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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and in particular the study of
dynamical systems In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
, 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 In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain c ...
or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the
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 th ...
.


Physical example

The gravitational
tidal force The tidal force or tide-generating force is the difference in gravitational attraction between different points in a gravitational field, causing bodies to be pulled unevenly and as a result are being stretched towards the attraction. It is the ...
s acting on the
rings of Saturn Saturn has the most extensive and complex ring system of any planet in the Solar System. The rings consist of particles in orbit around the planet made almost entirely of water ice, with a trace component of Rock (geology), rocky material. Parti ...
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 direction is along any radius, where forces stretch and pull particles apart. Two particles that start very near each other in
phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
will experience radial forces causing them to diverge, radially. These forces have a positive Lyapunov exponent; the trajectories lie on a hyperbolic manifold, and the movement of particles is essentially chaotic, wandering through the rings. The
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 modellin ...
is tangential to the rings, with particles experiencing neither compression nor stretching. This allows second-order gravitational forces to dominate, and so particles can be entrained by moons or moonlets in the rings, phase locking to them. The gravitational forces of the moons effectively provide a regularly repeating small kick, each time around the orbit, akin to a kicked rotor, such as found in a
phase-locked loop A phase-locked loop or phase lock loop (PLL) is a control system that generates an output signal whose phase is fixed relative to the phase of an input signal. Keeping the input and output phase in lockstep also implies keeping the input and ou ...
. The discrete-time motion of particles in the ring can be approximated by the
Poincaré map In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensiona ...
. The map effectively provides the transfer matrix of the system. The eigenvector associated with the largest eigenvalue of the matrix is the Frobenius–Perron eigenvector, which is also the
invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mappin ...
, ''i.e'' the actual density of the particles in the ring. All other eigenvectors of the transfer matrix have smaller eigenvalues, and correspond to decaying modes.


Definition

The following provides a definition for the case of a system that is either an
iterated function In mathematics, an iterated function is a function that is obtained by composing another function with itself two or several times. The process of repeatedly applying the same function is called iteration. In this process, starting from some ...
or has discrete-time dynamics. Similar notions apply for systems whose time evolution is given by a flow. Let X be a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, and f\colon X\to X a
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
. If p is a fixed point for f, the stable set of p is defined by :W^s(f,p) =\ and the unstable set of p is defined by :W^u(f,p) =\. Here, f^ denotes the inverse of the function f, i.e. f\circ f^=f^\circ f =id_, where id_ is the identity map on X. If p is a periodic point of least period k, then it is a fixed point of f^k, and the stable and unstable sets of p are defined by :W^s(f,p) = W^s(f^k,p) and :W^u(f,p) = W^u(f^k,p). Given a
neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
U of p, the local stable and unstable sets of p are defined by :W^s_(f,p,U) = \ and :W^u_(f,p,U) = W^s_(f^,p,U). If X is
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
, we can define the stable and unstable sets for any point by :W^s(f,p) = \ and :W^u(f,p) = W^s(f^,p), where d is a
metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...
for X. This definition clearly coincides with the previous one when p is a periodic point. Suppose now that X is a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
smooth manifold In mathematics, a differentiable manifold (also differential 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 ...
, and f is a \mathcal^k
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. Definit ...
, k\geq 1. If p is a hyperbolic periodic point, the stable manifold theorem assures that for some neighborhood U of p, the local stable and unstable sets are \mathcal^k embedded disks, whose
tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
s at p are E^s and E^u (the stable and unstable spaces of Df(p)), respectively; moreover, they vary continuously (in a certain sense) in a neighborhood of f in the \mathcal^k topology of \mathrm^k(X) (the space of all \mathcal^k diffeomorphisms from X to itself). Finally, the stable and unstable sets are \mathcal^k injectively immersed disks. This is why they are commonly called stable and unstable manifolds. This result is also valid for nonperiodic points, as long as they lie in some
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 th ...
(stable manifold theorem for hyperbolic sets).


Remark

If X is a (finite-dimensional)
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
and f an isomorphism, its stable and unstable sets are called stable space and unstable space, respectively.


See also

* Invariant manifold *
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 modellin ...
*
Limit set In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they c ...
*
Julia set In complex dynamics, the Julia set and the Classification of Fatou components, Fatou set are two complement set, complementary sets (Julia "laces" and Fatou "dusts") defined from a function (mathematics), function. Informally, the Fatou set of ...
* Slow manifold * Inertial manifold * Normally hyperbolic invariant manifold *
Lagrangian coherent structure Lagrangian coherent structures (LCSs) are distinguished surfaces of trajectories in a dynamical system that exert a major influence on nearby trajectories over a time interval of interest. The type of this influence may vary, but it invariably cr ...


References

* * * {{PlanetMath attribution, id=4357, title=Stable manifold Limit sets Dynamical systems Manifolds