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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 716024, 12 pages
High-Precision Continuation of Periodic Orbits
1Centro Universitario de la Defensa and IUMA, 50090 Zaragoza, Spain
2Departamento de Matemática Aplicada and IUMA, Universidad de Zaragoza, 50009 Zaragoza, Spain
Received 30 December 2011; Accepted 14 February 2012
Academic Editor: Muhammad Aslam Noor
Copyright © 2012 Ángeles Dena et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- R. Barrio, F. Blesa, and S. Serrano, “Bifurcations and safe regions in open Hamiltonians,” New Journal of Physics, vol. 11, Article ID 053004, 2009.
- R. Gilmore and M. Lefranc, The Topology of Chaos, John Wiley & Sons, New York, NY, USA, 2002.
- K. Pyragas, “Control of chaos via an unstable delayed feedback controller,” Physical Review Letters, vol. 86, no. 11, pp. 2265–2268, 2001.
- W. M. Zheng, “Predicting orbits of the Lorenz equation from symbolic dynamics,” Physica D, vol. 109, no. 1-2, pp. 191–198, 1997.
- D. A. Wisniacki, E. Vergini, R. M. Benito, and F. Borondo, “Signatures of homoclinic motion in quantum chaos,” Physical Review Letters, vol. 94, no. 5, Article ID 054101, 2005.
- A. D. Peters, C. Jaffé, and J. B. Delos, “Closed-orbit theory and the photodetachment cross section of H- in parallel electric and magnetic fields,” Physical Review A, vol. 56, no. 1, pp. 331–344, 1997.
- E. Kazantsev, “Sensitivity of the attractor of the barotropic ocean model to external influences: approach by unstable periodic orbits,” Nonlinear Processes in Geophysics, vol. 8, no. 4-5, pp. 281–300, 2001.
- J. Aguirre, R. L. Viana, and M. A. F. Sanjuán, “Fractal structures in nonlinear dynamics,” Reviews of Modern Physics, vol. 81, no. 1, pp. 333–386, 2009.
- R. P. Russell, “Global search for planar and three-dimensional periodic orbits near Europa,” Journal of the Astronautical Sciences, vol. 54, no. 2, pp. 199–226, 2006.
- R. Barrio, F. Blesa, and S. Serrano, “Periodic, escape and chaotic orbits in the Copenhagen and the (n + 1)-body ring problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2229–2238, 2009.
- S. C. Farantos, “Methods for locating periodic orbits in highly unstable systems,” Journal of Molecular Structure, vol. 341, no. 1-3, pp. 91–100, 1995.
- P. Schmelcher and F. K. Diakonos, “Detecting unstable periodic orbits of chaotic dynamical systems,” Physical Review Letters, vol. 78, no. 25, pp. 4733–4736, 1997.
- R. L. Davidchack and Y. C. Lai, “Efficient algorithm for detecting unstable periodic orbits in chaotic systems,” Physical Review E, vol. 60, no. 5 B, pp. 6172–6175, 1999.
- D. Viswanath, “The Lindstedt-Poincaré technique as an algorithm for computing periodic orbits,” SIAM Review, vol. 43, no. 3, pp. 478–495, 2001.
- M. Lara and J. Peláez, “On the numerical continuation of periodic orbits: an intrinsic, 3-dimensional, differential, predictor-corrector algorithm,” Astronomy and Astrophysics, vol. 389, no. 2, pp. 692–701, 2002.
- J. H. B. Deane and L. Marsh, “Unstable periodic orbit detection for ODEs with periodic forcing,” Physics Letters, Section A, vol. 359, no. 6, pp. 555–558, 2006.
- Y. Saiki, “Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors,” Nonlinear Processes in Geophysics, vol. 14, no. 5, pp. 615–620, 2007.
- R. Barrio and F. Blesa, “Systematic search of symmetric periodic orbits in 2DOF Hamiltonian systems,” Chaos, Solitons and Fractals, vol. 41, no. 2, pp. 560–582, 2009.
- A. Abad, R. Barrio, and A. Dena, “Computing periodic orbits with arbitrary precision,” Physical Review E, vol. 84, no. 1, Article ID 016701, 2011.
- J. N. L. Connor and D. Farrelly, “Uniform semiclassical and quantum calculations of Regge pole positions and residues for complex optical nuclear heavy-ion potentials,” Physical Review C, vol. 48, no. 5, pp. 2419–2432, 1993.
- D. Sokolovski, “Complex-angular-momentum analysis of atom-diatom angular scattering: zeros and poles of the S matrix,” Physical Review A, vol. 62, no. 2, Article ID 024702, 2000.
- G. L. Alfimov, D. Usero, and L. Vázquez, “On complex singularities of solutions of the equation ,” Journal of Physics A, vol. 33, no. 38, pp. 6707–6720, 2000.
- D. Viswanath and S. Şahutoǧlu, “Complex singularities and the lorenz attractor,” SIAM Review, vol. 52, no. 2, pp. 294–314, 2010.
- V. Gelfreich and C. Simó, “High-precision computations of divergent asymptotic series and homoclinic phenomena,” Discrete and Continuous Dynamical Systems - Series B, vol. 10, no. 2-3, pp. 511–536, 2008.
- D. Viswanath, “The fractal property of the Lorenz attractor,” Physica D, vol. 190, no. 1-2, pp. 115–128, 2004.
- H. B. Keller, Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems, Academic Press, 1977.
- J. Broeckhove, P. Kłosiewicz, and W. Vanroose, “Applying numerical continuation to the parameter dependence of solutions of the Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 234, no. 4, pp. 1238–1248, 2010.
- A. Abad, R. Barrio, F. Blesa, and M. Rodríguez, “TIDES Software,” http://gme.unizar.es/software/tides.
- A. Abad, R. Barrio, F. Blesa, and M. Rodríguez, “TIDES: a Taylor series Integrator for Differential Equations,” ACM Transactions on Mathematical Software. In press.
- J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, Pa, USA, 1997.
- L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, Philadelphia, Pa, USA, 1997.
- L. Fousse, G. Hanrot, V. Lefèvre, P. Pélissier, and P. Zimmermann, “MPFR: a multiple-precision binary floating-point library with correct rounding,” ACM Transactions on Mathematical Software, vol. 33, no. 2, article 13, p. 15, 2007.
- M. Hénon and C. Heiles, “The applicability of the third integral of motion: some numerical experiments,” The Astronomical Journal, vol. 69, pp. 73–79, 1964.
- H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Dover, New York, NY, USA, 1957.
- R. Barrio, “Sensitivity tools vs. Poincaré sections,” Chaos, Solitons and Fractals, vol. 25, no. 3, pp. 711–726, 2005.
- K. R. Meyer, “Generic bifurcation of periodic points,” Transactions of the American Mathematical Society, vol. 149, pp. 95–107, 1970.
- C. Wulff and A. Schebesch, “Numerical continuation of symmetric periodic orbits,” SIAM Journal on Applied Dynamical Systems, vol. 5, no. 3, pp. 435–475, 2006.
- E. Doedel, “AUTO: a program for the automatic bifurcation analysis of autonomous systems,” in Proceedings of the 10th Manitoba Conference on Numerical Mathematics and Computing, Vol. I (Winnipeg, Man., 1980), vol. 30, pp. 265–284, 1981.
- E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, B. Sandstede, and X. Wang, “AUTO 2000: continuation and bifurcation software for ordinary differential equations (with HomCont),” Tech. Rep., California Institute of Technology.
- A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs,” ACM Transactions on Mathematical Software, vol. 29, no. 2, pp. 141–164, 2003.
- B. Krauskopf, H. M. Osinga, and J. Galán-Vioque, Eds., Numerical Continuation Methods for Dynamical Systems, Springer, Dordrecht, The Netherlands, 2007.
- F. J. Muñoz-Almaraz, E. Freire, J. Galán, E. Doedel, and A. Vanderbauwhede, “Continuation of periodic orbits in conservative and Hamiltonian systems,” Physica D, vol. 181, no. 1-2, pp. 1–38, 2003.
- E. J. Doedel, V. A. Romanov, R. C. Paffenroth et al., “Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 17, no. 8, pp. 2625–2677, 2007.
- M. Hénon, “Exploration numérique du problème restreint. I. Masses égales ; orbites périodiques,” Annales d’Astrophysique, vol. 28, p. 499, 1965.