0000000001153109
AUTHOR
B. Campos
Links and Bifurcations in Nonsingular Morse–Smale Systems
Wada's theorem classifies the set of periodic orbits in NMS systems on S3 as links, that can be written in terms of six operations. This characterization allows us to study the topological restrictions that links require to suffer a given codimension one bifurcation. Moreover, these results are reproduced in the case of NMS systems with rotational symmetries, introducing new geometrical tools.
Bifurcations of Links of Periodic Orbits in Non-Singular Systems with Two Rotational Symmetries on S3
A topological characterization of all possible links composed of the periodic orbits of a Non Singular Morse-Smale flow on S3 has been made by M. Wada. The presence of symmetry forces the appearance of given types of links. In this paper we introduce a geometrical tool to represent these type of links when a symmetry around two axes is considered on NMS systems: mosaics. On the other hand, we use mosaics to study what kind of bifurcation can occur in this type of system maintaining the symmetry.
Bifurcations of links of periodic orbits in non-singular Morse–Smale systems with a rotational symmetry on S3
Abstract In this paper we consider a rotational symmetry on a non-singular Morse–Smale (NMS) system analyzing the restrictions this symmetry imposes on the links defined by the set of its periodic orbits and to the appearance of local generic codimension one bifurcations in the set of NMS flows on S 3 . The topological characterization is obtained by writing the involved links in terms of Wada operations. It is also obtained that symmetry implies that in general bifurcations have to be multiple. On the other hand, we also see that there exists a set of links that cannot be related to any other by sequences of this kind of bifurcation.
Bifurcations of Links of Periodic Orbits in Mathieu Systems
We prove that orbits escape from infinity, and that therefore the sphere S can be considered as its phase space. If the parameter δ is large enough, the system is non-singular MorseSmale, and its periodic orbits define a Hopf link. As δ decreases, the system undergoes some bifurcations that we describe geometrically. We relate the bifurcation orbits to periodic orbits continued from the linear Mathieu equation.
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PubMed: 26787795
Bifurcations of links of periodic orbits in non-singular Morse - Smale systems on
The set of periodic orbits of a non-singular Morse - Smale (NMS) flow on defines a link; a characterization of all possible links of NMS flows on has been developed by Wada. In the frame of codimension-one bifurcations, this characterization allows us to study the restrictions a link requires for suffering a given bifurcation. We also derive the topological description of the new link and the possibility of relating links by a chain of this type of bifurcation.