Topology and dynamics of a flow that has a non-saddle set or a W-set
DOI:
https://doi.org/10.12775/TMNA.2025.024Słowa kluczowe
Asymptotic behaviour, non-saddle sets, W-sets, Conley indexAbstrakt
The aim of this paper is to study dynamical and topological properties of a flow in the region of influence of an isolated non-saddle set or a $W$-set in a manifold. These are certain classes of compact invariant sets in whose vicinity the asymptotic behaviour of the flow is somewhat controlled. We are mainly concerned with global properties of the dynamics and establish cohomological relations between the non-saddle set and the manifold. As a consequence we obtain a dynamical classification of surfaces (orientable and non-orientable). We also examine robustness and bifurcation properties of non-saddle-sets and study in detail the behavior of $W$-sets in 2-manifolds.Bibliografia
K. Athanassopoulos, Explosions near isolated unstable attractors, Pacific J. Math. 210 (2003), no. 2, 201–214.
H. Barge, Čech cohomology, homoclinic trajectories and robustness of non-saddle sets, Discrete Contin. Dyn. Syst. 41 (2021), no. 6, 2677-2698.
H. Barge and J.M.R. Sanjurjo, J. Dynam. Differential Equations 30 (2018), no. 1, 257–272.
H. Barge and J.M.R. Sanjurjo, Dissonant points and the region of influence of nonsaddle sets, J. Differential Equations 268 (2020), no. 9, 5329–5352.
N.P. Bhatia, Attraction and nonsaddle sets in dynamical systems J. Differential Equations 8 (1970), 229–249.
N.P. Bhatia and G.P. Szegö, Stability Theory of Dynamical Systems, Classics in Mathematics, Springer, 2002.
K. Borsuk, Theory of Shape, Monografie Matematyczne, vol. 59, Polish Scientific Publishers, Warsaw, 1975.
W.C. Chewning and R.S. Owen, Local sections of flows on manifolds, Proc. Amer. Math. Soc. 49 (1975), 71–77.
R.C. Churchill, Isolated invariant sets in compact metric spaces, J. Differential Equations 12 (1972), 330-352.
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, RI, 1978.
C. Conley and R.W. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc. 158 (1971), 35–61.
J. Dydak and J. Segal, Shape Theory. An Introduction, Lecture Notes in Mathematics, vol. 688, Springer, 1978.
A. Giraldo, M.A. Morón, F.R. Ruiz del Portal and J.M.R. Sanjurjo, Some duality properties of non-saddle sets, Topology Appl. 113 (2001), no. 1–3, 51–59.
A. Giraldo and J.M.R. Sanjurjo, Topological robustness of non-saddle sets, Topology Appl. 156 (2009), no. 11, 1929–1936.
S. Mardešić and J. Segal, Shape Theory. The Inverse System Approach, North-Holland Mathematical Library, vol. 26, North-Holland Publishing Co., 1982.
M.A. Morón, J.J. Sánchez-Gabites and J.M.R. Sanjurjo, Topology and dynamics of unstable attractors, Fund. Math. 197 (2007), 239-252.
I. Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106 (1963), no. 2, 259–269.
D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), 1–41.
J.M.R. Sanjurjo, On the structure of uniform attractors, J. Math. Anal. Appl. 192 (1995), 519–528.
J.J. Sánchez-Gabites, Unstable attractors in manifolds, Trans. Amer. Math. Soc. 362 (2010), 3563–358
E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
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