In the theory of
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s, Peixoto's theorem, proved by
Maurício Peixoto
Maurício Matos Peixoto, (April 15, 1921, in Fortaleza, Ceará – April 28, 2019, in Rio de Janeiro), was a Brazilian engineer and mathematician. He pioneered the studies on structural stability, and was the author of Peixoto's theorem.
Biogra ...
, states that among all smooth
flows on
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s, i.e.
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
two-dimensional
manifolds,
structurally stable systems may be characterized by the following properties:
* The set of
non-wandering point
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 ...
s consists only of periodic orbits and fixed points.
* The set of
fixed points is finite and consists only of
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 hyperbo ...
s.
* Finiteness of attracting or repelling
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 a ...
s.
* Absence of
saddle
The saddle is a supportive structure for a rider of an animal, fastened to an animal's back by a girth. The most common type is equestrian. However, specialized saddles have been created for oxen, camels and other animals. It is not k ...
-to-saddle connections.
Moreover, they form an open set in the space of all flows endowed with
''C''1 topology.
See also
*
Andronov–Pontryagin criterion The Andronov–Pontryagin criterion is a necessary and sufficient condition for the stability of dynamical systems in the plane. It was derived by Aleksandr Andronov and Lev Pontryagin in 1937.
Statement
A dynamical system
: \dot = v(x),
w ...
References
*
Jacob Palis
__NOTOC__
Jacob Palis Jr. (born 15 March 1940) is a Brazilian mathematician and professor. Palis' research interests are mainly dynamical systems and differential equations. Some themes are global stability and hyperbolicity, bifurcations, a ...
, W. de Melo, ''Geometric Theory of Dynamical Systems''.
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, 1982
Stability theory
Theorems in dynamical systems
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