive energies.

Stable and Completely Unstable Periodic Points of Diffeomorphism of a Plane with a Heteroclinic Contour

The diffeomorphism of a plane into itself with three hyperbolic points is studied this paper. It is assumed that the heteroclinic points lie at the intersections of the unstable manifold of the first point and the stable manifold of the second point, of the unstable manifold of the second point and the stable manifold of the third point, of the unstable manifold of the third point and the stable manifold of the first point. The orbits of fixed and heteroclinic points form a heteroclinic contour. The case when stable and unstable manifolds intersect non-transversally at heteroclinic points is investigated. The points of tangency of finite order are firstly distinguished among the points of non-transversal intersection of a stable manifold with an unstable manifold; in this paper, such points are not considered. Diffeomorphism with a heteroclinic contour was studied in the works of L.P. Shilnikov, S.V. Gonchenko, and other authors, and it was assumed that the points of non-transversal intersection of stable and unstable manifolds are points of tangency of finite order. It follows from the works of these authors that a diffeomorphism exists for which there are stable and completely unstable periodic points in the neighborhood of the heteroclinic contour. It is assumed in the current paper that the points of non-transversal intersection of stable and unstable manifolds are not the points of tangency of finite order. It is demonstrated that two countable sets of periodic points may lie in the neighborhood of such a heteroclinic contour. One of these sets consists of stable periodic points whose characteristic exponents are separated from zero, and another set consists of completely unstable periodic points whose characteristic exponents are also separated from zero.

Joshua Parker and Ralph J. Poole, eds. Austria and America: 20th-Century Cross-Cultural Encounters

Joshua Parker and Ralph J. Poole, eds. Austria and America: 20th-Century Cross-Cultural Encounters. Zurich: Lit Verlag, 2017 (American Studies in Austria, Band 15) Pp. 178. ISBN: 9783643908124. Walter W. Hoelbling Given the manifold points of interaction between Austria and America throughout the 20th century, the 11 essays from the contributions of the 2015 conference hosted by the Salzburg Stefan Zweig Center are necessarily eclectic. Yet thanks to judicious selection by the editors, they ...

Chazy-Type Asymptotics and Hyperbolic Scattering for the n-Body Problem

We study solutions of the Newtonian $n$-body problem which tend to infinity hyperbolically, that is, all mutual distances tend to infinity with nonzero speed as $t \rightarrow +\infty$ or as $t \rightarrow -\infty$. In suitable coordinates, such solutions form the stable or unstable manifolds of normally hyperbolic equilibrium points in a boundary manifold "at infinity". We show that the flow near these manifolds can be analytically linearized and use this to give a new proof of Chazy's classical asymptotic formulas. We also address the scattering problem, namely, for solutions which are hyperbolic in both forward and backward time, how are the limiting equilibrium points related? After proving some basic theorems about this scattering relation, we use perturbations of our manifold at infinity to study scattering "near infinity", that is, when the bodies stay far apart and interact only weakly.

Artificial Intelligence in radiotherapy: state of the art and future directions.

Recent advances in computing capability allowed the development of sophisticated predictive models to assess complex relationships within observational data, described as Artificial Intelligence. Medicine is one of the several fields of application and Radiation oncology could benefit from these approaches, particularly in patients' medical records, imaging, baseline pathology, planning or instrumental data. Artificial Intelligence systems could simplify many steps of the complex workflow of radiotherapy such as segmentation, planning or delivery. However, Artificial Intelligence could be considered as a "black box" in which human operator may only understand input and output predictions and its application to the clinical practice remains a challenge. The low transparency of the overall system is questionable from manifold points of view (ethical included). Given the complexity of this issue, we collected the basic definitions to help the clinician to understand current literature, and overviewed experiences regarding implementation of AI within radiotherapy clinical workflow, aiming to describe this field from the clinician perspective.

The reverse Yang–Mills–Higgs flow in a neighbourhood of a critical point

The main result of this paper is a construction of solutions to the reverse Yang–Mills–Higgs flow converging in the $C^{\infty}$ topology to a critical point. The construction uses only the complex gauge group action, which leads to an algebraic classification of the isomorphism classes of points in the unstable set of a critical point in terms of a filtration of the underlying Higgs bundle. Analysing the compatibility this filtration with the canonical Harder–Narasimhan–Seshadri double filtration gives an algebraic criterion for two critical points to be connected by a flow line. As an application, we use this to construct Hecke modifications of Higgs bundles via the Yang–Mills–Higgs flow. When the Higgs field is zero (corresponding to the Yang–Mills flow), this criterion has a geometric interpretation in terms of secant varieties of the projectivisation of the underlying bundle inside the unstable manifold of a critical point, which gives a precise description of broken and unbroken flow lines connecting two critical points. For non-zero Higgs field, at generic critical points the analogous interpretation involves the secant varieties of the spectral curve of the Higgs bundle.

Earth Science Theoretical Models for Quantitative Analysis of Global Spatial Data

Geographical space is considered as a dual system of subspace of geophysical coordinates and subspace of component coordinates of the geosystems states. The theoretical and mathematical models of the interaction of these subspaces in terms of vector fields, wave equations and functions, eikonal equations and local sheaves of linear dependencies of meteorological parameters are developed. The climatic zonal norms are reconstructed in the form of points of the global manifold of indicators links according to the positions of the sheaf centers. This model implements the principle of geometrization of geographical knowledge as the chorological approach to justifying the local regularities of the interaction of natural characteristics in the geographical environment. The logical and statistical interrelation of the geographical phenomena of planetary and local levels is studied. Calculations used the average daily values of the scalar fields of atmospheric absolute temperature and specific humidity on the isobaric at-700 hPa surface for different months of 2014 according to logarithmically converted NCEP/NCAR Reanalysis Data. Latitudinal environmental invariants of air temperature and humidity are identified, and a map reflecting the zonal and azonal heterogeneity of the geographical environment is presented. This determines the main theme of earth science as the identification of the projection of space and global cycles onto the local system-forming patterns, which is necessary for the development of the theory, models and methods of scientific analysis of the correlation of scalar fields of geographical characteristics and spatial processes that cause them.

Stable and completely unstable periodic points of diffeomorphism of a plane with a heteroclinic contour

«Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия» — научно-теоретический рецензируемый журнал, публикующий исследования в области математики, механики и астрономии, а также математического моделирования.

Control in the Spaces of Ensembles of Points

We study the controlled dynamics of the ensembles of points of a Riemannian manifold $M$. Parameterized ensemble of points of $M$ is the image of a continuous map $\gamma:\Theta \to M$, where $\The...

Structure of the centre manifold of the $$L_1,L_2$$ collinear libration points in the restricted three-body problem

We present a global analysis of the center manifold of the collinear points in the circular restricted three-body problem. The phase-space structure is provided by a family of resonant 2-DOF Hamiltonian normal forms. The near 1:1 commensurability leads to the construction of a detuned Birkhoff-Gustavson normal form. The bifurcation sequences of the main orbit families are investigated by a geometric theory based on the reduction of the symmetries of the normal form, invariant under spatial mirror symmetries and time reversion. This global picture applies to any values of the mass parameter.

The large-scale general-relativistic correction for Newtonian mocks

We clarify the subtle issue of finding the correct mapping of Newtonian simulations to light-cone observables at very large distance scales. A faithful general-relativistic interpretation specifies a gauge, i.e. a chart that relates the simulation data to points of the space-time manifold. It has already been pointed out that the implicit gauge choice of Newtonian simulations is indeed different from the Poisson gauge that is commonly adopted for relativistic calculations, the difference being most significant at large scales. It is therefore inconsistent, for example, to predict weak-lensing observables from simulations unless this gauge issue is properly accounted for. Using perturbation theory as well as fully relativistic N-body simulations we quantify the systematic error introduced this way, and we discuss several solutions that would render the calculations relativistically self-consistent.

Low-Energy Sub-Optimal Low-Thrust Trajectories to Libration Points and Halo-Orbits

The paper considers the problem of calculating direct and low-energy, low-thrust trajectories to the libration points of the Earth–Moon system and to halo-orbits. A method for solving the problem is proposed. It consists of calculating stable manifolds of libration points or halo-orbits and calculating a low-thrust trajectory from an initial circular Earth orbit to the given point of this manifold using sub-optimal feedback control. With a fixed final mass of a spacecraft, the last stage of calculations is reduced to solving the Cauchy problem. The numerical examples are given for the calculation of direct and low-energy trajectories to libration points and to halo-orbits and the optimization of entry points to stable manifolds for low-energy trajectories.

Resilience Analysis for Competing Populations.

Ecological resilience refers to the ability of a system to retain its state when subject to state variables perturbations or parameter changes. While understanding and quantifying resilience is crucial to anticipate the possible regime shifts, characterizing the influence of the system parameters on resilience is the first step toward controlling the system to avoid undesirable critical transitions. In this paper, we apply tools of qualitative theory of differential equations to study the resilience of competing populations as modeled by the classical Lotka-Volterra system. Within the high interspecific competition regime, such model exhibits bistability, and the boundary between the basins of attraction corresponding to exclusive survival of each population is the stable manifold of a saddle point. Studying such manifold and its behavior in terms of the model parameters, we characterized the populations resilience: While increasing competitiveness leads to higher resilience, it is not always the case with respect to reproduction. Within a pioneering context where both populations initiate with few individuals, increasing reproduction of one population leads to an increase in its resilience; however, within an environment previously dominated by one population and then invaded by the other, an increase in the resilience of a population is obtained by decreasing its reproduction rate. Besides providing interesting insights for the dynamics of competing populations, this work brings near to each other the concepts of ecological resilience and the methods of differential equations and stimulates the development and application of new tools for ecological resilience.

Brigantium no século XXI: Aproximación aos últimos datos arqueolóxicos da ocupación galaicorromana na cidade de A Coruña (s. I-IV d.c.)

O seguinte artigo realiza unha revisión dos datos arqueolóxicos obtidos nos últimos 20 anos de arqueoloxía na cidade da Coruña, centrándonos no espazo que ocupou o vello asentamento de Brigantium. Se ben é certo que a ocupación romana na cidade vella coruñesa despertou o interese de múltiples investigadores durante as últimas décadas do pasado século (NAVEIRO LÓPEZ 1988; BELLO DIÉGUEZ 1991; BELLO DIÉGUEZ e VÁZQUEZ GÓMEZ 1994; PÉREZ LOSADA 2002; GONZÁLEZ GARCÍA 2003, LÓPEZ PÉREZ 2004; entre outros), o certo é que neste novo milenio o interese decaeu, en paradóxica relación co aumento exponencial da información arqueolóxica dispoñible.Nesta liña, os obxectivos principais son dous: realizar unha breve síntese que nos permita divulgar os datos arqueolóxicos máis relevantes obtidos nestas últimas dúas décadas, e actualizar o estado da cuestión sobre Brigantium; elaborando novas propostas que interpreten esta información, estudando o seu entramado “urbano” e elaborando hipóteses sobre os motivos que influíron na evolución do mesmo.