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August  2011, 30(3): 671-685. doi: 10.3934/dcds.2011.30.671

Persistent singular attractors arising from singular cycle under symmetric conditions

Mario E. Chávez-Gordillo 1, , Bernardo San Martín 1, and  Jaime Vera 2,

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

Received  May 2010 Revised  November 2010 Published  March 2011

We prove that generic symmetric $C^{r}$-vector field families on $\mathbb{R}^{3}$ unfolding a contracting singular cycle, exhibits singular attractors for a positive lebesgue measure set of parameter values. Essentially the cycle is formed by a real contracting singularity, like those in the geometric contracting Lorenz attractor, whose unstable branches go to periodic orbits in the cycle. We obtain a lower estimate for the density of this set at the first bifurcation value. Furthermore, for parameter values in this set the corresponding vector field admits a unique SRB measure, whose support coincides with the attractor.
Keywords: singular attractor, Singular cycle, Lorenz attractors., symmetric vector field.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Mario E. Chávez-Gordillo, Bernardo San Martín, Jaime Vera. Persistent singular attractors arising from singular cycle under symmetric conditions. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 671-685. doi: 10.3934/dcds.2011.30.671
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.  Google Scholar

[2]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Annals of Math. (2), 133 (1991), 73-169. doi: 10.2307/2944326.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Guckenheimer and A. Williams, Structural stability of Lorenz attractors, Publ. IHES, 50 (1979), 59-72. doi: 10.1007/BF02684769.  Google Scholar

[6]

R. Labarca, Bifurcation of contracting singular cycles, Ann. Scient. Ec. Norm. Sup. 4e Série, 28 (1995), 705-745.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. Mora and M. Viana, Abundance of strange attractors, Acta Math., 171 (1993), 1-71. doi: 10.1007/BF02392766.  Google Scholar

[14]

M. J. Pacífico and A. Rovella, Unfolding contracting singular cycles, Ann. Scient. Éc. Norm. Sup, 4e. Série, 26 (1993), 691-700.  Google Scholar

[15]

J. Palis, On Morse-Smale dynamical systems, Topology, 8 (1968), 385-404. doi: 10.1016/0040-9383(69)90024-X.  Google Scholar

[16]

J. Palis and S. Smale, Structural stability theorems, Global Analysis, Proc. Symp. in Pure Math., Vol XIV, American Mathematical Society, (1970), 223-231.  Google Scholar

[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.  Google Scholar

[18]

M. R. Rychlik, Lorenz attractors through Shilnikov-type bifurcation, Part I, Ergod. Th. & Dynam. Syst., 10 (1990), 793-821. doi: 10.1017/S0143385700005915.  Google Scholar

[19]

A. Rovella, The dynamics of perturbations of the contracting Lorenz attractor, Bol. Soc. Bras. Mat., 24 (1993), 233-259. doi: 10.1007/BF01237679.  Google Scholar

[20]

C. Robinson, Nonsymmetric Lorenz attractors from a homoclinic bifurcation, SIAM J. Math. Anal., 32 (2000), 119-141 (electronic). doi: 10.1137/S0036141098343598.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

B. San Martín, Contracting singular cycles, Ann. Inst. Henri Poincaré, 15 (1998), 651-659. doi: 10.1016/S0294-1449(98)80004-8.  Google Scholar

[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.  Google Scholar

[25]

M. Shub, "Global Stability of Dynamical Systems," Springer-Verlag, New York, 1987.  Google Scholar

[26]

M. Viana, "Stochastic Dynamics of Deterministic Systems," Lecture Notes XXI Braz. Math. Colloq., IMPA, Rio de Janeiro, 1997. Google Scholar

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.  Google Scholar

[2]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Annals of Math. (2), 133 (1991), 73-169. doi: 10.2307/2944326.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Guckenheimer and A. Williams, Structural stability of Lorenz attractors, Publ. IHES, 50 (1979), 59-72. doi: 10.1007/BF02684769.  Google Scholar

[6]

R. Labarca, Bifurcation of contracting singular cycles, Ann. Scient. Ec. Norm. Sup. 4e Série, 28 (1995), 705-745.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. Mora and M. Viana, Abundance of strange attractors, Acta Math., 171 (1993), 1-71. doi: 10.1007/BF02392766.  Google Scholar

[14]

M. J. Pacífico and A. Rovella, Unfolding contracting singular cycles, Ann. Scient. Éc. Norm. Sup, 4e. Série, 26 (1993), 691-700.  Google Scholar

[15]

J. Palis, On Morse-Smale dynamical systems, Topology, 8 (1968), 385-404. doi: 10.1016/0040-9383(69)90024-X.  Google Scholar

[16]

J. Palis and S. Smale, Structural stability theorems, Global Analysis, Proc. Symp. in Pure Math., Vol XIV, American Mathematical Society, (1970), 223-231.  Google Scholar

[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.  Google Scholar

[18]

M. R. Rychlik, Lorenz attractors through Shilnikov-type bifurcation, Part I, Ergod. Th. & Dynam. Syst., 10 (1990), 793-821. doi: 10.1017/S0143385700005915.  Google Scholar

[19]

A. Rovella, The dynamics of perturbations of the contracting Lorenz attractor, Bol. Soc. Bras. Mat., 24 (1993), 233-259. doi: 10.1007/BF01237679.  Google Scholar

[20]

C. Robinson, Nonsymmetric Lorenz attractors from a homoclinic bifurcation, SIAM J. Math. Anal., 32 (2000), 119-141 (electronic). doi: 10.1137/S0036141098343598.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

B. San Martín, Contracting singular cycles, Ann. Inst. Henri Poincaré, 15 (1998), 651-659. doi: 10.1016/S0294-1449(98)80004-8.  Google Scholar

[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.  Google Scholar

[25]

M. Shub, "Global Stability of Dynamical Systems," Springer-Verlag, New York, 1987.  Google Scholar

[26]

M. Viana, "Stochastic Dynamics of Deterministic Systems," Lecture Notes XXI Braz. Math. Colloq., IMPA, Rio de Janeiro, 1997. Google Scholar

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