×

On a computer-aided approach to the computation of Abelian integrals. (English) Zbl 1226.65102

Summary: An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian of degree four.

MSC:

65P30 Numerical bifurcation problems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37M20 Computational methods for bifurcation problems in dynamical systems
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
65G40 General methods in interval analysis

Software:

C-XSC; RODES; kepler98
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, New York (1983) · Zbl 0552.65041
[2] Chen, F., Li, C., Llibre, J., Zhang, Z.: A unified proof on the weak Hilbert 16th problem for n=2. J. Differ. Equ. 221(2), 309–342 (2006) · Zbl 1098.34024 · doi:10.1016/j.jde.2005.01.009
[3] Christopher, C., Li, C.: Limit Cycles of Differential Equations. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel (2007) · Zbl 1359.34001
[4] CXSC–C++ eXtension for Scientific Computation, version 2.0. Available from http://www.math.uni-wuppertal.de/org/WRST/xsc/cxsc.html
[5] Duff, G.F.D.: Limit-cycles and rotated vector fields. Ann. Math. (2) 57, 15–31 (2006) · Zbl 0050.09103 · doi:10.2307/1969724
[6] Dumortier, F., Li, C.: Perturbations from an elliptic Hamiltonian of degree four. I. Saddle loop and two saddle cycle. J. Differ. Equ. 176(1), 114–157 (2001) · Zbl 1004.34018 · doi:10.1006/jdeq.2000.3977
[7] Dumortier, F., Li, C.: Perturbations from an elliptic Hamiltonian of degree four. II. Cuspidal loop. J. Differ. Equ. 175(2), 209–243 (2001) · Zbl 1034.34036 · doi:10.1006/jdeq.2000.3978
[8] Dumortier, F., Li, C.: Perturbation from an elliptic Hamiltonian of degree four. III. Global centre. J. Differ. Equ. 188(2), 473–511 (2003) · Zbl 1056.34044 · doi:10.1016/S0022-0396(02)00110-9
[9] Dumortier, F., Li, C.: Perturbation from an elliptic Hamiltonian of degree four. IV. Figure eight-loop. J. Differ. Equ. 188(2), 512–554 (2003) · Zbl 1057.34015 · doi:10.1016/S0022-0396(02)00111-0
[10] Dumortier, F., Roussarie, R.: Abelian integrals and limit cycles. J. Differ. Equ. 227(1), 116–165 (2006) · Zbl 1111.34028 · doi:10.1016/j.jde.2005.08.015
[11] Écalle, J.: Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. [Introduction to Analyzable Functions and Constructive Proof of the Dulac Conjecture]. Actualités Mathématiques. [Current Mathematical Topics]. Hermann, Paris (1992) · Zbl 1241.34003
[12] Gabai, D., Meyerhoff, G.R., Thurston, N.: Homotopy hyperbolic 3–manifolds are hyperbolic. Ann. Math. (2) 157(2), 335–431 (2006) · Zbl 1052.57019 · doi:10.4007/annals.2003.157.335
[13] Guckenheimer, J.: Phase portraits of planar vector fields: computer proofs. Exp. Math. 4(2), 153–165 (1995) · Zbl 0856.34045 · doi:10.1080/10586458.1995.10504316
[14] Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences, vol. 42. Springer, New York (1983) · Zbl 0515.34001
[15] Guckenheimer, J., Malo, S.: Computer-generated proofs of phase portraits for planar systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 6(5), 889–892 (1996) · Zbl 0882.68133 · doi:10.1142/S0218127496000497
[16] Hales, T.C.: A proof of the Kepler conjecture. Ann. Math. (2) 162(3), 1065–1185 (2005) · Zbl 1096.52010 · doi:10.4007/annals.2005.162.1065
[17] Hammer, R., Hocks, M., Kulisch, U., Ratz, D.: C++ Toolbox for Verified Computing. Springer, New York (1995)
[18] Il’yashenko, Yu.S.: Finiteness Theorems for Limit Cycles. Translated from the Russian by H.H. McFaden. Translations of Mathematical Monographs, vol. 94. American Mathematical Society, Providence (1991)
[19] Il’yashenko, Yu.S.: Centennial history of Hilbert’s 16th problem. Bull., New Ser., Am. Math. Soc. 39(3), 301–354 (2002) · Zbl 1004.34017 · doi:10.1090/S0273-0979-02-00946-1
[20] Rigorous, S. Malo: Computer verification of planar vector field structure. Ph.D. thesis, Cornell University (1994)
[21] Moore, R.E.: Interval Analysis. Prentice Hall, Englewood Cliffs (1966) · Zbl 0176.13301
[22] Moore, R.E.: Methods and Applications of Interval Analysis. SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1979) · Zbl 0417.65022
[23] Neumaier, A.: Interval Methods for Systems of Equations. Encyclopedia of Mathematics and Its Applications, vol. 37. Cambridge Univ. Press, Cambridge (1990) · Zbl 0715.65030
[24] Petković, M.S., Petković, L.D.: Complex Interval Arithmetic and Its Applications. Mathematical Research, vol. 105. Wiley–VCH, Berlin (1998) · Zbl 0911.65038
[25] Petrov, G.S.: Nonoscillation of elliptic integrals. Funkc. Anal. Prilozh. 24(3), 45–50 (1990), 96 (Russian). Translation in Funct. Anal. Appl. 24(3), 205–210 (1991)
[26] Roussarie, R.: Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem. Progress in Mathematics, vol. 164. Birkhäuser Verlag, Basel (1998) · Zbl 0898.58039
[27] Tucker, W.: A rigorous ODE solver and Smale’s 14th problem. Found. Comput. Math. 2(1), 53–117 (2002) · Zbl 1047.37012
[28] Zhou, H., Xu, W., Li, S., Zhang, Y.: On the number of limit cycles of a cubic polynomials Hamiltonian system under quintic perturbation. Appl. Math. Comput. 190(1), 490–499 (2007) (English summary) · Zbl 1124.37030 · doi:10.1016/j.amc.2007.01.052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.