Periodic orbit continuation in multiple time scale systems

Numerical Continuation Methods for Dynamical Systems: Path following and boundary value problems

Volumn , Issue , 2007, Pages 253-267

  Guckenheimer, John ^a   Lamar, M Drew ^a  

^a Department of Obstetrics Gynecology   (United States)

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EID: 84892340565     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-1-4020-6356-5_8     Document Type: Chapter

References (28)

1. E. Allgower and K. Georg. Continuation and path following. Acta Numerica 1993, pages 1-64 (Cambridge University Press, Cambridge, 1993)
2. E. Benoit, J. L. Callot, F. Diener, and M. Diener. Chasse au canard. Collect. Math. 32:37-119, 1981.
3. K. Bold, C. Edwards, J. Guckenheimer, S. Guharay, K. Hoffman, J. Hubbard, R. Oliva, and W. Weckesser. The forced van der Pol equation II: Canards in the reduced system. SIAM J. Appl. Dyn. Syst. 2:570-608, 2003.
4. M. Cartwright and J. Littlewood On nonlinear differential equations of the second order: II the equation ÿ - kf (y, ẏ) ẏ + g (y, k) = p (t) = p1 (t) + kp2 (t), k> 0, f (y) ≥ 269-299 Ann. Math. 48:472-94, 1947.
5. [Addendum 50:504-505, 1949]
6. R. Clewley, M. D. La Mar, and E. Sherwood. "PyDSTool" available via http://sourceforge.net/projects/pydstool
7. B. Coomes, H. Kocak, and K. Palmer. Rigorous computational shadowing of orbits of ordinary differential equations. Numer. Math. 69:401-421 1995.
8. E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Yu. A. Kuznetsov, B. E. Oldeman, B. Sandstede, and X. J. Wang. Auto2000: Continuation and bifurcation software for ordinary differential equations. available via http://cmvl.cs.concordia.ca/.
9. F. Dumortier and R. Roussarie. Canard cycles and center manifolds. With an appendix by Cheng Zhi Li. Mem. Amer. Math. Soc. 121, no. 577, x+100 pp, 1996.
10. W. Eckhaus. Relaxation oscillations, including a standard chase on French ducks. Lecture Notes in Mathematics, Vol. 985, pages 449-494 (Springer-Verlag, 1983).
11. N. Fenichel, Geometric singular perturbation theory. J. Diff Eq. 31:53-98, 1979
12. J. Guckenheimer. Bifurcation and degenerate decomposition in multiple time scale dynamical systems. In J. Hogan, A. Champneys, B. Krauskopf, M. di Bernardo, E. Wilson, H. Osinga, and M. Homer, editors, Nonlinear Dynamics and Chaos: where do we go from here?, pages 1-21 (Institute of Physics Publishing, Bristol, 2002).
13. J. Guckenheimer. Bifurcations of relaxation oscillations. In Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, pages 295-316, NATO Sci. Ser. II Math. Phys. Chem., Vol. 137 (Kluwer Acad. Publ., Dordrecht, 2004).
14. J. Guckenheimer, K. Hoffman, and W. Weckesser. Numerical computation of canards. Internat. J. Bifurc. Chaos Appl. Sci. Engrg., 10:2669-2687, 2000.
15. J. Guckenheimer, K. Hoffman, and W. Weckesser. The Forced van der Pol Equation I: The Slow Flow and its Bifurcations. SIAM J. App. Dyn. Sys. 2:1-35, 2003.
16. J. Guckenheimer and B. Meloon. Computing periodic orbits and their bifurcations with automatic differentiation. SIAM J. Sci. Comp., 22:951-985, 2000.
17. J. Guckenheimer, M. Wechselberger, and L.-S. Young. Chaotic attractors of relaxation oscillators. Nonlinearity, 19:701-720, 2006.
18. R. Haiduc. Horseshoes in the forced van der Pol equation, PhD dissertation, Cornell University, 2005.
19. E. Hairer, S. P. Norsett and G. Wanner. Solving Ordinary Differential Equations I, 2nd. ed. (Springer-Verlag, 1993).
20. C. K. R. T. Jones. Geometric singular perturbation theory. In Dynamical Systems, Lecture Notes in Mathathematics, Vol 1609, pages 44-120 (Springerverlag, 1995).
21. N. Levinson. Perturbations of discontinuous solutions of non-linear systems of differential equations. Acta Math., 82:71-106, 1950.
22. 2) ẏ + y = bk cos (λt + α) for large k, and its generalizations. Acta Math., 97:267-308, 1957.
23. [Errata at Acta Math., 98:110, 1957]
24. J. Littlewood. On nonlinear differential equations of the second order. IV. The general equation ÿ - kf (y) ẏ + g (y) = bkp (φ), φ = t + α. Acta Math., 98:1-110, 1957.
25. E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov and N. Kh. Rhozov. Asymptotic Methods in Singularly Perturbed Systems. Monographs in Contemporary Mathematics, (Consultants Bureau, New York, A Division of Plenum Publishing Corporation, 1994).
26. E. Van Vleck. Numerical shadowing near hyperbolic trajectories. SIAM J. Sci. Comput. 16:1177-1189, 1995.
27. A. B. Vasil'eva. Asymptotic behaviour of solutions of certain problems for ordinary non-linear differential equations with a small parameter multiplying the highest derivatives. Russian Mathematical Surveys, 18:13-84, 1963. (Russian)
28. Uspehi Mat. Nauk, 18:15-86, 1963.