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Persistent singular attractors arising from singular cycle under symmetric conditions
1. | Departamento de Matemáticas, Universidad Católica del Norte, Av. Angamos 0610, Casilla 1280, Antofagasta, Chile, Chile |
2. | The Boeing Company, P.O.Box 3707 MC OX-CC Seattle, WA 98124-2207, United States |
References:
[1] |
M. Benedicks and L. Carleson, On iterations of $\ 1-ax^{2}$ on $(-1,1)$, Annals of Math. (2), 122 (1985), 1-25.
doi: 10.2307/1971367. |
[2] |
M. Benedicks and L. Carleson, The dynamics of the Hénon map, Annals of Math. (2), 133 (1991), 73-169.
doi: 10.2307/2944326. |
[3] |
R. Bamón, R. Labarca, R. Ma né and M. J. Pacífico, The explosion of singular cycles, Publications Mathématiques, I. H. E. S., 78 (1993), 207-232.
doi: 10.1007/BF02712919. |
[4] |
C. Bonatti, A. Pumariño and M. Viana, Lorenz attractors with arbitrary expanding dimension, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 883-888.
doi: 10.1016/S0764-4442(97)80131-0. |
[5] |
J. Guckenheimer and A. Williams, Structural stability of Lorenz attractors, Publ. IHES, 50 (1979), 59-72.
doi: 10.1007/BF02684769. |
[6] |
R. Labarca, Bifurcation of contracting singular cycles, Ann. Scient. Ec. Norm. Sup. 4e Série, 28 (1995), 705-745. |
[7] |
R. Labarca and M. J. Pacífico, Stability of singular horse-shoe, Topology, 25 (1986), 337-352.
doi: 10.1016/0040-9383(86)90048-0. |
[8] |
W. de Melo and S. Van Strien, "One-Dimensional Dynamics," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 25, Springer-Verlag, Berlin, (1993). |
[9] |
R. 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. |
[10] |
C. Morales, M. J. Pacifico and B. San Martín, Expanding Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. Anal., 36 (2005), 1836-1861 (electronic).
doi: 10.1137/S0036141002415785. |
[11] |
C. Morales, M. J. Pacifico and B. San Martín, Contracting Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. Anal., 38 (2006), 309-332 (electronic).
doi: 10.1137/S0036141004443907. |
[12] |
E. Muñoz, B. San Martín and J. Vera, Nonhyperbolic persistent attractors near the Morse-Smale boundary, Ann. I. H. Poincar\'e-AN, 20 (2003), 867-888. |
[13] |
L. Mora and M. Viana, Abundance of strange attractors, Acta Math., 171 (1993), 1-71.
doi: 10.1007/BF02392766. |
[14] |
M. J. Pacífico and A. Rovella, Unfolding contracting singular cycles, Ann. Scient. Éc. Norm. Sup, 4e. Série, 26 (1993), 691-700. |
[15] |
J. Palis, On Morse-Smale dynamical systems, Topology, 8 (1968), 385-404.
doi: 10.1016/0040-9383(69)90024-X. |
[16] |
J. Palis and S. Smale, Structural stability theorems, Global Analysis, Proc. Symp. in Pure Math., Vol XIV, American Mathematical Society, (1970), 223-231. |
[17] |
J. Palis and F. Takens, "Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors," Cambridge Studies in Advanced Mathematics, 35, Cambridge University Press, Cambridge, 1993. |
[18] |
M. R. Rychlik, Lorenz attractors through Shilnikov-type bifurcation, Part I, Ergod. Th. & Dynam. Syst., 10 (1990), 793-821.
doi: 10.1017/S0143385700005915. |
[19] |
A. Rovella, The dynamics of perturbations of the contracting Lorenz attractor, Bol. Soc. Bras. Mat., 24 (1993), 233-259.
doi: 10.1007/BF01237679. |
[20] |
C. Robinson, Nonsymmetric Lorenz attractors from a homoclinic bifurcation, SIAM J. Math. Anal., 32 (2000), 119-141 (electronic).
doi: 10.1137/S0036141098343598. |
[21] |
C. Robinson, Homoclinic bifurcation to a transitive attractor of Lorenz type II, SIAM J. Math. Anal., 23 (1992), 1255-1268.
doi: 10.1137/0523070. |
[22] |
C. Robinson, Homoclinic bifurcation to a transitive attractor of Lorenz type, Nonliniarity, 2 (1989), 495-518.
doi: 10.1088/0951-7715/2/4/001. |
[23] |
B. San Martín, Contracting singular cycles, Ann. Inst. Henri Poincaré, 15 (1998), 651-659.
doi: 10.1016/S0294-1449(98)80004-8. |
[24] |
B. San Martín, Saddle-focus singular cycles y prevalence of hyperbolicity, Ann. Inst. Henri Poincaré, 15 (1998), 623-649.
doi: 10.1016/S0294-1449(98)80003-6. |
[25] |
M. Shub, "Global Stability of Dynamical Systems," Springer-Verlag, New York, 1987. |
[26] |
M. Viana, "Stochastic Dynamics of Deterministic Systems," Lecture Notes XXI Braz. Math. Colloq., IMPA, Rio de Janeiro, 1997. |
show all references
References:
[1] |
M. Benedicks and L. Carleson, On iterations of $\ 1-ax^{2}$ on $(-1,1)$, Annals of Math. (2), 122 (1985), 1-25.
doi: 10.2307/1971367. |
[2] |
M. Benedicks and L. Carleson, The dynamics of the Hénon map, Annals of Math. (2), 133 (1991), 73-169.
doi: 10.2307/2944326. |
[3] |
R. Bamón, R. Labarca, R. Ma né and M. J. Pacífico, The explosion of singular cycles, Publications Mathématiques, I. H. E. S., 78 (1993), 207-232.
doi: 10.1007/BF02712919. |
[4] |
C. Bonatti, A. Pumariño and M. Viana, Lorenz attractors with arbitrary expanding dimension, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 883-888.
doi: 10.1016/S0764-4442(97)80131-0. |
[5] |
J. Guckenheimer and A. Williams, Structural stability of Lorenz attractors, Publ. IHES, 50 (1979), 59-72.
doi: 10.1007/BF02684769. |
[6] |
R. Labarca, Bifurcation of contracting singular cycles, Ann. Scient. Ec. Norm. Sup. 4e Série, 28 (1995), 705-745. |
[7] |
R. Labarca and M. J. Pacífico, Stability of singular horse-shoe, Topology, 25 (1986), 337-352.
doi: 10.1016/0040-9383(86)90048-0. |
[8] |
W. de Melo and S. Van Strien, "One-Dimensional Dynamics," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 25, Springer-Verlag, Berlin, (1993). |
[9] |
R. 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. |
[10] |
C. Morales, M. J. Pacifico and B. San Martín, Expanding Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. Anal., 36 (2005), 1836-1861 (electronic).
doi: 10.1137/S0036141002415785. |
[11] |
C. Morales, M. J. Pacifico and B. San Martín, Contracting Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. Anal., 38 (2006), 309-332 (electronic).
doi: 10.1137/S0036141004443907. |
[12] |
E. Muñoz, B. San Martín and J. Vera, Nonhyperbolic persistent attractors near the Morse-Smale boundary, Ann. I. H. Poincar\'e-AN, 20 (2003), 867-888. |
[13] |
L. Mora and M. Viana, Abundance of strange attractors, Acta Math., 171 (1993), 1-71.
doi: 10.1007/BF02392766. |
[14] |
M. J. Pacífico and A. Rovella, Unfolding contracting singular cycles, Ann. Scient. Éc. Norm. Sup, 4e. Série, 26 (1993), 691-700. |
[15] |
J. Palis, On Morse-Smale dynamical systems, Topology, 8 (1968), 385-404.
doi: 10.1016/0040-9383(69)90024-X. |
[16] |
J. Palis and S. Smale, Structural stability theorems, Global Analysis, Proc. Symp. in Pure Math., Vol XIV, American Mathematical Society, (1970), 223-231. |
[17] |
J. Palis and F. Takens, "Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors," Cambridge Studies in Advanced Mathematics, 35, Cambridge University Press, Cambridge, 1993. |
[18] |
M. R. Rychlik, Lorenz attractors through Shilnikov-type bifurcation, Part I, Ergod. Th. & Dynam. Syst., 10 (1990), 793-821.
doi: 10.1017/S0143385700005915. |
[19] |
A. Rovella, The dynamics of perturbations of the contracting Lorenz attractor, Bol. Soc. Bras. Mat., 24 (1993), 233-259.
doi: 10.1007/BF01237679. |
[20] |
C. Robinson, Nonsymmetric Lorenz attractors from a homoclinic bifurcation, SIAM J. Math. Anal., 32 (2000), 119-141 (electronic).
doi: 10.1137/S0036141098343598. |
[21] |
C. Robinson, Homoclinic bifurcation to a transitive attractor of Lorenz type II, SIAM J. Math. Anal., 23 (1992), 1255-1268.
doi: 10.1137/0523070. |
[22] |
C. Robinson, Homoclinic bifurcation to a transitive attractor of Lorenz type, Nonliniarity, 2 (1989), 495-518.
doi: 10.1088/0951-7715/2/4/001. |
[23] |
B. San Martín, Contracting singular cycles, Ann. Inst. Henri Poincaré, 15 (1998), 651-659.
doi: 10.1016/S0294-1449(98)80004-8. |
[24] |
B. San Martín, Saddle-focus singular cycles y prevalence of hyperbolicity, Ann. Inst. Henri Poincaré, 15 (1998), 623-649.
doi: 10.1016/S0294-1449(98)80003-6. |
[25] |
M. Shub, "Global Stability of Dynamical Systems," Springer-Verlag, New York, 1987. |
[26] |
M. Viana, "Stochastic Dynamics of Deterministic Systems," Lecture Notes XXI Braz. Math. Colloq., IMPA, Rio de Janeiro, 1997. |
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