Periodic Vector Fields Research Articles

Enumeration of Spin-Space Groups: Toward a Complete Description of Symmetries of Magnetic Orders

Symmetries of three-dimensional periodic scalar fields are described by 230 space groups (SGs). Symmetries of three-dimensional periodic (pseudo)vector fields, however, are described by the spin-space groups (SSGs), which were initially used to describe the symmetries of magnetic orders. In SSGs, the real-space and spin degrees of freedom are unlocked in the sense that an operation could have different spatial and spin rotations. SSGs give a complete symmetry description of magnetic structures and have natural applications in the band theory of itinerary electrons in magnetically ordered systems with weak spin-orbit coupling. , a concept raised recently that belongs to the symmetry-compensated collinear magnetic orders but has nonrelativistic spin plitting, is well described by SSGs. Because of the vast number and complicated group structures, SSGs have not yet been systematically enumerated. In this work, we exhaust SSGs based on the invariant subgroups of SGs, with spin operations constructed from three-dimensional (3D) real representations of the quotient groups for the invariant subgroups. For collinear and coplanar magnetic orders, the spin operations can be reduced into lower-dimensional real representations. As the number of SSGs is infinite, we consider only SSGs that describe magnetic unit cells up to 12 times crystal unit cells. We obtain 157 289 noncoplanar, 24 788 coplanar-noncollinear, and 1421 collinear SSGs. The enumerated SSGs are stored in an online database with a user-friendly interface. We develop an algorithm to identify SSGs for realistic materials and find SSGs for 1626 magnetic materials. We also discuss several potential applications of SSGs, including the representation theory, topological states protected by SSGs, structures of spin textures, and refinement of magnetic neutron diffraction patterns using SSGs. Our results serve as a solid starting point for further studies of symmetry and topology in magnetically ordered materials. Published by the American Physical Society 2024

Nonautonomous vector fields on $$S^3$$: Simple dynamics and wild embedding of separatrices

We construct new substantive examples of non-autonomous vector fields on 3-dimensional sphere having a simple dynamics but non-trivial topology. The construction is based on two ideas: the theory of diffeomorpisms with wild separatrix embedding (Pixton, Bonatti-Grines, etc.) and the construction of a non-autonomous suspension over a diffeomorpism (Lerman-Vainshtein). As a result, we get periodic, almost periodic or even nonrecurrent vector fields which have a finite number of special integral curves possessing exponential dichotomy on $\R$ such that among them there is one saddle integral curve (with an exponential dichotomy of the type (3,2)) having wildly embedded two-dimensional unstable separatrix and wildly embedded three-dimensional stable manifold. All other integral curves tend, as $t\to \pm \infty,$ to these special integral curves. Also we construct another vector fields having $k\ge 2$ special saddle integral curves with tamely embedded two-dimensional unstable separatrices forming mildly wild frames in the sense of Debrunner-Fox. In the case of periodic vector fields, corresponding specific integral curves are periodic with the period of the vector field, and they are almost periodic for the case of almost periodic vector fields.

Formula omitted]-contraction: Theory and applications

A dynamical system is called contractive if any two solutions approach one another at an exponential rate. More precisely, the dynamics contracts lines at an exponential rate. This property implies highly ordered asymptotic behavior including entrainment to time-varying periodic vector fields and, in particular, global asymptotic stability for time-invariant vector fields. Contraction theory has found numerous applications in systems and control theory because there exist easy to verify sufficient conditions, based on matrix measures, guaranteeing contraction. We provide a geometric generalization of contraction theory called k-contraction. A dynamical system is called k-contractive if the dynamics contracts k-parallelotopes at an exponential rate. For k=1 this reduces to standard contraction. We describe easy to verify sufficient conditions for k-contraction based on a matrix measure of the kth additive compound of the Jacobian of the vector field. We also describe applications of the seminal work of Muldowney and Li, that can be interpreted in the framework of 2-contraction, to systems and control theory.

A picture of the ODE's flow in the torus: From everywhere or almost-everywhere asymptotics to homogenization of transport equations

In this paper, we study various aspects of the ODE's flow induced by a periodic vector field b in the torus. We present a picture of all logical connections between: the everywhere or the a.e. asymptotics of the flow, the rectification of b, the ergodicity of the flow, the unit set condition for Herman's rotation set, the unit set condition for the set Db composed of the means of b related to the invariant measures being absolutely continuous with respect to Lebesgue's measure, the homogenization of the linear transport equation with oscillating velocity b(x/ε). The main result of the paper is that the a.e. asymptotics of the flow, the unit set condition for Db and the homogenization of the transport equation with divergence free b, are equivalent. Extending the two-dimensional results on Stepanoff flow to any dimension, we show that the flow may be ergodic without satisfying the everywhere asymptotics.

Invariant manifolds of parabolic fixed points (II). Approximations by sums of homogeneous functions

We study the computation of local approximations of invariant manifolds of parabolic fixed points and parabolic periodic orbits of periodic vector fields. If the dimension of these manifolds is two or greater, in general, it is not possible to obtain polynomial approximations. Here we develop an algorithm to obtain them as sums of homogeneous functions by solving suitable cohomological equations. We deal with both the differentiable and analytic cases. We also study the dependence on parameters. In the companion paper [4] these approximations are used to obtain the existence of true invariant manifolds close by. Examples are provided.

Invariant manifolds of parabolic fixed points (I). Existence and dependence on parameters

In this paper we study the existence and regularity of stable manifolds associated to fixed points of parabolic type in the differentiable and analytic cases, using the parametrization method.The parametrization method relies on a suitable approximate solution of a functional equation. In the case of parabolic points, if the manifolds have dimension two or higher, in general this approximation cannot be obtained in the ring of polynomials but as a sum of homogeneous functions and it is given in [4]. Assuming a sufficiently good approximation is found, here we provide an “a posteriori” result which gives a true invariant manifold close to the approximate one. In the differentiable case, in some cases, there is a loss of regularity.We also consider the case of parabolic periodic orbits of periodic vector fields and the dependence of the manifolds on parameters. Examples are provided.We apply our method to prove that in several situations, namely, related to the parabolic infinity in the elliptic spatial three body problem, these invariant manifolds exist and do have polynomial approximations.

Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions

The problem of determining a periodic Lipschitz vector field $b=(b_1, \dots, b_d)$ from an observed trajectory of the solution $(X_t: 0 \le t \le T)$ of the multi-dimensional stochastic differential equation \begin{equation*} dX_t = b(X_t)dt + dW_t, \quad t \geq 0, \end{equation*} where $W_t$ is a standard $d$-dimensional Brownian motion, is considered. Convergence rates of a penalised least squares estimator, which equals the maximum a posteriori (MAP) estimate corresponding to a high-dimensional Gaussian product prior, are derived. These results are deduced from corresponding contraction rates for the associated posterior distributions. The rates obtained are optimal up to log-factors in $L^2$-loss in any dimension, and also for supremum norm loss when $d \le 4$. Further, when $d \le 3$, nonparametric Bernstein-von Mises theorems are proved for the posterior distributions of $b$. From this we deduce functional central limit theorems for the implied estimators of the invariant measure $\mu_b$. The limiting Gaussian process distributions have a covariance structure that is asymptotically optimal from an information-theoretic point of view.

Isotropic Realizability of Fields and Reconstruction of Invariant Measures under Positivity Properties. Asymptotics of the Flow by a Non-Ergodic Approach

The paper is devoted to the isotropic realizability of a regular gradient field u or a more general vector field b, namely the existence of a continuous positive function σ such that σb is divergence free in R d or in an open set of R d. First, we prove that under some suitable positivity condition satisfied by u, the isotropic realizability of u holds either in R d if u does not vanish, or in the open sets {c j < u < c j+1 } if the c j are the critical values of u (including inf R d u and sup R d u) which are assumed to be in finite number. It turns out that this positivity condition is not sufficient to ensure the existence of a continuous positive invariant measure σ on the torus when u is periodic. Then, we establish a new criterium of the existence of an invariant measure for the flow associated with a regular periodic vector field b, which is based on the equality b · v = 1 in R d. We show that this gradient invertibility is not related to the classical ergodic assumption, but it actually appears as an alternative to get the asymptotics of the flow.

Quadrangulation through morse-parameterization hybridization

We introduce an approach to quadrilateral meshing of arbitrary triangulated surfaces that combines the theoretical guarantees of Morse-based approaches with the practical advantages of parameterization methods. We first construct, through an eigensolver followed by a few Gauss-Newton iterations, a periodic four-dimensional vector field that aligns with a user-provided frame field and/or a set of features over the input mesh. A field-aligned parameterization is then greedily computed along a spanning tree based on the Dirichlet energy of the optimal periodic vector field, from which quad elements are efficiently extracted over most of the surface. The few regions not yet covered by elements are then upsampled and the first component of the periodic vector field is used as a Morse function to extract the remaining quadrangles. This hybrid parameterization- and Morse-based quad meshing method is not only fast (the parameterization is greedily constructed, and the Morse function only needs to be upsampled in the few uncovered patches), but is guaranteed to provide a feature-aligned quad mesh with non-degenerate cells that closely matches the input frame field over an arbitrary surface. We show that our approach is much faster than Morse-based techniques since it does not require a densely tessellated input mesh, and is significantly more robust than parameterization-based techniques on models with complex features.

Navier-Stokes Three Dimensional Equations Solutions Volume Three

Solution of Navier-Stokes equation is found by introducing new method for solving differential equations. This new method is writing periodic scalar function in any dimensions and any dimensional vector fields as the sum of sine and cosine series with proper coefficients. The method is extension of Fourier series representation for one variable function to multi-variable functions and vector fields.Before solving Navier-Stokes equations we introduce a new technique for writing periodic scalar functions or vector fields as the sum of cosine and sine series with proper coefficients. Fourier series representation is background for our new technique.Periodic nature of initial velocity for Navier-Stokes problem helps us write the vector field in the form of cosine and sine series sum which simplify the problem.

On a conjecture of De Giorgi related to homogenization

For a periodic vector field F, let X^varepsilon solve the dynamical system dXεdt=FXεε.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{{\\hbox {d}}{\\hbox {X}}^{\\varepsilon }}{{\\hbox {d}}t} = {{F}}\\left( \\frac{{X}^{\\varepsilon }}{\\varepsilon }\\right) . \\end{aligned}$$\\end{document}In (Set Valued Anal 2(1–2):175–182, 1994) Ennio De Giorgi enquiers whether from the existence of the limit X^0(t):=lim nolimits _{varepsilon rightarrow 0} X^varepsilon (t) one can conclude that frac{{hbox {d}} X^0}{{hbox {d}}t}= {hbox {constant}}. Our main result settles this conjecture under fairly general assumptions on F, which in some cases may also depend on t-variable. Once the above problem is solved, one can apply the result to the corresponding transport equation, in a standard way. This is also touched upon in the text to follow.

On the inclusion of a map into a flow

We consider the problem of the inclusion of a diffeomorphism into a flow generated by an autonomous or time periodic vector field. We discuss various aspects of the problem, present a series of results (both known and new ones) and point out some unsolved problems.

Bifurcation of periodic solution in a Predator–Prey type of systems with non-monotonic response function and periodic perturbation

A Predator–Prey type of dynamical systems with non-monotonic response function and time-periodic perturbation is considered in this paper. We present a proof for the number of equilibria in the unperturbed system at some parts of the parameter space. The perturbed system is a dynamical system defined by a periodic vector field. We present an alternative proof for a classical result on the period of the periodic solution. By using a numerical continuation method AUTO (Doedel et al., 1986 [9]), we present a bifurcation analysis for periodic solution of the perturbed system where we found fold, cusp and Swallowtail bifurcations.

Averaging of incompressible flows on two-dimensional surfaces

We consider a process on a compact two-dimensional surface which consists of the fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise. It gives rise to a process on the graph naturally associated to the structure of the stream lines of the unperturbed flow. It has been shown by Freidlin and Wentzell that if the unperturbed motion is periodic for almost all the initial points, then the corresponding process on the graph weakly converges to a Markov process. We consider the situation when the unperturbed motion is not periodic and the flow has ergodic components of positive measure. We show that the process on the graph still converges to a Markov process, which spends a positive proportion of time in the vertices corresponding to the ergodic components of the flow. As shown in a companion paper (Deterministic and stochastic perturbations of area preserving flows on a two-dimensional torus, by D. Dolgopyat, M. Freidlin, and L. Koralov), these results allow one to describe the viscosity regularization of dissipative deterministic perturbations of incompressible flows. In the case of surfaces of higher genus, the limiting process may exhibit intermittent behavior in the sense that the time axis can be divided into the intervals of random lengths where the particle stays inside one ergodic component separated by the intervals where the particle moves deterministically between the components.

A set oriented definition of finite-time Lyapunov exponents and coherent sets

The problem of phase space transport, which is of interest from both the theoretical and practical point of view, has been investigated extensively using geometric and probabilistic methods. Two important tools to study this problem that have emerged in recent years are finite-time Lyapunov exponents (FTLE) and the Perron–Frobenius operator. The FTLE measures the averaged local stretching around reference trajectories. Regions with high stretching are used to identify phase space transport barriers. One probabilistic method is to consider the spectrum of the Perron–Frobenius operator of the flow to identify almost-invariant densities. These almost-invariant densities are used to identify almost invariant sets. In this paper, a set-oriented definition of the FTLE is proposed which is applicable to phase space sets of finite size and reduces to the usual definition of FTLE in the limit of infinitesimal phase space elements. This definition offers a straightforward connection between the evolution of probability densities and finite-time stretching experienced by phase space curves. This definition also addresses some concerns with the standard computation of the FTLE. For the case of autonomous and periodic vector fields we provide a simplified method to calculate the set-oriented FTLE using the Perron–Frobenius operator. Based on the new definition of the FTLE we propose a simple definition of finite-time coherent sets applicable to vector fields of general time-dependence, which are the analogues of almost-invariant sets in autonomous and time-periodic vector fields. The coherent sets we identify will necessarily be separated from one another by ridges of high FTLE, providing a link between the framework of coherent sets and that of codimension one Lagrangian coherent structures. Our identification of coherent sets is applied to three examples.

Normalizations with Exponentially Small Remainders for Nonautonomous Analytic Periodic Vector Fields

In this paper we deal with analytic nonautonomous vector fields with a periodic time-dependancy, that we study near an equilibrium point. In a first part, we assume that the linearized system is split in two invariant subspaces E0 and E1. Under light diophantine conditions on the eigenvalues of the linear part, we prove that there is a polynomial change of coordinates in E1 allowing to eliminate up to a finite polynomial order all terms depending only on the coordinate u0 of E0 in the E1 component of the vector field. We moreover show that, optimizing the choice of the degree of the polynomial change of coordinates, we get an exponentially small remainder. In the second part, we prove a normal form theorem with exponentially small remainder. Similar theorems have been proved before in the autonomous case : this paper generalizes those results to the nonautonomous periodic case.

The embedding flows of [formula omitted] hyperbolic diffeomorphisms

In [Weigu Li, J. Llibre, Xiang Zhang, Extension of Floquet's theory to nonlinear periodic differential systems and embedding diffeomorphisms in differential flows, Amer. J. Math. 124 (2002) 107–127] we proved that for a germ of C∞ hyperbolic diffeomorphisms F(x)=Ax+f(x) in (Rn,0), if A has a real logarithm with its eigenvalues weakly nonresonant, then F(x) can be embedded in a C∞ autonomous differential system. Its proof was very complicated, which involved the existence of embedding periodic vector field of F(x) and the extension of the Floquet's theory to nonlinear C∞ periodic differential systems. In this paper we shall provide a simple and direct proof to this last result.Next we shall show that the weakly nonresonant condition in the last result on the real logarithm of A is necessary for some C∞ diffeomorphisms F(x)=Ax+f(x) to have C∞ embedding flows.Finally we shall prove that a germ of C∞ hyperbolic diffeomorphisms F(x)=Ax+f(x) with f(x)=O(|x|2) in (R2,0) has a C∞ embedding flow if and only if either A has no negative eigenvalues or A has two equal negative eigenvalues and it can be diagonalizable.

Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method

The renormalization group (RG) method is one of the singular perturbation methods which is used in searching for asymptotic behavior of solutions of differential equations. In this article, time-independent vector fields and time (almost) periodic vector fields are considered. Theorems on error estimates for approximate solutions, existence of approximate invariant manifolds and their stability, and inheritance of symmetries from those for the original equation to those for the RG equation are proved. Further, it is proved that the RG method unifies traditional singular perturbation methods, such as the averaging method, the multiple time scale method, the (hyper)normal forms theory, the center manifold reduction, the geometric singular perturbation method, and the phase reduction. A necessary and sufficient condition for the convergence of the infinite order RG equation is also investigated.

Averaging of Hamiltonian flows with an ergodic component

We consider a process on $\mathbb{T}^{2}$, which consists of fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise. It gives rise to a process on the graph naturally associated to the structure of the stream lines of the unperturbed flow. It has been shown by Freidlin and Wentzell [Random Perturbations of Dynamical Systems, 2nd ed. Springer, New York (1998)] and [Mem. Amer. Math. Soc. 109 (1994)] that if the stream function of the flow is periodic, then the corresponding process on the graph weakly converges to a Markov process. We consider the situation where the stream function is not periodic, and the flow (when considered on the torus) has an ergodic component of positive measure. We show that if the rotation number is Diophantine, then the process on the graph still converges to a Markov process, which spends a positive proportion of time in the vertex corresponding to the ergodic component of the flow.

Multiscale Young Measures in almost Periodic Homogenization and Applications

We prove the existence of multiscale Young measures associated with almost periodic homogenization. We give applications of this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure, such as scalar conservation laws, nonlinear transport equations, Hamilton–Jacobi equations and fully nonlinear elliptic equations. Motivated by the application in nonlinear transport equations, we also prove basic results on flows generated by Lipschitz almost periodic vector fields, which are of interest in their own. In our analysis, an important role is played by the so-called Bohr compactification \({\mathbb {G}^N}\) of \({\mathbb {R}^N}\) ; this is a natural parameter space for the Young measures. Our homogenization results provide also the asymptotic behavior for the whole set of \({\mathbb {G}^N}\) -translates of the solutions, which is in the spirit of recent studies on the homogenization of stationary ergodic processes.