Simultaneous continuation of infinitely many sinks at homoclinic bifurcations

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July 2011, 29(3): 693-736. doi: 10.3934/dcds.2011.29.693

Simultaneous continuation of infinitely many sinks at homoclinic bifurcations

Eleonora Catsigeras ^1, , Marcelo Cerminara ^2, and Heber Enrich ^3,

1.	Instituto de Matemática y Estadística Prof. Rafael Laguardia (IMERL), Facultad de Ingeniería, Universidad de la República, Uruguay
2.	Instituto de Matemática y Estadistica Prof. Rafael Laguardia (IMERL), Facultad de Ingeniería, Universidad de la República, Uruguay
3.	Instituto de Matemática y Estadistica Prof. Rafael Laguardia (IMERL), Facultad de Ingeniería, Universidad de la República, Montevideo, Uruguay

Received June 2009 Revised August 2010 Published November 2010

Abstract
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We prove that the $C^3$ diffeomorphisms on surfaces, exhibiting infinitely many sinks near the generic unfolding of a quadratic homoclinic tangency of a dissipative saddle, can be perturbed along an infinite dimensional manifold of $C^3$ diffeomorphisms such that infinitely many sinks persist simultaneously. On the other hand, if they are perturbed along one-parameter families that unfold generically the quadratic tangencies, then at most a finite number of those sinks have continuation.

Keywords: Homoclinic Bifurcations., Infinitely many Sinks.

Mathematics Subject Classification: Primary: 37G25, 58F14; Secondary: 37D45, 37C2.

Citation: Eleonora Catsigeras, Marcelo Cerminara, Heber Enrich. Simultaneous continuation of infinitely many sinks at homoclinic bifurcations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 693-736. doi: 10.3934/dcds.2011.29.693

References:

[1]	E. Colli, Infinitely many coexisting strange attractors, Annales de l'I.H.P. Analyse non-linéaire, 15 (1998), 539-579.
[2]	M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
[3]	W. de Melo, Structural stability of diffeomorphisms on two-manifolds, Inventiones Math, 21 (1973), 233-246. doi: 10.1007/BF01390199.
[4]	A. Gorodetski and V. Kaloshin, How often surface diffeomorphisms have inifinitely sinks and hyperbolicity of periodic points near an homoclinic tangency, Adv. Math., 208 (2007), 710-797. doi: 10.1016/j.aim.2006.03.012.
[5]	I. Kan, H. Koçak and J. A. Yorke, Antimonotonicity: Concurrent creation and annihilation of periodic orbits, Ann. Math., 136 (1992), 219-252. doi: 10.2307/2946605.
[6]	S. Newhouse, Non density of Axiom A on $S^2$, Proc. A.M.S. Symp. Pure Math., 14 (1970), 191-202. doi: 10.1016/0040-9383(74)90034-2.
[7]	S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.
[8]	S. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. IHÉS, 50 (1979), 101-151.
[9]	J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors. Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque, 261 (2000), 335-347.
[10]	J. Palis and F. Takens, "Hyperbolicity and Sensitive Chaotic Dynanics of Homoclinic Bifurcations," University Press, Cambridge, 1993.
[11]	C. Robinson, Bifurcation to infinitely many sinks, Comm Math Phys., 90 (1983), 433-459. doi: 10.1007/BF01206892.
[12]	M. Shub, "Global Stability of Dynamical Systems," Springer Verlag, Berlin, New York, (1987), 23-27.
[13]	S. Smale, Diffeomorphisms with many periodic points, in " Differential and Combinatorial Topology," Princeton Univ. Press, (1965), 63-80.
[14]	L. Tedeschini-Lalli and J. A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks?, Comm. Math. Phys., 106 (1986), 635-657. doi: 10.1007/BF01463400.
[15]	J. A. Yorke and K. T. Alligood, Cascades of period doubling bifurcations: A prerequisite for horseshoes, Bull AMS, 9 (1983), 319-322. doi: 10.1090/S0273-0979-1983-15191-1.

show all references

References:

[1]	E. Colli, Infinitely many coexisting strange attractors, Annales de l'I.H.P. Analyse non-linéaire, 15 (1998), 539-579.
[2]	M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
[3]	W. de Melo, Structural stability of diffeomorphisms on two-manifolds, Inventiones Math, 21 (1973), 233-246. doi: 10.1007/BF01390199.
[4]	A. Gorodetski and V. Kaloshin, How often surface diffeomorphisms have inifinitely sinks and hyperbolicity of periodic points near an homoclinic tangency, Adv. Math., 208 (2007), 710-797. doi: 10.1016/j.aim.2006.03.012.
[5]	I. Kan, H. Koçak and J. A. Yorke, Antimonotonicity: Concurrent creation and annihilation of periodic orbits, Ann. Math., 136 (1992), 219-252. doi: 10.2307/2946605.
[6]	S. Newhouse, Non density of Axiom A on $S^2$, Proc. A.M.S. Symp. Pure Math., 14 (1970), 191-202. doi: 10.1016/0040-9383(74)90034-2.
[7]	S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.
[8]	S. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. IHÉS, 50 (1979), 101-151.
[9]	J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors. Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque, 261 (2000), 335-347.
[10]	J. Palis and F. Takens, "Hyperbolicity and Sensitive Chaotic Dynanics of Homoclinic Bifurcations," University Press, Cambridge, 1993.
[11]	C. Robinson, Bifurcation to infinitely many sinks, Comm Math Phys., 90 (1983), 433-459. doi: 10.1007/BF01206892.
[12]	M. Shub, "Global Stability of Dynamical Systems," Springer Verlag, Berlin, New York, (1987), 23-27.
[13]	S. Smale, Diffeomorphisms with many periodic points, in " Differential and Combinatorial Topology," Princeton Univ. Press, (1965), 63-80.
[14]	L. Tedeschini-Lalli and J. A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks?, Comm. Math. Phys., 106 (1986), 635-657. doi: 10.1007/BF01463400.
[15]	J. A. Yorke and K. T. Alligood, Cascades of period doubling bifurcations: A prerequisite for horseshoes, Bull AMS, 9 (1983), 319-322. doi: 10.1090/S0273-0979-1983-15191-1.

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