A rigorous study of possible configurations of limit cycles bifurcating from a hyper-elliptic Hamiltonian of degree five

Dynamical Systems

Volumn 24, Issue 2, 2009, Pages 237-247

  Johnson, Tomas ^a   Tucker, Warwick ^b  

^a UPPSALA UNIVERSITY   (Sweden)

^b UNIVERSITY OF BERGEN   (Norway)

Author keywords

Bifurcation theory; Interval analysis; Limit cycles; Planar Hamiltonian systems

Indexed keywords

BIFURCATION THEORY; DIFFERENTIAL SYSTEMS; INTERVAL ANALYSIS; LIMIT CYCLE; LIMIT CYCLES; PLANAR HAMILTONIAN SYSTEMS;

BIFURCATION (MATHEMATICS);

HAMILTONIANS;

EID: 70349510621     PISSN: 14689367     EISSN: 14689375     Source Type: Journal    
DOI: 10.1080/14689360802641206     Document Type: Article

References (32)

1. Yu.S. Il'yashenko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc. (N.S.) 39(3) (2002), pp. 301-354.
2. R. Roussarie, Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem, Progress in Mathematics, Vol.164, Birkhäuser Verlag, Basel, 1998.
3. J. É calle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac (French) [Introduction to Analyzable Functions and Constructive Proof of the Dulac Conjecture]. Actualités Mathématiques. [Current Mathematical Topics] Hermann, Paris, 1992.
4. Yu.S. Il'yashenko, Finiteness Theorems for Limit Cycles, Translated from the Russian by H. H. McFaden. Translations of Mathematical Monographs, Vol.94, American Mathematical Society, Providence, RI, 1991.
5. C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007.
6. F. Chen, C. Li, J. Llibre, and Z. Zhang, A unified proof on the weak Hilbert 16th problem for n=2, J. Differ. Equ. 221(2) (2006), pp. 309-342.
7. F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four. I. Saddle loop and two saddle cycle, J. Differ. Equ. 176(1) (2001), pp. 114-157. (Pubitemid 33082627)
8. F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four. II. Cuspidal loop, J. Differ. Equ. 175(2) (2001), pp. 209-243. (Pubitemid 33082423)
9. F. Dumortier and C. Li, Perturbation from an elliptic Hamiltonian of degree four. III. Global centre, J. Differ. Equ. 188(2) (2003), pp. 473-511. (Pubitemid 36269953)
10. F. Dumortier and C. Li, Perturbation from an elliptic Hamiltonian of degree four. IV. Figure eight-loop, J. Differ. Equ. 188(2) (2003), pp. 512-554. (Pubitemid 36269954)
11. T. Johnson and W. Tucker, On a computer-aided approach to the computation of Abelian integrals, (submitted).
12. J. Guckenheimer, Phase portraits of planar vector fields: computer proofs, Experiment. Math. 4(2) (1995), pp. 153-165.
13. J. Guckenheimer and S. Malo, Computer-generated proofs of phase portraits for planar systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6(5) (1996), pp. 889-892.
14. S. Malo, Rigorous computer verification of planar vector field structure, Ph.D. thesis Cornell University, Cornell University, 1994.
15. J. Llibre and G. Rodríguez, Configurations of limit cycles and planar polynomial vector fields, J. Differ. Equ. 198(2) (2004), pp. 374-380.
16. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, Vol.42, Springer-Verlag, New York, 1983.
17. D. Novikov and S. Yakovenko, Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), pp. 55-65.
18. D. Gabai, G.R. Meyerhoff, and N. Thurston, Homotopy hyperbolic 3-manifolds are hyperbolic, Ann. of Math. (2) 157(2) (2003), pp. 335-431.
19. T.C. Hales, A proof of the Kepler conjecture, Ann. of Math. (2) 162(3) (2005), pp. 1065-1185.
20. W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math. 2(1) (2002), pp. 53-117.
21. G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, New York, 1983.
22. R.E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1966.
23. R.E. Moore, Methods and Applications of Interval Analysis, SIAM Studies in Applied Mathematics, Philadelphia, 1979.
24. A. Neumaier, Interval Methods for Systems of Equations, Encyclopedia of Mathematics and its Applications vol. 37, Cambridge Univ. Press, Cambridge, 1990.
25. M.S. Petkovićand L.D. Petković, Complex Interval Arithmetic and its Applications, Mathematical Research, Vol.105, Wiley-VCH Verlag Berlin GmbH, Berlin, 1998.
26. C. Li and K. Lu, The period function of hyperelliptic Hamiltonians of degree 5 with real critical points, Nonlinearity 21(3) (2008), pp. 465-483.
27. L. Gavrilov and I.D. Iliev, Complete hyperelliptic integrals of the first kind and their nonoscillation, Trans. Amer. Math. Soc. 356(3) (2004), pp. 1185-1207.
28. M. Caubergh and F. Dumortier, Hilbert's 16th problem for classical Liénard equations of even degree, J. Differ. Equ. 244(6) (2008), pp. 1359-1394.
29. R. Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst. 17(2) (2007), pp. 441-448.
30. Emil Horozov and Iliya D. Iliev, Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians, Nonlinearity 11(6) (1998), pp. 1521-1537. (Pubitemid 128406182)
31. CXSC - C++ eXtension for Scientific Computation, version 2.11. (accessed 09.03.2006) Available at http://www.math.uni-wuppertal.de/org/WRST/xsc/cxsc. html.
32. R. Hammer, M. Hocks, U. Kulisch, and D. Ratz, C++ Toolbox for Verified Computing, Springer-Verlag, New York, 1995.