American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020381

Čech cohomology, homoclinic trajectories and robustness of non-saddle sets

* Corresponding author: Héctor Barge

Received  February 2020 Revised  October 2020 Published  November 2020

Fund Project: The author is partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (grant PGC2018-098321-B-I00)

In this paper we study flows having an isolated non-saddle set. We see that the global structure of a flow having an isolated non-saddle set $K$ depends on the way $K$ sits in the phase space at the cohomological level. We construct flows on surfaces having isolated non-saddle sets with a prescribed global structure. We also study smooth parametrized families of flows and continuations of isolated non-saddle sets.

Citation: Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020381
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Flow on a double torus having an isolated non-saddle set $K'$ comprised of stationary points that is a sphere with the interiors of four closed topological disks removed. The region of influence of $K'$ is the double torus with the fixed points $p_1$ and $p_2$ removed. $\mathcal{I}(K')\setminus K'$ has two connected components $C'_1$ and $C'_2$ with local complexities $0$ and $2$ respectively
Flow on a double torus which has an isolated non-saddle set $K$ comprised of stationary points that is a sphere with the interiors of four disjoint closed topological disks removed. The region of influence of $K$ is the whole double torus and $\mathcal{I}(K)\setminus K$ has two components $C_1$ and $C_2$ with local complexity $1$
Model for a degenerate saddle fixed point
Flow on $S^1\times[0,1]$ which has $S^1\times \{0\}$ as a repelling circle of fixed points, $S^1\times\{1\}$ as an attracting circle of fixed points and the point $\{z\}\times\{1/2\}$ as a degenerate saddle fixed point
Flow defined on a sphere with the interior of four closed topological disks removed. This flow has a Morse decomposition $\{M_1,M_2,M_3\}$ where $M_1$ is the attracting outer circle of fixed points, $M_2$ is the union of two topologically hyperbolic saddle fixed points and $M_3$ is the union of the three repelling inner circles of fixed points
A sphere in $\mathbb{R}^3$ embedded in such a way that the height function with respect to some plane has five maxima at height $c$, four saddle critical points at height $b$ and one minimum at height $a$
An isolated non-saddle circle which continues to a family of saddle sets which are contractible
Flow on a double torus having an isolated non-saddle set $K''$ whose region of influence has complexity zero but after an arbitrarily small perturbation becomes topologically equivalent to the flow depicted in Figure 2
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