In the theory of

dynamical system 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 ...

s, Peixoto's theorem, proved by

Maurício Peixoto Maurício Matos Peixoto, (April 15, 1921 – April 28, 2019), was a Brazilian engineer and mathematician. He pioneered the studies on structural stability, and was the author of Peixoto's theorem. Biography Maurício Peixoto, born in Fortaleza ...

, 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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

two-dimensional

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, structurally stable systems may be characterized by the following properties: * The set of non-wandering points consists only of periodic orbits and fixed points. * The set of

fixed points Fixed may refer to: * ''Fixed'' (EP), EP by Nine Inch Nails * ''Fixed'' (film), an upcoming animated film directed by Genndy Tartakovsky * Fixed (typeface), a collection of monospace bitmap fonts that is distributed with the X Window System * Fi ...

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 hyperbol ...

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 ...

s. * Absence of

saddle A 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 know ...

-to-saddle connections. Moreover, they form an open set in the space of all flows endowed with ''C''¹ topology.

References

* Jacob Palis, 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 in ...

, 1982 Stability theory Theorems in dynamical systems {{math-physics-stub

See also

References