# 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
##### References:

show all references

##### References:
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
 [1] Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054 [2] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 [3] Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004 [4] Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139 [5] Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354 [6] Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088 [7] Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053 [8] Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020404 [9] Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021010 [10] Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176 [11] Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108 [12] Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169 [13] Sumit Kumar Debnath, Pantelimon Stǎnicǎ, Nibedita Kundu, Tanmay Choudhury. Secure and efficient multiparty private set intersection cardinality. Advances in Mathematics of Communications, 2021, 15 (2) : 365-386. doi: 10.3934/amc.2020071 [14] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [15] Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 [16] Balázs Kósa, Karol Mikula, Markjoe Olunna Uba, Antonia Weberling, Neophytos Christodoulou, Magdalena Zernicka-Goetz. 3D image segmentation supported by a point cloud. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 971-985. doi: 10.3934/dcdss.2020351 [17] Jérôme Lohéac, Chaouki N. E. Boultifat, Philippe Chevrel, Mohamed Yagoubi. Exact noise cancellation for 1d-acoustic propagation systems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020055 [18] Waixiang Cao, Lueling Jia, Zhimin Zhang. A $C^1$ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327 [19] Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102 [20] Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $p$-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293

2019 Impact Factor: 1.338

## Tools

Article outline

Figures and Tables