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

Héctor Barge

doi:10.3934/dcds.2020381

Article Contents

2021, Volume 41, Issue 6: 2677-2698. Doi: 10.3934/dcds.2020381

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

Héctor Barge^,

E.T.S. Ingenieros informáticos, Universidad Politécnica de Madrid, 28660 Madrid, (España), Spain

^* Corresponding author: Héctor Barge
^* Corresponding author: Héctor Barge

Received: February 2020

Revised: October 2020

Early access: November 2020

Published: June 2021

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

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Abstract

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.

Keywords:

Received February 2020; 1st revision June 2020; final revision October 2020ech cohomology,
non-saddle set,
homoclinic trajectory,
dissonant point,
robustness,
isolating block.

Mathematics Subject Classification: Primary: 37B35, 37B25; Secondary: 37B99.

Citation:

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Figure 2. 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

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Figure 1. 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 $

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Figure 3. Model for a degenerate saddle fixed point

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Figure 4. 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

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Figure 5. 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

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Figure 6. 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 $

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Figure 7. An isolated non-saddle circle which continues to a family of saddle sets which are contractible

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Figure 8. 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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