Search results for "Morse code"
showing 8 items of 8 documents
Principal Poincar\'e Pontryagin Function associated to some families of Morse real polynomials
2014
It is known that the Principal Poincar\'e Pontryagin Function is generically an Abelian integral. We give a sufficient condition on monodromy to ensure that it is an Abelian integral also in non generic cases. In non generic cases it is an iterated integral. Uribe [17, 18] gives in a special case a precise description of the Principal Poincar\'e Pontryagin Function, an iterated integral of length at most 2, involving logarithmic functions with only one ramification at a point at infinity. We extend this result to some non isodromic families of real Morse polynomials.
Bifurcations of links of periodic orbits in non-singular Morse - Smale systems on
1997
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.
Strongly invertible links and divides
2008
Abstract To a proper generic immersion of a finite number of copies of the unit interval in a 2-disc, called a divide, A’Campo associates a link in S 3 . From the more general notion of ordered Morse signed divides, one obtains a braid presentation of links of divides. In this paper, we prove that every strongly invertible link is isotopic to the link of an ordered Morse signed divide. We give fundamental moves for ordered Morse signed divides and show that strongly invertible links are equivalent if and only if we can pass from one ordered Morse signed divide to the other by a sequence of such moves. Then we associate a polynomial to an ordered Morse signed divide, invariant for these move…
Minimal Morse flows on compact manifolds
2006
Abstract In this paper we prove, using the Poincare–Hopf inequalities, that a minimal number of non-degenerate singularities can be computed in terms only of abstract homological boundary information. Furthermore, this minimal number can be realized on some manifold with non-empty boundary satisfying the abstract homological boundary information. In fact, we present all possible indices and types (connecting or disconnecting) of singularities realizing this minimal number. The Euler characteristics of all manifolds realizing this minimal number are obtained and the associated Lyapunov graphs of Morse type are described and shown to have the lowest topological complexity.
Lyapunov graphs for circle valued functions
2018
International audience; Conley index theory is used to obtain results for flows associated to circular Lyapunov functions defined on general compact smooth n-manifolds. This is done in terms of their underlying circular Lyapunov digraphs, which are generalizations of Morse digraphs, by extensively studying their combinatorics, invariants and realizability.
Bifurcations of links of periodic orbits in non-singular Morse–Smale systems with a rotational symmetry on S3
2000
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.
Singular levels and topological invariants of Morse Bott integrable systems on surfaces
2016
Abstract We classify up to homeomorphisms closed curves and eights of saddle points on orientable closed surfaces. This classification is applied to Morse Bott foliations and Morse Bott integrable systems allowing us to define a complete invariant. We state also a realization Theorem based in two transformations and one generator (the foliation of the sphere with two centers).
Links and Bifurcations in Nonsingular Morse–Smale Systems
1997
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.