Persistent singular attractors arising from singular cycle under symmetric conditions

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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

Abstract
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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 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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