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Normally Hyperbolic Invariant Manifold
A normally hyperbolic invariant manifold (NHIM) is a natural generalization of a hyperbolic fixed point and a hyperbolic set. The difference can be described heuristically as follows: For a manifold \Lambda to be normally hyperbolic we are allowed to assume that the dynamics of \Lambda itself is neutral compared with the dynamics nearby, which is not allowed for a hyperbolic set. NHIMs were introduced by Neil Fenichel in 1972. In this and subsequent papers, Fenichel proves that NHIMs possess stable and unstable manifolds and more importantly, NHIMs and their stable and unstable manifolds persist under small perturbations. Thus, in problems involving perturbation theory, invariant manifolds exist with certain hyperbolicity properties, which can in turn be used to obtain qualitative information about a dynamical system.A. Katok and B. Hasselblatt''Introduction to the Modern Theory of Dynamical Systems'', Cambridge University Press (1996), Definition Let ''M'' be a compact sm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Fixed Point
In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperbolas. This fails to hold in general. Strogatz notes that "hyperbolic is an unfortunate name—it sounds like it should mean 'saddle point'—but it has become standard." Several properties hold about a neighborhood of a hyperbolic point, notably * A stable manifold and an unstable manifold exist, * Shadowing occurs, * The dynamics on the invariant set can be represented via symbolic dynamics, * A natural measure can be defined, * The system is structurally stable. Maps If T \colon \mathbb^ \to \mathbb^ is a ''C''1 map and ''p'' is a fixed point then ''p'' is said to be a hyperbolic fixed point when the Jacobian matrix \operatorname T (p) has no eigenvalues on the complex unit circle. One example of a map whose only fixed point is h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of manifold the tangent spaces and have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle , see Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle. of the tangent spaces of M . That is, : \begin TM &= \bigsqcup_ T_xM \\ &= \bigcup_ \left\ \times T_xM \\ &= \bigcup_ \left\ \\ &= \left\ \end where T_x M denotes the tangent space to M at the point x . So, an el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Set
In dynamical systems theory, a subset Λ of a smooth manifold ''M'' is said to have a hyperbolic structure with respect to a smooth map ''f'' if its tangent bundle may be split into two invariant subbundles, one of which is contracting and the other is expanding under ''f'', with respect to some Riemannian metric on ''M''. An analogous definition applies to the case of flows. In the special case when the entire manifold ''M'' is hyperbolic, the map ''f'' is called an Anosov diffeomorphism. The dynamics of ''f'' on a hyperbolic set, or hyperbolic dynamics, exhibits features of local structural stability and has been much studied, cf. Axiom A. Definition Let ''M'' be a compact smooth manifold, ''f'': ''M'' → ''M'' a diffeomorphism, and ''Df'': ''TM'' → ''TM'' the differential of ''f''. An ''f''-invariant subset Λ of ''M'' is said to be hyperbolic, or to have a hyperbolic structure, if the restriction to Λ of the tangent bundle of ''M'' admits a sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Fixed Point
In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperbolas. This fails to hold in general. Strogatz notes that "hyperbolic is an unfortunate name—it sounds like it should mean 'saddle point'—but it has become standard." Several properties hold about a neighborhood of a hyperbolic point, notably * A stable manifold and an unstable manifold exist, * Shadowing occurs, * The dynamics on the invariant set can be represented via symbolic dynamics, * A natural measure can be defined, * The system is structurally stable. Maps If T \colon \mathbb^ \to \mathbb^ is a ''C''1 map and ''p'' is a fixed point then ''p'' is said to be a hyperbolic fixed point when the Jacobian matrix \operatorname T (p) has no eigenvalues on the complex unit circle. One example of a map whose only fixed point is h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Center Manifold
In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modelling. Center manifolds play an important role in bifurcation theory because interesting behavior takes place on the center manifold and in multiscale mathematics because the long time dynamics of the micro-scale often are attracted to a relatively simple center manifold involving the coarse scale variables. Informal description Saturn's rings capture much center-manifold geometry. Dust particles in the rings are subject to tidal forces, which act characteristically to "compress and stretch". The forces compress particle orbits into the rings, stretch particles along the rings, and ignore small shifts in ring radius. The compressing direction defines the stable manifold, the stretching direction defining the unstable manifold, and the ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Manifold
In mathematics, and in particular the study of dynamical systems, the idea of ''stable and unstable sets'' or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set. Physical example The gravitational tidal forces acting on the rings of Saturn provide an easy-to-visualize physical example. The tidal forces flatten the ring into the equatorial plane, even as they stretch it out in the radial direction. Imagining the rings to be sand or gravel particles ("dust") in orbit around Saturn, the tidal forces are such that any perturbations that push particles above or below the equatorial plane results in that particle feeling a restoring force, pushing it back into the plane. Particles effectively oscillate in a harmonic well, damped by collisions. The stable direction is perpendicular to the ring. The unstable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Metric
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifold, manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport. Any smooth surface in three-dimensional Eucl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Manifold
In mathematics, and in particular the study of dynamical systems, the idea of ''stable and unstable sets'' or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set. Physical example The gravitational tidal forces acting on the rings of Saturn provide an easy-to-visualize physical example. The tidal forces flatten the ring into the equatorial plane, even as they stretch it out in the radial direction. Imagining the rings to be sand or gravel particles ("dust") in orbit around Saturn, the tidal forces are such that any perturbations that push particles above or below the equatorial plane results in that particle feeling a restoring force, pushing it back into the plane. Particles effectively oscillate in a harmonic well, damped by collisions. The stable direction is perpendicular to the ring. The unstable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions. Formal definition In the following we assume all manifolds are differentiable manifolds of class C^r for a fixed r\geq 1, and all morphisms are differentiable of class C^r. Immersed submanifolds An immersed submanifold of a manifold M is the image S of an immersion map f: N\rightarrow M; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (one-to-one) – it can have self-intersections. More narrowly, one can require that the map f: N\rightarrow M be an injection (one-to-one), in which we call it an injective immersion, and define an immersed submanifold to be the image subset S together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Set
In dynamical systems theory, a subset Λ of a smooth manifold ''M'' is said to have a hyperbolic structure with respect to a smooth map ''f'' if its tangent bundle may be split into two invariant subbundles, one of which is contracting and the other is expanding under ''f'', with respect to some Riemannian metric on ''M''. An analogous definition applies to the case of flows. In the special case when the entire manifold ''M'' is hyperbolic, the map ''f'' is called an Anosov diffeomorphism. The dynamics of ''f'' on a hyperbolic set, or hyperbolic dynamics, exhibits features of local structural stability and has been much studied, cf. Axiom A. Definition Let ''M'' be a compact smooth manifold, ''f'': ''M'' → ''M'' a diffeomorphism, and ''Df'': ''TM'' → ''TM'' the differential of ''f''. An ''f''-invariant subset Λ of ''M'' is said to be hyperbolic, or to have a hyperbolic structure, if the restriction to Λ of the tangent bundle of ''M'' admits a sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pushforward (differential)
In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that \varphi\colon M\to N is a smooth map between smooth manifolds; then the differential of \varphi at a point x, denoted \mathrm d\varphi_x, is, in some sense, the best linear approximation of \varphi near x. It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of M at x to the tangent space of N at \varphi(x), \mathrm d\varphi_x\colon T_xM \to T_N. Hence it can be used to ''push'' tangent vectors on M ''forward'' to tangent vectors on N. The differential of a map \varphi is also called, by various authors, the derivative or total derivative of \varphi. Motivation Let \varphi: U \to V be a Smooth function#Smooth functions on and between manifolds, smooth map from an Open subset#Euclidean space, open subset U of \R^m to an open subset V of \R^n. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definition Given two differentiable manifolds M and N, a Differentiable manifold#Differentiability of mappings between manifolds, continuously differentiable map f \colon M \rightarrow N is a diffeomorphism if it is a bijection and its inverse f^ \colon N \rightarrow M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. Two C^r-differentiable manifolds are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. Diffeomorphisms of subsets of manifolds Given a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
