In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point. It roughly states that the existence of a local diffeomorphism near a fixed point implies the existence of a local stable center manifold containing that fixed point. This manifold has dimension equal to the number of eigenvalues of the Jacobian matrix of the fixed point that are less than 1. [1]
Let
be a smooth map with hyperbolic fixed point at . We denote by the stable set and by the unstable set of .
The theorem [2] [3] [4] states that
Accordingly is a stable manifold and is an unstable manifold .
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries.
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold.
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector diverge at a rate given by
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero, but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function has a critical point at that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the -direction.
In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations.
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
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.
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.
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.
Numerical continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations,
In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearisation—a natural simplification of the system—is effective in predicting qualitative patterns of behaviour. The theorem owes its name to Philip Hartman and David M. Grobman.
In dynamical systems theory, an area of pure mathematics, a Morse–Smale system is a smooth dynamical system whose non-wandering set consists of finitely many hyperbolic equilibrium points and hyperbolic periodic orbits and satisfying a transversality condition on the stable and unstable manifolds. Morse–Smale systems are structurally stable and form one of the simplest and best studied classes of smooth dynamical systems. They are named after Marston Morse, the creator of the Morse theory, and Stephen Smale, who emphasized their importance for smooth dynamics and algebraic topology.
In applied mathematics, Arnold diffusion is the phenomenon of instability of integrable Hamiltonian systems. The phenomenon is named after Vladimir Arnold who was the first to publish a result in the field in 1964. More precisely, Arnold diffusion refers to results asserting the existence of solutions to nearly integrable Hamiltonian systems that exhibit a significant change in the action variables.
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 to be normally hyperbolic we are allowed to assume that the dynamics of 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.
In mathematics, the slow manifold of an equilibrium point of a dynamical system occurs as the most common example of a center manifold. One of the main methods of simplifying dynamical systems, is to reduce the dimension of the system to that of the slow manifold—center manifold theory rigorously justifies the modelling. For example, some global and regional models of the atmosphere or oceans resolve the so-called quasi-geostrophic flow dynamics on the slow manifold of the atmosphere/oceanic dynamics, and is thus crucial to forecasting with a climate model.
In dynamical systems, a branch of mathematics, an invariant manifold is a topological manifold that is invariant under the action of the dynamical system. Examples include the slow manifold, center manifold, stable manifold, unstable manifold, subcenter manifold and inertial manifold.
