Tutorials and ReviewsNo Access

BIFURCATIONS AND CHAOS IN HAMILTONIAN SYSTEMS

R. BARRIO

GME, Departamento Matemática Aplicada, Spain

IUMA, Spain

Search for more papers by this author

F. BLESA

GME, Departamento Física Aplicada, Universidad de Zaragoza, E-50009 Zaragoza, Spain

Search for more papers by this author

, and

S. SERRANO

GME, Departamento Matemática Aplicada, Spain

Search for more papers by this author

https://doi.org/10.1142/S0218127410026496Cited by:24 (Source: Crossref)

Abstract

This paper deals with the use of recent computational techniques in the numerical study of qualitative properties of two degrees of freedom of Hamiltonian systems. These numerical methods are based on the computation of the OFLI2 Chaos Indicator, the Crash Test and exit basins and the skeleton of symmetric periodic orbits. As paradigmatic examples, three classical problems are studied: the Copenhagen and the (n + 1)-body ring problems and the Hénón–Heiles Hamiltonian. All the numerical integrations have been done by using the state-of-the-art numerical library TIDES based on the extended Taylor series method.

Keywords:

References

Abad, A., Barrio, R., Blesa, F. & Rodríguez, M. [2009a] "Tides: A Taylor series integrator for differential Equations package," preprint . Google Scholar
Abad, A., Barrio, R., Blesa, F. & Rodríguez, M. [2009b] "TIDES software," http://gme.unizar.es/software/tides . Google Scholar
J. Aguirre, J. C. Vallejo and M. A. F. Sanjuán, Phys. Rev. E 64, 066208 (2001), DOI: 10.1103/PhysRevE.64.066208. Crossref, Web of Science, Google Scholar
J. Aguirre and M. A. F. Sanjuán, Phys. Rev. E 67, 056201 (2003), DOI: 10.1103/PhysRevE.67.056201. Crossref, Web of Science, Google Scholar
J. Aguirre, R. L. Viana and M. A. F. Sanjuán, Rev. Mod. Phys. 81, 333 (2009), DOI: 10.1103/RevModPhys.81.333. Crossref, Web of Science, Google Scholar
M. Arribas and A. Elipe, Mech. Res. Comm. 31, 1 (2004), DOI: 10.1016/S0093-6413(03)00086-7. Crossref, Web of Science, Google Scholar
D. H. Bailey, ACM Trans. Math. Softw. 21, 379 (1995), DOI: 10.1145/212066.212075. Crossref, Web of Science, Google Scholar
R. Barrio, F. Blesa and M. Lara, Comput. Math. Appl. 50, 93 (2005), DOI: 10.1016/j.camwa.2005.02.010. Crossref, Web of Science, Google Scholar
R. Barrio, Appl. Math. Comput. 163, 525 (2005), DOI: 10.1016/j.amc.2004.02.015. Crossref, Web of Science, Google Scholar
R. Barrio, Chaos Solit. Fract. 25, 711 (2005), DOI: 10.1016/j.chaos.2004.11.092. Crossref, Web of Science, Google Scholar
R. Barrio, SIAM J. Sci. Comput. 27, 1929 (2006), DOI: 10.1137/030601892. Crossref, Web of Science, Google Scholar
R. Barrio, Int. J. Bifurcation and Chaos 16, 2777 (2006). Link, Web of Science, Google Scholar
R. Barrio, F. Blesa and S. Serrano, Chaos Solit. Fract. 36, 1067 (2008), DOI: 10.1016/j.chaos.2006.08.007. Crossref, Web of Science, Google Scholar
R. Barrio, F. Blesa and S. Serrano, EPL 82, 10003 (2008), DOI: 10.1209/0295-5075/82/10003. Crossref, Web of Science, Google Scholar
R. Barrio and F. Blesa, Chaos Solit. Fract. 41, 560 (2009), DOI: 10.1016/j.chaos.2008.02.032. Crossref, Web of Science, Google Scholar
R. Barrio, F. Blesa and S. Serrano, Commun. Nonlin. Sci. Numer. Simul. 14, 2229 (2009), DOI: 10.1016/j.cnsns.2008.07.007. Crossref, Web of Science, Google Scholar
R. Barrio, F. Blesa and S. Serrano, New J. Phys. 11, 053004 (2009), DOI: 10.1088/1367-2630/11/5/053004. Crossref, Web of Science, Google Scholar
R. Barrio, W. Borczyk and S. Breiter, Chaos Solit. Fract. 40, 1697 (2009), DOI: 10.1016/j.chaos.2007.09.084. Crossref, Web of Science, Google Scholar
M. Brack, Found. Phys. 31, 209 (2001), DOI: 10.1023/A:1017582218587. Crossref, Web of Science, Google Scholar
R. C. Churchill, G. Pecelli and D. L. Rod, Arch. Rat. Mech. Anal. 73, 313 (1980), DOI: 10.1007/BF00247673. Crossref, Web of Science, Google Scholar
P. M. Cincotta, C. M. Giordano and C. Simó, Phys. D 182, 151 (2003), DOI: 10.1016/S0167-2789(03)00103-9. Crossref, Web of Science, Google Scholar
G. Corliss and Y. F. Chang, ACM Trans. Math. Softw. 8, 114 (1982), DOI: 10.1145/355993.355995. Crossref, Web of Science, Google Scholar
M. A. M. de Aguiaret al., Ann. Phys. 180, 167 (1987). Crossref, Web of Science, Google Scholar
A. P. S. de Moura and P. S. Letelier, Phys. Lett. A 256, 362 (1999), DOI: 10.1016/S0375-9601(99)00209-1. Crossref, Web of Science, Google Scholar
M. Dellnitz, I. Melbourne and J. E. Marsden, Nonlinearity 5, 979 (1992), DOI: 10.1088/0951-7715/5/4/008. Crossref, Web of Science, Google Scholar
H. R. Dullin and A. Wittek, J. Phys. A 28, 7157 (1995), DOI: 10.1088/0305-4470/28/24/015. Crossref, Google Scholar
M. Fouchardet al., Celes. Mech. Dyn. Astron. 83, 205 (2003), DOI: 10.1023/A:1020199201070. Crossref, Web of Science, Google Scholar
C. Froeschlé and E. Lega, Celes. Mech. Dyn. Astron. 78, 167 (2000). Crossref, Web of Science, Google Scholar
Z. Galias and P. Zgliczyński, Phys. D 115, 165 (1998), DOI: 10.1016/S0167-2789(97)00233-9. Crossref, Web of Science, Google Scholar
C. Goudas and K. Papadakis, Astrophys. Space Sci. 306, 41 (2006), DOI: 10.1007/s10509-006-9232-7. Crossref, Web of Science, Google Scholar
J. Guckenheimer and B. Meloon, SIAM J. Sci. Comput. 22, 951 (2000), DOI: 10.1137/S1064827599359278. Crossref, Web of Science, Google Scholar
M. Guzzo, E. Lega and C. Froeschlé, Phys. D 163, 1 (2003), DOI: 10.1016/S0167-2789(01)00383-9. Crossref, Web of Science, Google Scholar
K. G. Hadjifotinou and T. J. Kalvouridis, Int. J. Bifurcation and Chaos 15, 2681 (2005). Link, Web of Science, Google Scholar
E. Hairer , S. P. Nørsett and G. Wanner , Solving Ordinary Differential Equations. I , Springer Series in Computational Mathematics 8 ( Springer-Verlag , Berlin , 1993 ) . Google Scholar
M. Hénon and C. Heiles, Astronom. J. 69, 73 (1964). Crossref, Web of Science, Google Scholar
M. Hénon, Ann. Astrophys. 28, 992 (1965). Google Scholar
M. Hénon, Astron. Astrophys. 28, 415 (1973). Web of Science, Google Scholar
A. Jorba and M. Zou, Experim. Math. 14, 99 (2005). Crossref, Web of Science, Google Scholar
J. M. Mao and J. B. Delos, Phys. Rev. A 45, 1746 (1992), DOI: 10.1103/PhysRevA.45.1746. Crossref, Web of Science, Google Scholar
K. Meyer, Trans. Am. Math. Soc. 149, 95 (1970). Crossref, Web of Science, Google Scholar
J. Nagler, Phys. Rev. E 69, 066218 (2004), DOI: 10.1103/PhysRevE.69.066218. Crossref, Web of Science, Google Scholar
J. Nagler, Phys. Rev. E 71, 026227 (2005), DOI: 10.1103/PhysRevE.71.026227. Crossref, Web of Science, Google Scholar
K. Papadakis, C. Goudas and G. Katsiaris, Astrophys. Space Sci. 295, 375 (2005), DOI: 10.1007/s10509-005-6564-7. Crossref, Web of Science, Google Scholar
K. Papadakis and C. Goudas, Astrophys. Space Sci. 305, 99 (2006), DOI: 10.1007/s10509-006-9046-7. Crossref, Web of Science, Google Scholar
H. Salo and C. F. Yoder, Astron. Astrophys. 205, 309 (1988). Web of Science, Google Scholar
D. J. Scheeres and N. X. Vinh, Celes. Mech. Dyn. Astronom. 51, 83 (1991), DOI: 10.1007/BF02426672. Crossref, Web of Science, Google Scholar
Ch. Skokoset al., J. Phys. A 37, 6269 (2004), DOI: 10.1088/0305-4470/37/24/006. Crossref, Google Scholar
Ch. Skokos, Lecture Notes in Physics 790 (2010) pp. 63–135. Google Scholar
C. Simó, Mem. Real Acad. Cienc. Artes Barcelona 61, 3 (2003). Google Scholar
D. K. Umberger and J. D. Farmer, Phys. Rev. Lett. 55, 661 (1985), DOI: 10.1103/PhysRevLett.55.661. Crossref, Web of Science, Google Scholar