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On the oscillation behavior of solutions to the one-dimensional heat equation
July
2019, 39(7): 4057-4071.
doi: 10.3934/dcds.2019163
Asymptotically sectional-hyperbolic attractors
Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile |
The notion of asymptotically sectional-hyperbolic set was recently introduced. The main feature is that any point outside of the stable manifolds of its singularities has arbitrarily large hyperbolic times. In this paper we prove the existence, on any three-dimensional Riemannian manifold, of attractors with Rovella-like singularities satisfying this kind of hyperbolicity. Furthermore, we prove that asymptotically sectional-hyperbolic Lyapunov-stable sets, under certain conditions, have positive topological entropy.
Keywords:
Flow,
sectional-hyperbolic,
asymptotically sectional-hyperbolic,
attractors,
topological entropy.
Mathematics Subject Classification:
Primary: 54H20, 37B25; Secondary: 37B40.
Citation:
B. San Martín, Kendry J. Vivas. Asymptotically sectional-hyperbolic attractors. Discrete & Continuous Dynamical Systems - A,
2019, 39
(7)
: 4057-4071.
doi: 10.3934/dcds.2019163
![]()
References:
| [1] |
V. S. Afraimovic, V. V. Bykov and L. P. Shilnikov,
The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR, 234 (1977), 336-339.
|
| [2] |
A. Arbieto, A. M. Lopez and C. A. Morales,
Homoclinic classes for sectional-hyperbolic sets, Kyoto J. Math, 56 (2016), 531-538.
doi: 10.1215/21562261-3600157. |
| [3] |
S. Bautista,
The geometric Lorenz attractor is a homoclinic class, Bol. Mat., 11 (2004), 69-78.
|
| [4] |
S. Bautista and C. A. Morales,
On the intersection of sectional-hyperbolic sets, J. Mod. Dyn., 9 (2015), 203-218.
doi: 10.3934/jmd.2015.9.203. |
| [5] |
C. Bonatti, A. Pumariño and M. Viana, Lorenz attractors with arbitrary expanding dimension, International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), 39–44, World Sci. Publ., River Edge, NJ, 2000.
doi: 10.1016/S0764-4442(97)80131-0. |
| [6] |
J. Carmona, D. Carrasco-Olivera and B. San Martín,
On the $C^1$ robust transitivity of the geometric Lorenz attractor, J. Differential Equations, 262 (2017), 5928-5938.
|
| [7] |
E. Catsigeras, M. Cerminara and H. Enrich,
The Pesin entropy formula for $C^1$ diffeomorphisms with dominated splitting, Ergodic Theory Dynam. Systems, 35 (2015), 737-761.
doi: 10.1017/etds.2013.93. |
| [8] |
S. Gan and L. Wen,
Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math., 164 (2006), 279-315.
doi: 10.1007/s00222-005-0479-3. |
| [9] |
S. Gähler,
Lineare 2-normierte Räume, Math. Nachr., 28 (1964), 1-43.
doi: 10.1002/mana.19640280102. |
| [10] |
J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, (1976), 368–381.
doi: 10.1007/978-1-4612-6374-6_25. |
| [11] |
J. Guckenheimer and R. F. Williams,
Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72.
|
| [12] |
A. Kawaguchi,
On areal spaces. I. Metric tensors in $n$-dimensional spaces based on the notion of two-dimensional area, Tensor N.S., 1 (1950), 14-45.
|
| [13] |
R. Labarca and M. J. Pacifico,
Stability of singularity horseshoes, Topology, 25 (1986), 337-352.
doi: 10.1016/0040-9383(86)90048-0. |
| [14] |
E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), 130-141. |
| [15] |
R. J. Metzger,
Stochastic stability for contracting Lorenz maps and flows, Comm. Math. Phys., 212 (2000), 277-296.
doi: 10.1007/s002200000220. |
| [16] |
R. J. Metzger,
Sinai-Ruelle-Bowen measures for contracting Lorenz maps and flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 247-276.
doi: 10.1016/S0294-1449(00)00111-6. |
| [17] |
R. J. Metzger and C. A. Morales,
The Rovella attractor is a homoclinic class, Bull. Braz. Math. Soc., 37 (2006), 89-101.
doi: 10.1007/s00574-006-0005-2. |
| [18] |
R. J. Metzger and C. A. Morales,
Sectional-hyperbolic systems, Ergodic Theory Dynam. Systems, 28 (2008), 1587-1597.
doi: 10.1017/S0143385707000995. |
| [19] |
C. A. Morales, M. J. Pacifico and and E. R. Pujals,
Singular hyperbolic systems, Proc. Amer. Math. Soc., 127 (1999), 3393-3401.
doi: 10.1090/S0002-9939-99-04936-9. |
| [20] |
C. A. Morales, M. J. Pacifico and B. San Martín,
Contracting Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. Anal., 38 (2006), 309-332.
doi: 10.1137/S0036141004443907. |
| [21] |
C. A. Morales and B. San Martín,
Contracting singular horseshoe, Nonlinearity, 30 (2017), 4208-4219.
doi: 10.1088/1361-6544/aa864e. |
| [22] |
C. A. Morales and M. Vilches,
On 2-Riemannian manifolds, SUT J. Math., 46 (2010), 119-153.
|
| [23] |
E. M. Muñoz, B. San Martín and J. A. Vera,
Nonhyperbolic persistent attractors near the Morse-Smale boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 867-888.
doi: 10.1016/S0294-1449(03)00015-5. |
| [24] |
M. J. Pacifico and M. Todd,
Thermodynamic formalism for contracting Lorenz flows, J. Stat. Phys., 139 (2010), 159-176.
doi: 10.1007/s10955-010-9939-2. |
| [25] |
R. C. Robinson, An Introduction to Dynamical Systems: Continuous and Discrete, 2$^nd$ edition, American Mathematical Society, Providence, 2012. |
| [26] |
A. Rovella,
The dynamics of perturbations of the contracting Lorenz attractor, Bol. Soc. Brasil. Mat., 24 (1993), 233-259.
doi: 10.1007/BF01237679. |
| [27] |
D. V. Turaev and L. P. Shilnikov,
An example of a wild strange attractor, Sb. Math., 189 (1998), 291-314.
doi: 10.1070/SM1998v189n02ABEH000300. |
| [28] |
R. F. Williams,
The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 73-99.
|
show all references
References:
| [1] |
V. S. Afraimovic, V. V. Bykov and L. P. Shilnikov,
The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR, 234 (1977), 336-339.
|
| [2] |
A. Arbieto, A. M. Lopez and C. A. Morales,
Homoclinic classes for sectional-hyperbolic sets, Kyoto J. Math, 56 (2016), 531-538.
doi: 10.1215/21562261-3600157. |
| [3] |
S. Bautista,
The geometric Lorenz attractor is a homoclinic class, Bol. Mat., 11 (2004), 69-78.
|
| [4] |
S. Bautista and C. A. Morales,
On the intersection of sectional-hyperbolic sets, J. Mod. Dyn., 9 (2015), 203-218.
doi: 10.3934/jmd.2015.9.203. |
| [5] |
C. Bonatti, A. Pumariño and M. Viana, Lorenz attractors with arbitrary expanding dimension, International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), 39–44, World Sci. Publ., River Edge, NJ, 2000.
doi: 10.1016/S0764-4442(97)80131-0. |
| [6] |
J. Carmona, D. Carrasco-Olivera and B. San Martín,
On the $C^1$ robust transitivity of the geometric Lorenz attractor, J. Differential Equations, 262 (2017), 5928-5938.
|
| [7] |
E. Catsigeras, M. Cerminara and H. Enrich,
The Pesin entropy formula for $C^1$ diffeomorphisms with dominated splitting, Ergodic Theory Dynam. Systems, 35 (2015), 737-761.
doi: 10.1017/etds.2013.93. |
| [8] |
S. Gan and L. Wen,
Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math., 164 (2006), 279-315.
doi: 10.1007/s00222-005-0479-3. |
| [9] |
S. Gähler,
Lineare 2-normierte Räume, Math. Nachr., 28 (1964), 1-43.
doi: 10.1002/mana.19640280102. |
| [10] |
J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, (1976), 368–381.
doi: 10.1007/978-1-4612-6374-6_25. |
| [11] |
J. Guckenheimer and R. F. Williams,
Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72.
|
| [12] |
A. Kawaguchi,
On areal spaces. I. Metric tensors in $n$-dimensional spaces based on the notion of two-dimensional area, Tensor N.S., 1 (1950), 14-45.
|
| [13] |
R. Labarca and M. J. Pacifico,
Stability of singularity horseshoes, Topology, 25 (1986), 337-352.
doi: 10.1016/0040-9383(86)90048-0. |
| [14] |
E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), 130-141. |
| [15] |
R. J. Metzger,
Stochastic stability for contracting Lorenz maps and flows, Comm. Math. Phys., 212 (2000), 277-296.
doi: 10.1007/s002200000220. |
| [16] |
R. J. Metzger,
Sinai-Ruelle-Bowen measures for contracting Lorenz maps and flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 247-276.
doi: 10.1016/S0294-1449(00)00111-6. |
| [17] |
R. J. Metzger and C. A. Morales,
The Rovella attractor is a homoclinic class, Bull. Braz. Math. Soc., 37 (2006), 89-101.
doi: 10.1007/s00574-006-0005-2. |
| [18] |
R. J. Metzger and C. A. Morales,
Sectional-hyperbolic systems, Ergodic Theory Dynam. Systems, 28 (2008), 1587-1597.
doi: 10.1017/S0143385707000995. |
| [19] |
C. A. Morales, M. J. Pacifico and and E. R. Pujals,
Singular hyperbolic systems, Proc. Amer. Math. Soc., 127 (1999), 3393-3401.
doi: 10.1090/S0002-9939-99-04936-9. |
| [20] |
C. A. Morales, M. J. Pacifico and B. San Martín,
Contracting Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. Anal., 38 (2006), 309-332.
doi: 10.1137/S0036141004443907. |
| [21] |
C. A. Morales and B. San Martín,
Contracting singular horseshoe, Nonlinearity, 30 (2017), 4208-4219.
doi: 10.1088/1361-6544/aa864e. |
| [22] |
C. A. Morales and M. Vilches,
On 2-Riemannian manifolds, SUT J. Math., 46 (2010), 119-153.
|
| [23] |
E. M. Muñoz, B. San Martín and J. A. Vera,
Nonhyperbolic persistent attractors near the Morse-Smale boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 867-888.
doi: 10.1016/S0294-1449(03)00015-5. |
| [24] |
M. J. Pacifico and M. Todd,
Thermodynamic formalism for contracting Lorenz flows, J. Stat. Phys., 139 (2010), 159-176.
doi: 10.1007/s10955-010-9939-2. |
| [25] |
R. C. Robinson, An Introduction to Dynamical Systems: Continuous and Discrete, 2$^nd$ edition, American Mathematical Society, Providence, 2012. |
| [26] |
A. Rovella,
The dynamics of perturbations of the contracting Lorenz attractor, Bol. Soc. Brasil. Mat., 24 (1993), 233-259.
doi: 10.1007/BF01237679. |
| [27] |
D. V. Turaev and L. P. Shilnikov,
An example of a wild strange attractor, Sb. Math., 189 (1998), 291-314.
doi: 10.1070/SM1998v189n02ABEH000300. |
| [28] |
R. F. Williams,
The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 73-99.
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