In
dynamical systems theory
Dynamical systems theory is an area of mathematics used to describe the behavior of complex systems, complex dynamical systems, usually by employing differential equations by nature of the ergodic theory, ergodicity of dynamic systems. When differ ...
, an area of
pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications ...
, a Morse–Smale system is a smooth dynamical system whose
non-wandering set
In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is the opposi ...
consists of finitely many
hyperbolic equilibrium 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 hyperbol ...
s and
hyperbolic
Hyperbolic may refer to:
* of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics
** Hyperbolic geometry, a non-Euclidean geometry
** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...
periodic orbit
In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
Iterated functions
Given ...
s and satisfying a transversality condition on the
stable
A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles
There are many different types of stables in use tod ...
and unstable
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
s. Morse–Smale systems are
structurally stable
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 (to be exact ''C''1-small perturbations).
Examples of such q ...
and form one of the simplest and best studied classes of smooth dynamical systems. They are named after
Marston Morse
Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) was an American mathematician best known for his work on the ''calculus of variations in the large'', a subject where he introduced the technique of differential topology now known a ...
, the creator of the
Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differenti ...
, and
Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty ...
, who emphasized their importance for smooth dynamics and
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
.
Definition
Consider a smooth and
complete vector field ''X'' defined on a compact differentiable manifold ''M'' with dimension ''n''. The flow defined by this vector field is a Morse-Smale system if
# ''X'' has only a finite number of singular points (i.e. equilibrium points of the flow), and all of them are
hyperbolic equilibrium points.
# ''X'' has only a finite number of periodic orbits, and all of them are
hyperbolic
Hyperbolic may refer to:
* of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics
** Hyperbolic geometry, a non-Euclidean geometry
** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...
periodic orbit
In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
Iterated functions
Given ...
s.
# The
limit sets
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2009 ...
of all orbits of X tends to a singular point or a periodic orbit.
# The stable and unstable manifolds of the singular points and periodic orbits intersect transversely. In other words, if
is a singular point (or periodic orbit) and
(respectively,
) its stable (respectively, unstable) manifold, then
implies that the corresponding tangent spaces of the stable and unstable manifold satisfy
.
Examples
* Any
Morse function
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differenti ...
''f'' on 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 ...
Riemannian manifold
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 surf ...
''M'' defines a gradient vector field. If one imposes the condition that the
unstable
In dynamical systems instability means that some of the outputs or internal state (controls), states increase with time, without bounds. Not all systems that are not Stability theory, stable are unstable; systems can also be marginal stability ...
and
stable
A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles
There are many different types of stables in use tod ...
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
s of the
critical points intersect transversely, then the gradient vector field and the corresponding smooth
flow form a Morse–Smale system. The finite set of
critical points of ''f'' forms the non-wandering set, which consists entirely of fixed points.
*
Gradient-like dynamical systems are a particular case of Morse–Smale systems.
* For Morse–Smale systems on the 2D-sphere all equilibrium points and periodical orbits are
hyperbolic
Hyperbolic may refer to:
* of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics
** Hyperbolic geometry, a non-Euclidean geometry
** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...
; there are no
separatrice loops.
Properties
* By
Peixoto's theorem
In the theory of dynamical systems, Peixoto's theorem, proved by Maurício Peixoto, states that among all smooth flows on surfaces, i.e. compact two-dimensional manifolds, structurally stable systems may be characterized by the following properti ...
, the vector field on a 2D manifold is structurally stable if and only if this field is Morse-Smale.
* Consider a Morse-Smale system defined on compact differentiable manifold ''M'' with dimension ''n'', and let the index of an equilibrium point (or a periodic orbit) be defined as the dimension of its associated unstable manifold. In Morse-Smale systems, the indices of the equilibrium points (and periodic orbits) are related with the topology of ''M'' by the Morse-Smale inequalities. Precisely, define ''m
i'' as the sum of the number of equilibrium points with index ''i'' and the number of periodic orbits with indices ''i'' and ''i'' + 1, and ''b
i'' as the ''i''-th
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
of ''M''. Then the following inequalities are valid:
:
Notes
References
*
*
Dynamical systems
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