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Cross section
Mathematical science terms
Not all smooth flows have a cross section, and an obvious necessary condition is that the flow cannot have singularities. The cross section was introduced by Poincare,(J.-)H. The relation between smooth flow and differential homeomorphism generating discrete dynamical system can be established by cross section.
- Chinese name
- Cross section
- Foreign name
- cross-section
- Scope of application
- Mathematical science
- Generic type
- Mathematical science terms
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Consider the torus by differential equations
(α is a real number)
Smooth flow
φ take a cross section of the torus
(Figure 1), each orbit of φ intersects with C transversally, and the orbits starting from C intersect with C in both positive and negative directions.
The first return map on C is induced by φ
, due to
give
So that
The smallest t value that holds), f is a differential homeomorphism on C. In general, C on a manifold M
r
The cross section of the flow φ (corresponding vector field X) is a closed submanifold with codimension 1
, it meets:
1. Σ intersects X transversally;
(2) Every track of φ departing from Σ has its future and past intersecting with Σ;
3. Every orbit of φ intersects Σ.
Figure 1. Cross section
Figure 1. Cross section
Set C
r
The flow φ has cross section Σ, and the first return mapping can be defined above
f is C
r
Differential homeomorphism, thus having a cross-section of C
r
The flow induces a C in the cross section
r
Differential homeomorphism
.
However, not all smooth flows have a cross section, and an obvious necessary condition is that the flow cannot have singularities. The cross section was introduced by Poincare,(J.-)H.
The relation between smooth flow and differential homeomorphism generating discrete dynamical system can be established by cross section.
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