Genericity of historic behavior for maps and flows

A blender is a hyperbolic set with a stable or unstable invariant manifold that behaves as a geometric object of a dimension larger than that of the respective manifold itself. Blenders have been constructed in diffeomorphisms with a phase space of dimension at least three. We consider here the question of how one can identify, characterize and also visualize the underlying hyperbolic set of a given diffeomorphism to verify whether it actually is a blender or not. More specifically, we employ advanced numerical techniques for the computation of global manifolds to identify the hyperbolic set and its stable and unstable manifolds in an explicit Hénon-like family of three-dimensional diffeomorphisms. This allows to determine and illustrate whether the hyperbolic set is a blender; in particular, we consider as a distinguishing feature the self-similar structure of the intersection set of the respective global invariant manifold with a plane. By checking and illustrating a denseness property, we are able to identify a parameter range over which the hyperbolic set is a blender, and we discuss and illustrate how the blender disappears.

The paper deals with the design of the material laying trajectory on the technological mandrel surface and the complex geometric shape mandrel surface model representation in automated systems for preparing control programs for winding composite materials products. The technological mandrel surface model can be represented as an irregular spatial point set. It is proposed to use an approach based on small surface segment approximation and modification of the differential equations system intended for integration laying material trajectories over an analytically given surface. The developed mathematical apparatus for constructing curves on surfaces given by irregular spatial point set makes it possible to construct smooth class-second curves without the need to first obtain the complete surface model. The proposed solutions adequacy are confirmed by the numerical experiments carried out on modeling the filament laying trajectory on various complex-profile mandrels given by an irregular point set and by the results of winding prepared control programs running out on a real multiaxis machine.KeywordsWindingMandrel surfacePoint cloudComposite material productCNC winding machineLaying material trajectory

The increasing use of unstructured grids for reservoir modeling motivates the development of geostatistical techniques to populate them with properties such as facies proportions, porosity and permeability. Unstructured grids are often populated by upscaling high-resolution regular grid models, but the size of the regular grid becomes unreasonably large to ensure that there is sufficient resolution for small unstructured grid elements. The properties could be modeled directly on the unstructured grid, which leads to an irregular configuration of points in the three-dimensional reservoir volume. Current implementations of Gaussian simulation for geostatistics are for regular grids. This paper addresses important implementation details involved in adapting sequential Gaussian simulation to populate irregular point configurations including general storage and computation issues, generating random paths for improved long range variogram reproduction, and search strategies including the superblock search and the k-dimensional tree. An efficient algorithm for computing the variogram of very large irregular point sets is developed for model checking.

Kinematic features of three–dimensional mixing by advection of passive particles in time–periodic flows are the primary subject of this study. A classification of periodic points, providing important information about the mixing properties of a flow, is presented, and the dynamics of the Poincare map in the vicinity of periodic points is analysed for all identified types. Three examples of Stokes flow in a finite cylindrical cavity with discontinuous periodic motion of its end walls are used to illustrate the determination of both periodic lines and isolated periodic points in the flow domain. The stable and unstable manifolds of points on the periodic lines create two surfaces in the flow. A numerical technique based on tracking of a material surface is presented to study the manifold surfaces and their intersections. It is illustrated with numerical examples that flows with periodic lines possess only quasi–two–dimensional mechanisms of chaotic advection.

Crowd flows prediction is an important problem for traffic management and public safety. Graph Convolutional Network (GCN), known for its ability to effectively capture and utilize topological information, has demonstrated significant advancements in addressing this problem. However, GCN-based models were often based on predefined crowd-flow graphs via historical movement behaviors of human beings and traffic vehicles, which ignored the abnormal changes in crowd flows. In this study, we propose a multi-scale fusion GCN-based framework with Tucker decomposition named mTDNet to enhance dynamic GCN for crowd flows prediction. Following the paradigm of extant methods, we also employ the predefined crowd-flow graphs as a part of mTDNet to effectively capture the historical movement behaviors of crowd flows. To capture the abnormal changes, we propose a Tucker decomposition-based network with the product of the adjacency matrix of historical movement pattern graphs and an Adaptive Learning Tensor ( ALT ) by reconstructing the crowd flows. Particularly, we utilize the Tucker decomposition scheme to decompose ALT , which enhances the dynamic learning of graph structures, allowing for effective capturing of the dynamic changes in crowd flow, including abnormal changes. Furthermore, a multi-scale 3DGCN is utilized to mine and fuse the multi-scale spatio-temporal information from crowd flows, to further boost the mTDNet prediction performance. Experiments conducted on two real-world datasets showed that the proposed mTDNet surpasses other crowd flow prediction methods.

Diffeomorphisms of a multidimensional space into itself with a hyperbolic fixed point are discussed in this paper. It is assumed that at the intersection of stable and unstable manifolds, there are points that are different from the hyperbolic point. Such points are called homoclinic and are divided into transversal and non-transversal, depending on the behavior of stable and unstable manifolds. It follows from articles by S. Newhouse, L. P. Shil’nikov, B. F. Ivanov, and others, that with a certain method of tangency of a stable manifold with an unstable one, the neighborhood of a non-transversal homoclinic point contains an infinite number of stable periodic points, but at least one of the characteristic exponents at these points tends to zero with increasing period. The present study is a continuation of previous studies by the author. In previously published papers, restrictions were imposed on the eigenvalues of the Jacobi matrix of the original diffeomorphism at a hyperbolic point. More precisely, it was assumed that either all eigenvalues are real and the Jacobi matrix is diagonal, or the matrix has only one real eigenvalue less than one in modulus, while all other eigenvalues are various complex integers greater than one in modulus. Within this framework, conditions are obtained for the presence of an infinite set of stable periodic points with characteristic exponents separated from zero in an arbitrary neighborhood of a non-transversal homoclinic point. It is assumed in this paper that the Jacobi matrix of a diffeomorphism has an arbitrary set of eigenvalues at a hyperbolic point. In this case, the conditions are obtained for the existence of an infinite set of stable periodic points whose characteristic exponents are separated from zero in the neighborhood of the non-transversal homoclinic point. The conditions are imposed, first of all, on the method of tangency of a stable manifold with an unstable one; however, the proof of the theorem essentially uses the properties of the eigenvalues of the Jacobi matrix at a hyperbolic point.

The dynamical properties of Lorenz attractors have been studied in a number of recent papers. While the Lorenz dynamical system is a flow in 3-space, a major tool for its study has been the kneading invariant of a piecewise monotonic map of an interval to itself. This is related to a semi-flow on a 2 dimensional branched manifold. The semi-flow can be used to define a flow which admits a geometric realisation in 3-space. The main results can be traced through references (1), (5) and (6). It is known that if the slope of the map always exceeds √2 then the periodic points are dense. That this cannot be true without some restriction on the kneading function is clear from Parry's measure theoretic results for piecewise linear kneading functions (4). In the first part of this paper I prove some analogous topological results concerning the condition that the kneading function is locally eventually onto, and concerning the non-wandering set and set of periodic points of the discrete semi-flow defined by the function. In the second part of the paper I give alternative proofs of results on kneading invariants in a more general context than Rand's (5). The new methods of proof are needed in the third part where kneading functions on intervals are used to define kneading functions on product spaces of intervals with compact metric spaces.

We study here the invariance of topological distributionally scrambled sets for maps on uniform spaces (not necessarily compact or metrizable). For a uniformly continuous surjective self-map defined on a uniformly locally compact Hausdorff uniform space having topological weak specification, a fixed point, and countably many periodic points with distinct periods, we prove that the map admits an invariant topological distributionally scrambled set of type 1. Further, if the uniform space is second countable, the map admits a dense Mycielski invariant topological distributionally scrambled set of type 1.

A continuous function on the product of compact metric spaces to itself and back to the same space is known as a genera system. Where each element's orbit is an infinite sequence and where the first two elements are the same as given and next to it depends on the two elements prior to it form stronger conditions for the orbit. We could define an m-step dynamical system by extending the definition of a compact metric space to its m-times product. Because the system's current state frequently depends directly on the conditions of previous terms, this system appears more realistic. The basic theorems regarding periodic points and their related points, such as fixed point, limit points, recurrent points, will be proved in this paper. We also define the topological transitivity and its properties. In the end we find periodic points with periods one and two for affine maps and periodicity of tent map in dynamical system and generalised dynamical system.

We introduce a type of zero-dimensional dynamical system (a pair consisting of a totally disconnected compact metrizable space along with a homeomorphism of that space), which we call “fiberwise essentially minimal”, that is a class that includes essentially minimal systems and systems in which every orbit is minimal. We prove that the associated crossed product [Formula: see text]-algebra of such a system is an A[Formula: see text]-algebra. Under the additional assumption that the system has no periodic points, we prove that the associated crossed product [Formula: see text]-algebra has real rank zero, which tells us that such [Formula: see text]-algebras are classifiable by [Formula: see text]-theory. The associated crossed product [Formula: see text]-algebras to these nontrivial examples are of particular interest because they are non-simple (unlike in the minimal case).

The central limit theorem and ergodicity

In this paper, we consider linear skew-product semiflows on bundles of Banach fibers over a locally compact metric space. Our aim is to give discrete-time theorems for the existence of global and pointwise continuous-time dichotomies with no invariant unstable manifolds. We involve here a concept of exponential dichotomy for skew-product semiflows weaker than the concept used by Sacker and Sell (J Differ Equ 15:429–458, 1974; 22:478–522, 1976) and Magalhaes (in: Chow, Hale (eds) Dynamics of infinite dimensional systems, NATO Advanced Science Institutes Series F: Computer and Systems Sciences, Springer, New York, vol 37, pp 161–168, 1987); our definition (of no past exponential dichotomy) follows roughly the definition given by Chow and Leiva (Proc Am Math Soc 124:1071–1081, 1996) in the sense that we allow the unstable subspace to have infinite dimension. The main improvement is that we go even more general and we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable subspace is invariant under the cocycle). Roughly speaking, we prove that if the solution of the corresponding inhomogeneous variational difference equation belongs to any sequence space (on which the right shift is an isometry) for every inhomogeneity from the same class of sequence spaces, then the continuous-time solutions of the variational homogeneous differential equation will exhibit a (no past) exponential dichotomic behavior. This approach has many advantages among which we emphasize on the facts that the above condition is very general (since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical p-summable spaces, sequence Orlicz spaces, etc.). Since we use a discrete-time technique we are not forced to require any continuity or measurability hypotheses on the trajectories of the exponentially bounded cocycle. Also, it is worth to mention that from discrete-time conditions we get informations about the continuous-time behavior of the solutions of differential variational equations.

We show that a $C^{1}$-generic non-partially hyperbolic symplectic diffeomorphism $f$ has topological entropy equal to the supremum of the sum of the positive Lyapunov exponents of its hyperbolic periodic points. Moreover, we also prove that $f$ has topological entropy approximated by the topological entropy of $f$ restricted to basic hyperbolic sets. In particular, the topological entropy map is lower semicontinuous in a $C^{1}$-generic set of symplectic diffeomorphisms far from partial hyperbolicity.

We introduce the notion of quasi-expansion in the context of polynomial diffeomorphisms of C 2 . Like hyperbolic diffeomorphisms, quasi-expanding maps have uniformly large multipliers at saddle points. On the other hand, unlike the hyperbolic case, quasi-expanding maps can have tangencies between stable and unstable manifolds. We characterize quasi-expansion in a number of ways and develop some of the structure they possess. Quasi-expansion was motivated by the study of real polynomial diffeomorphisms of maximal entropy, and our study of maximal entropy diffeomorphisms relies on the results of this paper.

We consider the spaceN ofC 2 twist maps that satisfy the following requirements. The action is the sum of a purely quadratic term and a periodic potential times a constantk (hereafter called the nonlinearity). The potential restricted to the unit circle is bimodal, i.e. has one local minimum and one local maximum. The following statements are proven for maps inN with nonlinearityk large enough. The intersection of the unstable and stable invariant manifolds to the hyperbolic minimizing periodic points contains minimizing homoclinic points. Consider two finite pieces of these manifolds that connect two adjacent homoclinic minimizing points (hereafter called fundamental domains). We prove that all such fundamental domains have precisely one point in their intersection (the Single Intersection theorem).In addition, we show that limit points of minimizing points are recurrent, which implies that Aubry Mather sets (with irrational rotation number) are contained in diamonds formed by local stable and unstable manifolds of nearby minimizing periodic orbits (the Diamond Configuration theorem). Another corollary concerns the intersection of the minimax orbits with certain symmetry lines of the map.