Topological conjugacy of non-singular flows with two limit cycles on $S^2 times S^1$

Abstract

In the paper, non-singular flows with two limit cycles on the manifold $S^2 times S^1$ are considered. For such flows, a classification is obtained up to topological conjugacy, and it is shown that they have a functional modulus of stability. Since the functional modulus of stability takes on its own value for each fixed argument, the presence of such modulus implies the presence of an infinite number of numerical moduli of stability. To obtain this result, linearization is carried out in the neighbourhoods of two limit cycles using the construction from the work by M. Irwin. A result is obtained on the presence of a two-dimensional foliation in a neighborhood of the limit cycle; this foliation is invariant up to topological conjugacy. Existence of the functional modulus of stability follows from the presence of such foliations. Namely, when considering the intersection of two foliations and, accordingly, two linearizations acting in the basins of two limit cycles, the desired functional modulus is a map describing the relative position of the foliation layer in the neighborhood of the first limit cycle relative to the layer of the second limit cycle. The results are used from the work by Pochinka O. V. and Shubin D. D. on exactly two classes of topological equivalence of flows in the class under consideration and on description of their differences. The work includes figure which shows 2 classes of topological conjugacy of flows from the classes studied. Also there is a figure which shows the process of gluing $mathbb R^3$ into a manifold with a stable limit cycle. Moreover, the construction of a solid torus is shown. The figures show consistent and inconsistent orientation of limit cycles, as well as invariant foliations. Also there is a figure which shows the functional modulus.

Keywords

• topological conjugacy”, “modulus of stability”, “non-singular flow”, “limit cycle