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June  2021, 41(6): 2677-2698. 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  June 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (6) : 2677-2698. doi: 10.3934/dcds.2020381
##### References:
 [1] K. Athanassopoulos, Explosions near isolated unstable attractors, Pacific J. Math., 210 (2003), 201-214.  doi: 10.2140/pjm.2003.210.201. [2] K. Athanassopoulos, Remarks on the region of attraction of an isolated invariant set, Colloq. Math., 104 (2006), 157-167.  doi: 10.4064/cm104-2-1. [3] H. Barge, Regular blocks and Conley index of isolated invariant continua in surfaces, Nonlinear Anal., 146 (2016), 100-119.  doi: 10.1016/j.na.2016.08.023. [4] H. Barge and J. M. R. Sanjurjo, Unstable manifold, Conley index and fixed points of flows, J. Math. Anal. Appl., 420 (2014), 835-851.  doi: 10.1016/j.jmaa.2014.06.016. [5] H. Barge and J. M. R. Sanjurjo, Bifurcations and attractor-repeller splittings of non-saddle sets, J. Dyn. Diff. Equat., 30 (2018), 257-272.  doi: 10.1007/s10884-017-9569-3. [6] H. Barge and J. M. R. Sanjurjo, Dissonant points and the region of influence of non-saddle sets, J. Differential Equations, 268 (2020), 5329-5352.  doi: 10.1016/j.jde.2019.11.012. [7] N. P. Bhatia, Attraction and nonsaddle sets in dynamical systems, J. Differential Equations, 8 (1970), 229-249.  doi: 10.1016/0022-0396(70)90003-3. [8] N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems, Classics in Mathematics, Springer-Verlag, Berlin, 2002. [9] K. Borsuk, Theory of Shape, Monografie Matematyczne 59. Polish Scientific Publishers, Warsaw, 1975. [10] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics 38, American Mathematical Society, Providence, R.I., 1978. [11] C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc., 158 (1971), 35-61.  doi: 10.1090/S0002-9947-1971-0279830-1. [12] A. Giraldo, M. A. Morón, F. R. Ruiz del Portal and J. M. R. Sanjurjo, Some duality properties of nonsaddle sets, Topology Appl., 113 (2001), 51-59.  doi: 10.1016/S0166-8641(00)00017-1. [13] A. Giraldo and J. M. R. Sanjurjo, Topological robustness of non-saddle sets, Topology Appl., 156 (2009), 1929-1936.  doi: 10.1016/j.topol.2009.03.020. [14] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. [15] S. Hu, Theory of Retracts, Wayne State University Press, 1965. [16] L. Kapitanski and I. Rodnianski, Shape and Morse theory of attractors, Comm. Pure Appl. Math., 53 (2000), 218-242.  doi: 10.1002/(SICI)1097-0312(200002)53:2<218::AID-CPA2>3.0.CO; 2-W. [17] S. Mardešić, Equivalence of singular and Čech homology for ANR-s. {A}pplication to unicoherence, Fund. Math., 46 (1958), 29-45.  doi: 10.4064/fm-46-1-29-45. [18] S. Mardešić and J. Segal, Shape Theory. The Inverse System Approach, North-Holland Mathematical Library, 26. North-Holland Publishing Co., 1982. [19] M. A. Morón, J. J. Sánchez-Gabites and J. M. R. Sanjurjo, Topology and dynamics of unstable attractors, Fund. Math., 197 (2007), 239-252.  doi: 10.4064/fm197-0-11. [20] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc., 291 (1985), 1-41.  doi: 10.1090/S0002-9947-1985-0797044-3. [21] J. J. Sánchez-Gabites, Dynamical systems and shapes, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 102 (2008), 127-159.  doi: 10.1007/BF03191815. [22] J. J. Sánchez-Gabites, Aplicaciones de Topología Geométrica y Algebraica al Estudio de Flujos Continuos en Variedades, Ph.D thesis, Universidad Complutense de Madrid, 2009. [23] J. J. Sánchez-Gabites, Unstable attractors in manifolds, Trans. Amer. Math. Soc., 362 (2010), 3563-3589.  doi: 10.1090/S0002-9947-10-05061-0. [24] J. J. Sánchez-Gabites, An approach to the shape Conley index without index pairs, Rev. Mat. Complut., 24 (2011), 95-114.  doi: 10.1007/s13163-010-0031-x. [25] J. M. R. Sanjurjo, On the structure of uniform attractors, J. Math. Anal. Appl., 192 (1995), 519-528.  doi: 10.1006/jmaa.1995.1186. [26] J. M. R. Sanjurjo, Stability, attraction and shape: A topological study of flows, in Topological Methods in Nonlinear Analysis, volume 12 of Lect. Notes Nonlinear Anal., Juliusz Schauder Cent. Nonlinear Stud., Toruń, (2011), 93–122. [27] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. [28] T. Ura, On the flow outside a closed invariant set; stability, relative stability and saddle sets, Contrib. Differential Equations, 3 (1963), 249-294.

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##### References:
 [1] K. Athanassopoulos, Explosions near isolated unstable attractors, Pacific J. Math., 210 (2003), 201-214.  doi: 10.2140/pjm.2003.210.201. [2] K. Athanassopoulos, Remarks on the region of attraction of an isolated invariant set, Colloq. Math., 104 (2006), 157-167.  doi: 10.4064/cm104-2-1. [3] H. Barge, Regular blocks and Conley index of isolated invariant continua in surfaces, Nonlinear Anal., 146 (2016), 100-119.  doi: 10.1016/j.na.2016.08.023. [4] H. Barge and J. M. R. Sanjurjo, Unstable manifold, Conley index and fixed points of flows, J. Math. Anal. Appl., 420 (2014), 835-851.  doi: 10.1016/j.jmaa.2014.06.016. [5] H. Barge and J. M. R. Sanjurjo, Bifurcations and attractor-repeller splittings of non-saddle sets, J. Dyn. Diff. Equat., 30 (2018), 257-272.  doi: 10.1007/s10884-017-9569-3. [6] H. Barge and J. M. R. Sanjurjo, Dissonant points and the region of influence of non-saddle sets, J. Differential Equations, 268 (2020), 5329-5352.  doi: 10.1016/j.jde.2019.11.012. [7] N. P. Bhatia, Attraction and nonsaddle sets in dynamical systems, J. Differential Equations, 8 (1970), 229-249.  doi: 10.1016/0022-0396(70)90003-3. [8] N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems, Classics in Mathematics, Springer-Verlag, Berlin, 2002. [9] K. Borsuk, Theory of Shape, Monografie Matematyczne 59. Polish Scientific Publishers, Warsaw, 1975. [10] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics 38, American Mathematical Society, Providence, R.I., 1978. [11] C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc., 158 (1971), 35-61.  doi: 10.1090/S0002-9947-1971-0279830-1. [12] A. Giraldo, M. A. Morón, F. R. Ruiz del Portal and J. M. R. Sanjurjo, Some duality properties of nonsaddle sets, Topology Appl., 113 (2001), 51-59.  doi: 10.1016/S0166-8641(00)00017-1. [13] A. Giraldo and J. M. R. Sanjurjo, Topological robustness of non-saddle sets, Topology Appl., 156 (2009), 1929-1936.  doi: 10.1016/j.topol.2009.03.020. [14] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. [15] S. Hu, Theory of Retracts, Wayne State University Press, 1965. [16] L. Kapitanski and I. Rodnianski, Shape and Morse theory of attractors, Comm. Pure Appl. Math., 53 (2000), 218-242.  doi: 10.1002/(SICI)1097-0312(200002)53:2<218::AID-CPA2>3.0.CO; 2-W. [17] S. Mardešić, Equivalence of singular and Čech homology for ANR-s. {A}pplication to unicoherence, Fund. Math., 46 (1958), 29-45.  doi: 10.4064/fm-46-1-29-45. [18] S. Mardešić and J. Segal, Shape Theory. The Inverse System Approach, North-Holland Mathematical Library, 26. North-Holland Publishing Co., 1982. [19] M. A. Morón, J. J. Sánchez-Gabites and J. M. R. Sanjurjo, Topology and dynamics of unstable attractors, Fund. Math., 197 (2007), 239-252.  doi: 10.4064/fm197-0-11. [20] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc., 291 (1985), 1-41.  doi: 10.1090/S0002-9947-1985-0797044-3. [21] J. J. Sánchez-Gabites, Dynamical systems and shapes, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 102 (2008), 127-159.  doi: 10.1007/BF03191815. [22] J. J. Sánchez-Gabites, Aplicaciones de Topología Geométrica y Algebraica al Estudio de Flujos Continuos en Variedades, Ph.D thesis, Universidad Complutense de Madrid, 2009. [23] J. J. Sánchez-Gabites, Unstable attractors in manifolds, Trans. Amer. Math. Soc., 362 (2010), 3563-3589.  doi: 10.1090/S0002-9947-10-05061-0. [24] J. J. Sánchez-Gabites, An approach to the shape Conley index without index pairs, Rev. Mat. Complut., 24 (2011), 95-114.  doi: 10.1007/s13163-010-0031-x. [25] J. M. R. Sanjurjo, On the structure of uniform attractors, J. Math. Anal. Appl., 192 (1995), 519-528.  doi: 10.1006/jmaa.1995.1186. [26] J. M. R. Sanjurjo, Stability, attraction and shape: A topological study of flows, in Topological Methods in Nonlinear Analysis, volume 12 of Lect. Notes Nonlinear Anal., Juliusz Schauder Cent. Nonlinear Stud., Toruń, (2011), 93–122. [27] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. [28] T. Ura, On the flow outside a closed invariant set; stability, relative stability and saddle sets, Contrib. Differential Equations, 3 (1963), 249-294.
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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