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

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

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.
Citation: Mario E. Chávez-Gordillo, Bernardo San Martín, Jaime Vera. Persistent singular attractors arising from singular cycle under symmetric conditions. Discrete and 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.

[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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