A survey of isochronous centers

Qualitative Theory of Dynamical Systems

Volumn 1, Issue 1, 1999, Pages 1-70

  Chavarriga, Javier ^a   Sabatini, Marco ^b  

^a UNIVERSITY OF LLEIDA   (Spain)

^b UNIVERSITY OF TRENTO   (Italy)

Author keywords

center focus problem; Integrability; isochronicity

Indexed keywords

EID: 0002294533     PISSN: 15755460     EISSN: 16623592     Source Type: Journal    
DOI: 10.1007/BF02969404     Document Type: Article

References (116)

1. A. Algaba, E. Freire andE. Gamero, Hypernormal form for the Hopf-zero bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 8 (1998), 1857-1887.
2. A. Algaba, E. Freire and E. Gamero, Isochronicity via normal form, Preprint, Universidad de Sevilla.
3. V. V. Amelkin, Isochronism of a center for two-dimensional analytic differential systems. Differential equations13 (1977), 667-674.
4. A. A. Andronov, A. A. Vitt andS. E. Khaikin, Theory of Oscilators, Dover Publications, New York, 1966.
5. V. I. Arno'ld,Équations Différentielles Ordinaires, Éditions Mir, Moscou, 1974.
6. V. I. Arno'ld, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989.
7. D. Bambusi, G. Cicogna, G. Gaeta andG. Marmo, Normal forms, symmetry and linearization of dynamical systems, J. Phys. A: Math. Gen. 31, (1998), 5065-5082.
8. H. Bass, E. H. Connell andD. Wright, The Jacobian Conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (1982), 287-330.
9. N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sb. 30 (1952), 181-196 (in russian); Amer. Math. Soc. Transl. 100 (1954), 397-413.
10. L. Cesari, Asymptolic behaviour and slabilily problems in ordinary differential equations, Springer-Verlag, Berlin, 1963.
11. J. Chavarriga, Integrable systems in the plane with a center type linear part, App. Math. 22 (1994), 285-309.
12. J. Chavarriga and I. A. García, Isochronous centers of cubic reversible systems, Lecture Notes in Physics518, Dynamical Systems, Plasmas and Gravitation, Springer-Verlag (1999), 255-268.
13. J. Chavarriga, H. Giacomini andJ. Giné, The null divergence factor, Publ. Mat. 41 (1997), 41-56.
14. J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, On the integrability of two-dimensional flows. To appear in Journal of Differential Equations.
15. J. Chavarriga andJ. Giné, Integrability of a linear center perturbed by fourth degree homogeneous polynomial, Publ. Mat. 40 (1996), 21-39.
16. J. Chavarriga andJ. Giné, Integrability of a linear center perturbed by fifth degree homogeneous polynomial, Publ. Mat. 41 (1997), 335-356.
17. J. Chavarriga andJ. Giné, Integrability of cubic systems with degenerate infinity, Differential Equations and Dynamical Systems. 6, 4 (1998), 425-438.
18. J. Chavarriga, J. Giné and I. A. García, Isochronous centers of cubic systems with degenerate infinity, Differential Equations and Dynamical Systems, 7, 1 (1999).
19. J. Chavarriga, J. Giné andI. A. García, Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial, Bulletin des Sciences Mathématiques, 123 (1999), 77-96.
20. J. Chavarriga, J. Giné and I. A. García, Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomial, Preprint, Universitat de Lleida.
21. J. Chavarriga, J. Llibre andJ. Sotomayor, Algebraic solutions for polynomial systems with emphasis in the quadratic case, Expo. Math. 15 (1997), 161-173.
22. J. Chazy, Sur la thèorie des centres, C. R. Acad. Sci. Paris221 (1947), 7-10.
23. Chen Guang-Qing and Liang Zhao-Jun, Affine classification for the quadratic vector fields without the critical points at infinity, J. of Math. Anal. and Appl. (1993), 62-72.
24. L. A. Cherkas, Conditions for a Liénard equation to have a center, Differential Equations12 (1977), 201-206.
25. C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations69 (1987), 310-321.
26. C. Chicone andF. Dumortier, A quadratic system with a nonmonotonic period function, Proc. of AMS102 (1988), 706-710.
27. C. Chicone andM. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989), 433-486.
28. C. Christopher andJ. Devlin, Isochronous centres in planar polynomial systems, SIAM Jour. Math. Anal. 28 (1997), 162-177.
29. C. Christopher, J. Devlin and N. G. Lloyd, On the classification of Liénard systems with amplitude-independent periods, Preprint.
30. C. Christopher, J. Devlin, N. G. Lloyd, J. M. Pearson andN. Yasmin, Quadratic-like cubic systems, Differential Equations and Dynamical Systems, 5, 3-4, (1997), 329-345.
31. A. Cima, A. Gasull, V. Mañosa andF. Mañosas, Algebraic properties of the Liapunov and period constants, Rocky Mount. Jour. Math. 27 (1997), 471-501.
32. A. Cima, F. Mañosas and J. Villadelprat, Isochronicity for several classes of Hamiltonian systems, Preprint, Universitat Autònoma de Barcelona (1998).
33. C. B. Collins, The period function of some polynomial systems of arbitrary degree, Diff. and Int. Eq. 9 (1996), 251-266.
34. C. B. Collins, Conditions for a centre in a simple class of cubic systems, Diff. and Int. Eq. 10 (1997), 333-356.
35. R. Conti, On isochronous centers of cubic systems, Revue Roum. de Math. Pures et Appl. XXXIX (1994), 295-301.
36. 2, Lecture Notes in Pure and Appl. Math. 152 (1994), 21-31. M. Dekker, New York.
37. R. Conti, Centers of planar polynomial systems. A review, To appear.
38. W. Coppel andL. Gavrilov, The period function of a hamiltonian quadratic system, Diff. and Int. Eq. 6 (1993), 1357-1365.
39. G. Darboux, Mémoire sur les équations differéntielles algébriques du premier ordre et du premier degré, Bull. Sci. Math. (2)2 (1878), 60-96.
40. J. Devlin, Coexisting isochronous and nonisochronous centres, Bull. Lond. Math. Soc. 28 (1996), 495-500.
41. G. F. D. Duff, Limit cycles and rotated vector fields, Ann. Math. 57 (1953), 15-31.
42. H. Dulac, Détermination et intégration d'une certaine classe d'équations différentielles ayant pour point singulier un centre, Bull. Sci. Math. (2)32 (1908), 230-252.
43. M. Farkas, Periodic Motions, Applied Mathematical Sciences, 104, Springer Verlag, Berlin, (1994).
44. M. Frommer,Über das Auftreten von Wirbeln und Strudeln (geschlossener und spiraliger Integralkurven) in der Umgebung rationalre Unbestimmtkeitssellen, Math. Ann. 109 (1934), 395-424.
45. nd Catalan Days of Applied Mathematics (1995) edited by M. Sofonea and J. N. Corvellec. Perpignan (1996).
46. A. Gasull, A. Guillamón, V. Mañosa andF. MañosasThe period function for Hamiltonian systems with homogeneous nonlinearities, J. Differential Equations139 (1997), 237-260.
47. A. Gasull andR. Prohens, Quadratic and cubic systems with degenerate infinity, J. of Math. Anal. Appl. 198 (1996), 25-34.
48. A. Gasull andJ. Torregrosa, Center problem for several differential equations via Cherkas method, J. of Math. Anal. Appl. 228 (1998), 322-343.
49. L. Gavrilov, Isochronicity of plane polynomial Hamillonian systems, Nonlinearity10 (1997), 433-448.
50. J. Gregor, Dynamical systems with regular hand-side, Pokroky MFAIII (1958), 153-160.
51. O. Hajek, Notes on meromorphic dynamical systems I, Czechoslovak Math. J. 16 (1966), 14-27.
52. O. Hajek, Notes on meromorphic dynamical systems, II, Czechoslovak, Math. J. 16 (1966), 27-35.
53. Hsu andSze-Bi, A remark on the period of the periodic solution in the Lotka-Volterra system, Jour. Math. Anal. Appl. 95 (1983), 428-436.
54. M. C. Irwin, Smooth Dynamical Systems, Academic Press, London, 1980.
55. X. Jim andD. Wang, On Kukles' conditions for the existence of a centre, Bull. London Math. Soc. 22 (1990), 1-4.
56. W. Kapteyn, On the centra of the integral curves which satisfy differential equations of the first order and the first degree, Konikl, Akademic von Wetenschappen to Amsterdam, Proceedings of the Section of Science, 13. 2 (1911), 1241-1252; 14. 2 (1911), 1185-1195.
57. O. A. Kononova, On the isochronicity of the center of a system of nonlinear oscillations, (Russian) Vestnik Beloruss. Gos. Univ. Ser. I Fiz. Mat. Mekh. (1990), 73-74, 81.
58. I. S. Kukles, Sur quelques cas de distinction entre un foyer et un centre, Dokl. Akad. Nauk. SSSR42 (1944), 208-211.
59. J. J. Levin andS. S. Siiatz, Nonlinear oscillations of fixed period, Jour. Math. Anal. Appl. 7 (1963), 284-288.
60. M. A. Liapunov, Problème général de la slabilité du mouvement Ann. of Math. Stud. 17, Princeton University Press, 1947.
61. Li Ciiengziii, Two problems of planar quadratic systems, Sci. Sinica Ser. A26 (1983), 471-481.
62. N. G. Lloyd, Small amplitude limit cycles of polynomial differential equations, Lect. Notes in Maths. 1032, Ordinary differential Equations and Operators (1983), 346-356.
63. N. G. Lloyd andJ. M. Pearson, Computing centre conditions for certain cubic systems, Journal of Computational and Applied Mathematics40 (1992), 323-336.
64. W. S. Loud, Behavior of the period of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations3 (1964), 21-36.
65. N. A. Lukasiievicii, Isochronicity of center for certain systems of differential equations, Diff. Uravn. 1 (1965), 220-226.
66. V. A. Lunkevicii andK. S. Sidirskii, Integrals of a general quadratic differential system in cases of a center, Diff. Equations18 (1982), 563-568.
67. P. Mardešić, L. Moser-Jauslin andC. Rousseau, Darboux linearization and isochronous centers with a rational first integral, J. Differential Equations134 (1997), 216-268.
68. P. Mardešić, C. Rousseau andB. Toni, Linearization of isochronous centers, J. Differential Equations121 (1995), 67-108.
69. L. Mazzi andM. Sabatini, A characterization of centres via first integrals, J. Differential Equations76 (1988), 222-237.
70. L. Mazzi and M. Sabatini, Commutators and linearizations of isochronous centers, Preprint, Università degli studi di Trento, UTM 482 (1996), presented at the Symposium on Planar Vector Fields, Lleida, November 24-27, 1996. To appear in Rend. Acc. Lincci.
71. G. Meisters, Jacobian problems in differential equations and algebraic geometry, Rocky Mount. J. Math. 12 (1982) 679-705.
72. V. V. Nemitskij andV. V. Stepanov, Qualitative Theory of Differential Equations, Princeton Univ. Press, Princeton, 1960.
73. C. Obi, Analytical theory of nonlinear oscillations, VII, The periods of the periodic solutions of the equation $$\ddot x + g(x) = 0$$, J. Math. Anal. Appl. 55 (1976), 295-301.
74. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986.
75. Z. Opial, Sur les périodes des solutions de l'équation différentiellex″g(x), Ann. Polon. Math. 10 (1961), 49-72.
76. I. A. Pleshkan, A new method of investigating the isochronicity of a system of two differential equations, Differential Equations5 (1969), 796-802.
77. I. I. Pleshkan, On the isochronicity of systems of two differential equations whose right-hand sides contain common factors, Differential Equations23 (1987), 657-667.
78. H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. de Mathématiques (3)7 (1881), 375-422; 8 (1882), 251-296; Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris 1951, 3-84.
79. H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. de Mathématiques Pures et Appliquées (4)1 (1885), 167-244; Ocuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris 1951, 95-114.
80. N. V. Pyziikova andL. P. CiierenkovaNecessary conditions for isochronicity of the center of a differential system with a nonlinearity of the third power, (Russian. English summary) Dokl. Akad. Nauk BSSR29 (1985), 980-981, 1052.
81. F. Rothe, The periods of the Lotka-Volterra system, J. Reine Angew. Math. (1985), 355 129-138.
82. C. Rousseau andB. Toni, Local bifurcations in vector fields with homogeneous nonlinearities of the third degree, Canad. Math. Bull. (1993), 36 473-484.
83. C. Rousseau andB. Toxi, Local bifurcations of critical periods in the reduced Kukles sytems, Can. J. Math. 49 (1997) 338-358.
84. M. Sabatini, Qualitative analysis of commuting flows on two-dimensional manifolds, Proc. of the Conference Equadiff 95, Lisboa, World Scientific Publ. Co., 494-497.
85. M. Sabatini, Quadratic isochronous centers commute, Preprint, Università degli studi di Trento, UTM 461 (1995) - To appear in Applicationes Mathematicae.
86. M. Sabatini, Characterizing isochronous centres by Lie brackets, Diff. Equations and Dyn. Syst., 5 (1997), 91 99.
87. M. Sabatini, Dynamics of commuting systems on two-dimensional manifolds, Ann. Mat. Pura Appl. (IV)CLXXIII (1997), 213-232.
88. M. Sabatini, A connection between isochronous Hamiltonian centres and the Jacobian Conjecture, Nonlinear Analysis, 34 (1998), 829-838.
89. M. Sabatini, The time of commuting systems (preliminary version), Preprint, Università degli studi di Trento (1996), presented at the Symposium on Planar Vector Fields, Lleida, November 24-27, 1996.
90. M. Sabatini, On the period function of Liśystems, J. Differential Equations, 152 (1999), 467-487.
91. E. Sáez andI. Szántó, One-parameter family of cubic Kolmogorov system with an isochronous center, Collect. Math., 48 (1997), 297 301.
92. G. Sansone andR. Conti, Non-linear differential equations, Revised edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 67, The Macmillan Co., New York 1964.
93. R. Schaaf, A class of Hamiltonian systems with increasing periods, Jour. Reine Ang. Math. 363 (1985), 96 109.
94. B. Schuman, Sur la forme normale de Birkhoff et les centres isochrones, C. R. Acad. Sci. Paris, 322 (1996), 21-24.
95. Siii Songling, A method of constructing cycles without contact around a weak focus, J. Differential Equations41 (1981), 301-312.
96. K. S. Sibirskii, Algebraic invariants for differential equations and matrices, Kshiner Shtiint-sa, 1976 (in Russian).
97. K. S. Sibirskii, On the number of limit cycles in the neighbourhood of a singular point, Differential Equations1 (1965), 36-47.
98. J. Sokulski, The beginning of classification of Darboux integrals for cubic systems with center.
99. J. Sotomayor, Liçóes de Equaçóes Diferenciais Ordinárias, IMPA, Rio de Janciro, 1979.
100. J. Sotomayor, Curvas definidas por equaçóes diferenciais no plano, IMPA, Rio de Janciro, 1981.
101. M. Urabe, Potential forces which yield periodic motions of a fixed period, Jour. Math. Mech. 10 (1961), 569-578.
102. M. Urabe, The potential force yielding a periodic motion whose period is an arbitrary continuous function of the amplitude of the velocity, Arch. Rational Mech. and Anal. 11 (1962), 27-33.
103. J. Villadelprat, Index of vector fields on manifolds and isochronicity for planar hamiltonian differential systems, Ph. d. Thesis, U. A. B., Facultat de Ciències, Departament de Matemàtiques, Barcelona, 1999.
104. M. Villarini, Regularity properties of the period function near a centre of planar vector fields, Nonlinear Analysis19 (1992), 787-803.
105. M. Villarini, Smooth linearizations of centres, Preprint, Università degli studi di Firenze (1996).
106. E. P. Volokitin andV. V. IvanovIsochronicity and commutation of polynomial vector fields. Siberian Math. Journal, 40 (1999), 23-38.
107. A. P. Vorob'ev, On isochronous systems of two differential equations, Dokladi Akademii Nauk Belorusskoi SSR, 7 (1963), 155-156 (in russian).
108. A. P. Vorob'ev, The construction of isochronous systems of two differential equations, Dokladi Akademii Nauk Belorusskoi SSR7 (1963), 513 515 (in russian).
109. A. P. Vorob'ev, Qualitative investigation in the large of integral curves of isochronous systems of two differential equations, Diff. Uravn. 1 (1965), 439-441. (in russian).
110. A. P. Vorob'ev, Sufficiency conditions for isochronism of canonical systems of two differential equations, Diff. Uravn. 1 (1965), 582-584 (in russian).
111. J. Waldvogel, The period in the Volterra-Lotka predator-prey model, SIAM Jour. Numer. Anal. 20 (1983), 1264-1272.
112. N. Yasmin, Closed orbits of certain two-dimensional cubic systems, Ph. D. thesis University College of Wales, Aberystwyth, (1989).
113. G. Zampieri, On the periodic oscillations of x″=g(x), Journal of Differential Equations, 78 (1989) 74-88.
114. H. Żola{ogonek}dek, Quadratic systems with center and their perturbations, J. Differential Equations109 (1994), 223-273.
115. H. Żola{ogonek}dek, On certain generalization of the Bautin's Thorem, Nonlinearity7 (1994), 273 279.
116. H. Żola{ogonek}dek, The classification of reversible cubic systems with center, Topological Methods in Nonlinear Analysis4 (1994) 79-136.