A construction of three-dimensional vector fields which have a codimension two heteroclinic loop at Glendinning-Sparrow T-point

1993 ◽  
Vol 44 (3) ◽  
pp. 510-536 ◽  
Author(s):  
Hiroshi Kokubu
Keyword(s):  
Vector Fields ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Heteroclinic Loop ◽  
Codimension Two

Unique Normal Form and the Associated Coefficients for a Class of Three-Dimensional Nilpotent Vector Fields

2017 ◽  
Vol 27 (14) ◽  
pp. 1750224
Author(s):  
Jing Li ◽  
Liying Kou ◽  
Duo Wang ◽  
Wei Zhang
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
Three Dimensional ◽  
Recursive Formula ◽  
Higher Order ◽  
Dimensional Vector ◽  
Symbolic Calculations

In this paper, we mainly focus on the unique normal form for a class of three-dimensional vector fields via the method of transformation with parameters. A general explicit recursive formula is derived to compute the higher order normal form and the associated coefficients, which can be achieved easily by symbolic calculations. To illustrate the efficiency of the approach, a comparison of our result with others is also presented.

scholarly journals NONORIENTABLE MANIFOLDS IN THREE-DIMENSIONAL VECTOR FIELDS

2003 ◽  
Vol 13 (03) ◽  
pp. 553-570 ◽  
Author(s):  
HINKE M. OSINGA
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Period Doubling ◽  
Building Blocks ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Three Dimensional Vector Field

It is well known that a nonorientable manifold in a three-dimensional vector field is topologically equivalent to a Möbius strip. The most frequently used example is the unstable manifold of a periodic orbit that just lost its stability in a period-doubling bifurcation. However, there are not many explicit studies in the literature in the context of dynamical systems, and so far only qualitative sketches could be given as illustrations. We give an overview of the possible bifurcations in three-dimensional vector fields that create nonorientable manifolds. We mainly focus on nonorientable manifolds of periodic orbits, because they are the key building blocks. This is illustrated with invariant manifolds of three-dimensional vector fields that arise from applications. These manifolds were computed with a new algorithm for computing two-dimensional manifolds.

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