Peixoto's theorem
Encyclopedia
In the theory of dynamical system
s, Peixoto theorem, proved by Maurício Peixoto
, states that among all smooth flows
on surface
s, i.e. compact
two-dimensional manifold
s, structurally stable systems may be characterized by the following properties:
Moreover, they form an open set in the space of all flows endowed with C1 topology.
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s, Peixoto theorem, proved by Maurício Peixoto
Maurício Peixoto
Maurício Matos Peixoto, born on April 15 of 1921, in Fortaleza, Ceará, is a Brazilian engineer who pursued a bright career as a mathematician. He pioneered the studies on structural stability, and is the author of Peixoto's theorem....
, states that among all smooth flows
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over...
on surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
s, i.e. compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
two-dimensional manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
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 pointsFixed point (mathematics)In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
is finite and consists only of hyperbolic equilibrium pointHyperbolic equilibrium pointIn 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...
s. - Finiteness of attracting or repelling periodic orbits.
- Absence of saddleSaddle pointIn mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
-to-saddle connections.
Moreover, they form an open set in the space of all flows endowed with C1 topology.