Robust cycles unfolding from conservative bifocal homoclinic orbits

Dynamical Systems

Volumn 31, Issue 4, 2016, Pages 546-579

  Barrientos, Pablo G ^a   Ibáñez, Santiago ^b   Rodríguez, J Ángel ^b  

^a FLUMINENSE FEDERAL UNIVERSITY   (Brazil)

^b UNIVERSITY OF OVIEDO   (Spain)

Author keywords

Bifocal homoclinic orbits; Hamiltonian vector fields; nilpotent singularities; robust heterodimensional cycles; robust homoclinic tangencies; strong homoclinic bifurcation

Indexed keywords

BIFURCATION (MATHEMATICS); HAMILTONIANS;

HAMILTONIAN VECTOR FIELDS; HETERODIMENSIONAL CYCLES; HOMOCLINIC BIFURCATIONS; HOMOCLINIC ORBITS; HOMOCLINIC TANGENCIES; NILPOTENT;

DYNAMICAL SYSTEMS;

EID: 84995955723     PISSN: 14689367     EISSN: 14689375     Source Type: Journal    
DOI: 10.1080/14689367.2016.1170763     Document Type: Article

References (61)

1. Anosov DV. Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat Inst Steklov. 1967;90:3–210.
2. Smale S. Differentiable dynamical systems. Bull Amer Math Soc. 1967;73:747–817.
3. 1 stability conjecture. Publications Mathématiques de l’Institut des Hautes Études Scientifiques. 1987;66(1):161–210. Available from:http://dx.doi.org/10.1007/BF02698931
4. Moreira CG. There are no C1-stable intersections of regular Cantor sets. Acta Math. 2011;206(2):311–323.
5. Mora L, Viana M. Abundance of strange attractors. Acta Math. 1993;171(1):1–71. Available from:http://dx.doi.org/10.1007/BF02392766.
6. 2. Proc Amer Math Soc Sympos Pure Math. 1970;14:191–202.
7. Newhouse SE. Diffeomorphisms with infinitely many sinks. Topology. 1974;13:9–18.
8. Colli E. Infinitely many coexisting strange attractors. Ann Inst H Poincaré Anal Non Linéaire. 1998;15(5):539–579. Available from:http://dx.doi.org/10.1016/S0294-1449(98)80001-2.
9. Palis J, Viana M. High dimension diffeomorphisms displaying infinitely many periodic attractors. Ann of Math (2). 1994;140(1):207–250. Available from:http://dx.doi.org/10.2307/2118546.
10. Viana M. Strange attractors in higher dimensions. Bol Soc Brasil Mat (NS). 1993;24(1):13–62. Available from:http://dx.doi.org/10.1007/BF01231695.
11. Leal B. High dimension diffeomorphisms exhibiting infinitely many strange attractors. Ann Inst H Poincaré Anal Non Linéaire. 2008;25(3):587–607. Available from:http://dx.doi.org/10.1016/j.anihpc.2007.03.002.
12. Abraham R, Smale S. Nongenericity of Ω-stability. In:Global analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968). Providence, R.I.:Amer. Math. Soc.; 1970. p. 5–8.
13. Simon CP. A 3-dimensional Abraham-Smale example. Proc Amer Math Soc. 1972;34:629–630.
14. Bonatti C, Díaz LJ. Persistent nonhyperbolic transitive diffeomorphisms. Ann Math (2). 1996;143(2):357–396.
15. Nassiri M, Pujals ER. Robust transitivity in hamiltonian dynamics. Ann Sci Éc Norm Supér. 2012;45(2):191–239.
16. Alligood KT, Sander E, Yorke JA. Crossing bifurcations and unstable dimension variability. Phys Rev Lett. 2006 Jun;96:244103. Available from:http://link.aps.org/doi/10.1103/PhysRevLett.96.244103
17. Turaev DV, Shil'nikov LP. An example of a wild strange attractor. Mat Sb. 1998;189(2):137–160. Available from:http://dx.doi.org/10.1070/SM1998v189n02ABEH000300
18. Gonchenko SV, Ovsyannikov II, Simó C, Turaev D. Three-dimensional Hénon-like maps and wild Lorenz-like attractors. Int J Bifur Chaos Appl Sci Engrg. 2005;15(11):3493–3508; Available from:http://dx.doi.org/10.1142/S0218127405014180
19. Li D, Turaev DV. Existence of heterodimensional cycles near Shilnikov loops, ArXiv 2015. Available from:http://arxiv.org/abs/1512.01280
20. Hittmeyer S, Krauskopf B, Osinga HM. From wild Lorenz-like to wild Rovella-like dynamics. Dyn Syst. 2015;30(4):525–542. Available from:http://dx.doi.org/10.1080/14689367.2015.1081677
21. Ibáñez S, Rodríguez JA. Shil′nikov configurations in any generic unfolding of the nilpotent singularity of codimension three on. J Differ Equ. 2005;208(1):147–175. Available from:http://dx.doi.org/10.1016/j.jde.2003.08.006
22. Barrientos PG, Ibáñez S, Rodríguez JÁ. Heteroclinic cycles arising in generic unfoldings of nilpotent singularities. J Dynam Differ Equ. 2011;23(4):999–1028. Available from:http://dx.doi.org/10.1007/s10884-011-9230-5
23. Pumariño A, Rodríguez JA. Coexistence and persistence of strange attractors. Vol. 1658 of Lecture Notes in Mathematics. Berlin:Springer-Verlag; 1997.
24. Dumortier F, Ibáñez S, Kokubu H, et al. About the unfolding of a Hopf-zero singularity. Discrete Contin Dyn Syst. 2013;33(10):4435–4471. Available from:http://dx.doi.org/10.3934/dcds.2013.33.4435
25. Drubi F, Ibáñez S, Rodríguez JÁ. Coupling leads to chaos. J Differ Equ. 2007;239(2):371–385. Available from:http://dx.doi.org/10.1016/j.jde.2007.05.024
26. Shil'nikov LP. Existence of a denumerable set of periodic motions in a four dimensional space in an extended neighbourhood of a saddle-focus. Soviet Math Dokl. 1967;8:54–58.
27. Fowler AC, Sparrow CT. Bifocal homoclinic orbits in four dimensions. Nonlinearity. 1991;4(4):1159–1182. Available from:http://stacks.iop.org/0951-7715/4/1159
28. Ibáñez S, Rodrigues A. On the dynamics near a homoclinic network to a bifocus:switching and horseshoes. Int J Bifur Chaos Appl Sci Engrg. 2015;25(11):1530030. Available from:http://dx.doi.org/10.1142/S021812741530030X
29. Devaney RL. Homoclinic orbits in Hamiltonian systems. J Differ. Equ. 1976;21(2):431–438.
30. Lerman LM. Complex dynamics and bifurcations in a Hamiltonian system having a transversal homoclinic orbit to a saddle focus. Chaos. 1991;1(2):174–180; Available from:http://dx.doi.org/10.1063/1.165859
31. Lerman LM. Homo- and heteroclinic orbits, hyperbolic subsets in a one-parameter unfolding of a Hamiltonian system with heteroclinic contour with two saddle-foci. Regul Khaoticheskaya Din. 1997;2(3–4):139–155; V. I. Arnol′d (on the occasion of his 60th birthday) Russian.
32. Lerman LM. Dynamical phenomena near a saddle-focus homoclinic connection in a Hamiltonian system. J Stat Phys. 2000;101(1–2):357–372; dedicated to Grégoire Nicolis on the occasion of his sixtieth birthday (Brussels, 1999). Available from:http://dx.doi.org/10.1023/A:1026411506781
33. 1-generic dynamics. J Inst Math Jussieu. 2008;7(3):469–525. Available from:http://dx.doi.org/10.1017/S1474748008000030
34. 1-homoclinic tangencies. Trans Amer Math Soc. 2012;264:5111–5148.
35. Härterich J. Cascades of reversible homoclinic orbits to a saddle-focus equilibrium. Phys D. 1998;112(1–2):187–200.
36. Hirsch MW, Pugh CC, Shub M. Invariant manifolds. Lecture notes in mathematics, Vol. 583. Berlin:Springer-Verlag; 1977.
37. Bonatti C, Díaz LJ, Kiriki S. Stabilization of heterodimensional cycles. Nonlinearity. 2012;25(4):931–960. Available from:http://stacks.iop.org/0951-7715/25/i=4/a=931
38. Barrientos PG, Ki Y, Raibekas A. Symbolic blender-horseshoes and applications. Nonlinearity. 2014;27(12):2805–2839. Available from:http://stacks.iop.org/0951-7715/27/i=12/a=2805
39. Bonatti C, Díaz LJ, Viana M. Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective. Vol. 102, Encyclopaedia of Mathematical Sciences. Mathematical Physics III. Berlin:Springer-Verlag; 2005.
40. Barrientos PG, Raibekas A. Robust cycles and tangencies of large codimension, ArXiv 2015. Available from:http://arxiv.org/abs/1509.05325
41. Moser J. On the generalization of a theorem of A. Liapounoff. Comm Pure Appl Math. 1958;11:257–271.
42. Lyčagin VV. Sufficient orbits of a group of contact diffeomorphisms. Math USSR Sbornik. 1977;33(2):223–242.
43. Banyaga A, de la Llave R, Wayne CE. Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem. J Geom Anal. 1996;6(4):613–649 (1997). Available from:http://dx.doi.org/10.1007/BF02921624
44. Bronstein IU, Kopanskii AY. Normal forms of vector fields satisfying certain geometric conditions. In:Broer HW, Hoveijn I, Takens F, van Gils SA, editors. Nonlinear dynamical systems and chaos (Groningen, 1995). Vol. 19 of Progress in Nonlinear Differential Equations and Applications; Basel:Birkhäuser; 1996. p. 79–101.
45. Duarte P. Plenty of elliptic islands for the standard family of area preserving maps. Ann Inst H Poincaré Anal Non Linéaire. 1994;11(4):359–409.
46. Katok A, Hasselblatt B. Introduction to the modern theory of dynamical systems. Vol. 54, Encyclopedia of Mathematics and its Applications. Cambridge:Cambridge University Press; 1995; with a supplementary chapter by Katok and Leonardo Mendoza. Available from:http://dx.doi.org/10.1017/CBO9780511809187
47. Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Vol. 42, Applied Mathematical Sciences. New York, NY:Springer-Verlag; 1990.
48. Belyakov LA. Bifurcations of systems with a homoclinic curve of the saddle-focus with a zero saddle value. Mat Zametki. 1984;36(5):681–689.
49. Yorke JA, Alligood KT. Cascades of period-doubling bifurcations:a prerequisite for horseshoes. Bull Amer Math Soc (NS). 1983;9(3):319–322. Available from:http://dx.doi.org/10.1090/S0273-0979-1983-15191-1
50. Gonchenko S, Shilnikov L. On two-dimensional area-preserving maps with homoclinic tangencies that have infinitely many generic elliptic periodic points. J Math Sci. 2005;128(2):2767–2773. Available from:http://dx.doi.org/10.1007/s10958-005-0228-6
51. Delshams A, Gonchenko M, Gonchenko S. On dynamics and bifurcations of area-preserving maps with homoclinic tangencies. 2014. Available from:http://arxiv.org/abs/1407.5473.
52. Amick CJ, Toland JF. Homoclinic orbits in the dynamic phase-space analogy of an elastic strut. Eur J Appl Math. 1992;3(2):97–114. Available from:http://dx.doi.org/10.1017/S0956792500000735
53. Champneys AR, Toland JF. Bifurcation of a plethora of multi-modal homoclinic orbits for autonomous Hamiltonian systems. Nonlinearity. 1993;6(5):665–721. Available from:http://stacks.iop.org/0951-7715/6/665
54. Buffoni B, Champneys AR, Toland JF. Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system. J Dynam Differ Equ. 1996;8(2):221–279; Available from:http://dx.doi.org/10.1007/BF02218892
55. Buffoni B. Periodic and homoclinic orbits for Lorentz-Lagrangian systems via variational methods. Nonlinear Anal. 1996;26(3):443–462. Available from:http://dx.doi.org/10.1016/0362-546X(94)00290-X
56. Champneys AR. Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics. Phys D. 1998;112(1–2):158–186 Available from:http://dx.doi.org/10.1016/S0167-2789(97)00209-1
57. Hofer H, Wysocki K. First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. Ann. 1990;288(3):483–503. Available from:http://dx.doi.org/10.1007/BF01444543
58. Zhang W, Krauskopf B, Kirk V. How to find a codimension-one heteroclinic cycle between two periodic orbits. Discrete Contin Dyn Syst. 2012;32(8):2825–2851. Available from:http://dx.doi.org/10.3934/dcds.2012.32.2825
59. Pumariño A, Tatjer JC. Dynamics near homoclinic bifurcations of three-dimensional dissipative diffeomorphisms. Nonlinearity. 2006;19(12):2833–2852. Available from:http://dx.doi.org/10.1088/0951-7715/19/12/006
60. Pumariño A, Tatjer JC. Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphisms. Discrete Contin Dyn Syst Ser B. 2007;8(4):971–1005 (electronic). Available from:http://dx.doi.org/10.3934/dcdsb.2007.8.971
61. Pumariño A, Rodríguez JÁ, Tatjer JC, Vigil E. Chaotic dynamics for two-dimensional tent maps. Nonlinearity. 2015;28(2):407–434. Available from:http://dx.doi.org/10.1088/0951-7715/28/2/407