Horseshoes in the forced van der Pol system

Nonlinearity

Volumn 22, Issue 1, 2009, Pages 213-237

  Haiduc, Radu ^a  

^a Department of Molecular Biology and Genetics   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 58149392828     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/22/1/011     Document Type: Article

References (38)

1. Arnold V I, Afrajmovich V S, Il'yashenko Yu S and Shil'nikov L P 1999 Bifurcation Theory and Catastrophe Theory (Berlin: Springer) (translated from the 1986 Russian original by N D Kazarinoff) reprint of the 1994 Dynamical Systems V (Encyclopaedia of Mathematical Sciences 5) (Berlin: Springer) MR 95c:58058 (English edition from the series Encyclopaedia of Mathematical Sciences)
2. Benoît I 1990 Canards et enlacements Inst. Hautes Itudes Sci. Publ. Math. 63-91 (1991)
3. 3 et leurs canards Third Schnepfenried Geometry Conf. (Schnepfenried, France, 1982) (Astérisque vol 109) vol 2 (Paris: Soc. Math. France) pp 159-191
4. Bold K, Edwards C, Guckenheimer J, Guharay S, Hoffman K, Oliva R and Weckesser W The forced van der Pol equation: II. Canards in the reduced system at press
5. Callot J-L, Diener F and Diener M 1978 Le problème de la 'chasse au canard' C. R. Acad. Sci. Paris Sér. A-B A 286 1059-61
6. Canalis-Durand M 1991 Formal expansion of van der Pol equation canard solutions are Gevrey Dynamic Bifurcations (Luminy, 1990) (Lecture Notes in Mathematics vol 1493) (Berlin: Springer) pp 29-39
7. Cartwright M L and Littlewood J E 1945 On non-linear differential equations of the second order: I. The equation large J. Lond. Math. Soc. 20 180-9
8. Dumortier F and Roussarie R 1996 Canard cycles and center manifolds Mem. Amer. Math. Soc. 121 x+100 (with an appendix by Cheng Zhi Li)
9. Fenichel N 1971/1972 Persistence and smoothness of invariant manifolds for flows Indiana Univ. Math. J. 21 193-226
10. Fenichel N 1979 Geometric singular perturbation theory for ordinary differential equations J. Diff. Eqns 31 53-98
11. Grasman J 1987 Asymptotic Methods for Relaxation Oscillations and Applications (Applied Mathematical Sciences vol 63) (New York: Springer)
12. Guckenheimer J 1995 Phase portraits of planar vector fields: computer proofs Exp. Math. 4 153-65
13. Guckenheimer J, Hoffman K and Weckesser W 2000 Numerical computation of canards Int. J. Bifurcation Chaos Appl. Sci. Eng. 10 2669-87
14. Guckenheimer J, Hoffman K and Weckesser W 2003 The forced van der Pol equation: I. The slow flow and its bifurcations SIAM J. Appl. Dyn. Syst. 2 1-35 (electronic)
15. Guckenheimer J and Holmes P 1990 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences vol 42) (New York: Springer) (revised and corrected reprint of the 1983 original)
16. Haiduc R 2005 Horseshoes in the forced van der Pol equation PhD in Mathematics Cornell University, Ithaca, New York 14853
17. Hargreaves G I 2002 Interval analysis in MATLAB Numerical Analysis Report No 416 Manchester Centre for Computational Mathematics, Manchester, England http://www.maths.man.ac.uk/ nareports/narep416.pdf
18. Ilyashenko Yu and Li W 1999 Nonlocal Bifurcations (Mathematical Surveys and Monographs vol 66) (Providence, RI: American Mathematical Society)
19. INTEGRO http://www.cam.cornell.edu/∼roj/odetools/
20. INTLAB http://www.ti3.tu-harburg.de/∼rump/intlab/
21. Jones C K R T 1995 Geometric singular perturbation theory Dynamical Systems (Montecatini Terme, 1994) (Lecture Notes in Mathematics vol 1609) (Berlin: Springer) pp 44-118
22. Levi M 1981 Qualitative analysis of the periodically forced relaxation oscillations Mem. Am. Math. Soc. 32 vi+147
23. Levinson N 1949 A second order differential equation with singular solutions Ann. Math. (2) 50 127-53
24. Mishchenko E F and Rozov N Kh 1980 Differential Equations with Small Parameters and Relaxation Oscillations (Mathematical Concepts and Methods in Science and Engineering vol 13) (New York: Plenum Press) (translated from the Russian by F. M. C. Goodspeed)
25. Moore R E 1966 Interval Analysis (Englewood Cliffs, NJ: Prentice-Hall)
26. Moser J 2001 Stable and Random Motions in Dynamical Systems (Princeton Landmarks in Mathematics) (Princeton, NJ: Princeton University Press) (with special emphasis on celestial mechanics, reprint of the 1973 original, with a foreword by Philip J Holmes)
27. Robbin J W 1971 A structural stability theorem Ann. Math. (2) 94 447-93
28. Rossetto B, Lenzini T, Ramdani S and Suchey G 1998 Slow-fast autonomous dynamical systems Int. J. Bifurcation Chaos Appl. Sci. Eng. 8 2135-45
29. Rump S M 1999 Intlab-Interval Laboratory http://www.ti3.tu-harburg.de/ publications/Intlab.ps.gz
30. Smale S 1965 Diffeomorphisms with many periodic points Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (Princeton, NJ: Princeton University Press) pp 63-80
31. Smale S 1998 Finding a horseshoe on the beaches of Rio Math. Intelligencer 20 39-44
32. Szmolyan P and Wechselberger M 2001 Canards in J. Diff. Eqns 177 419-53
33. Szmolyan P and Wechselberger M 2004 Relaxation oscillations in J. Diff. Eqns 200 69-104
34. Tucker W 1999 The Lorenz attractor exists C. R. Acad. Sci. Paris Sér. I Math. 328 1197-202
35. van der Pol B 1926 On 'relaxation oscillations' I Phil. Mag. 2 978-92
36. van der Pol B and van der Mark J 1927 Frequency demultiplication Nature 120 363-4
37. van der Pol B and van der Mark J 1928 The heartbeat considered as a relaxation oscillation, and an electrical model of the heart Phil. Mag. Suppl. 6 763-75
38. Ye Y Q, Cai S L, Chen L S, Huang K C, Luo D J, Ma Z E, Wang E N, Wang M S and Yang X A 1986 Theory of Limit Cycles (Translations of Mathematical Monographs vol 66) 2nd edn (Providence, RI: American Mathematical Society) (translated from the Chinese by Chi Y Lo)