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Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case
Čech cohomology, homoclinic trajectories and robustness of non-saddle sets
E.T.S. Ingenieros informáticos, Universidad Politécnica de Madrid, 28660 Madrid, (España), Spain |
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.
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.
|
show all references
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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