The four positive vortices problem : regions of chaotic behavior and the non-integrability
Annales de l'I.H.P. Physique théorique, Tome 59 (1993) no. 1, pp. 99-115. 
@article{AIHPA_1993__59_1_99_0,
     author = {Castilla, M. S. A. C. and Moauro, Vinicio and Negrini, Piero and Oliva, Waldyr Muniz},
     title = {The four positive vortices problem : regions of chaotic behavior and the non-integrability},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     pages = {99--115},
     publisher = {Gauthier-Villars},
     volume = {59},
     number = {1},
     year = {1993},
     mrnumber = {1244184},
     zbl = {0792.76010},
     language = {en},
     url = {http://archive.numdam.org/item/AIHPA_1993__59_1_99_0/}
}
TY  - JOUR
AU  - Castilla, M. S. A. C.
AU  - Moauro, Vinicio
AU  - Negrini, Piero
AU  - Oliva, Waldyr Muniz
TI  - The four positive vortices problem : regions of chaotic behavior and the non-integrability
JO  - Annales de l'I.H.P. Physique théorique
PY  - 1993
SP  - 99
EP  - 115
VL  - 59
IS  - 1
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/item/AIHPA_1993__59_1_99_0/
LA  - en
ID  - AIHPA_1993__59_1_99_0
ER  - 
%0 Journal Article
%A Castilla, M. S. A. C.
%A Moauro, Vinicio
%A Negrini, Piero
%A Oliva, Waldyr Muniz
%T The four positive vortices problem : regions of chaotic behavior and the non-integrability
%J Annales de l'I.H.P. Physique théorique
%D 1993
%P 99-115
%V 59
%N 1
%I Gauthier-Villars
%U http://archive.numdam.org/item/AIHPA_1993__59_1_99_0/
%G en
%F AIHPA_1993__59_1_99_0
Castilla, M. S. A. C.; Moauro, Vinicio; Negrini, Piero; Oliva, Waldyr Muniz. The four positive vortices problem : regions of chaotic behavior and the non-integrability. Annales de l'I.H.P. Physique théorique, Tome 59 (1993) no. 1, pp. 99-115. http://archive.numdam.org/item/AIHPA_1993__59_1_99_0/

[A - P] H. Aref and N. Pomphrey, Integrable and chaotic Motions of Four Vortices I. The Case of Identical Vortices, Proc. R. Soc. London, Ser. A, Vol. 380, 1982, pp. 359-387. | MR | Zbl

[C-M] A.J. Chorin and J.E. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer Verlag, 1979. | MR | Zbl

[H] P.J. Holmes, Averaging and Chaotic Motions in Forced Oscillations, S.I.A.M. J. Appl. Math., Vol. 38, 1, 1980, pp. 65-79. Errata, Vol. 40, 1, 1981, pp. 167-168. | MR | Zbl

[K] K.M. Khanin, Quasi-Periodic Motions on Vortex Systems, Physics, Vol. 4D, 1982, pp. 261-269. | MR | Zbl

[K-C] J. Koiller and S. Carvalho, Non-Integrability of the 4-Vortex System, Analytical Proof, Comm. Math. Phys., Vol. 120, 1989, pp. 643-652. | MR | Zbl

[M-C] C. Marchioro and M. Pulvirenti, Vortex Methods in Two Dimensional Fluid Dynamics, Edizioni Klim, Roma, 1983. | MR | Zbl

[M] V.K. Melnikov, On the Stability of the Center for Time-Periodic Perturbations, Trans. Moscow Math. Soc., Vol. 12, 1, 1963, pp. 1-57. | MR | Zbl

[Mo] J. Moser, Stable and Random Motions in Dynamical Systems, Princeton Univ. Press, Princeton N.J., Univ. of Tokyo Press, Tokyo, 1973. | MR | Zbl

[O1] W.M. Oliva, Integrability Problems in Hamiltonian Systems, Ravello (Italy), XVI Summer School on Mathematical Physics, 1991.

[O2] W.M. Oliva, On the chaotic behaviour and non integrability of the four vortices problems, Ann. Inst. H. Poincaré, Vol. 55, 21, 1991, pp. 707-718. | Numdam | MR | Zbl

[S] S. Smale, The Mathematics of Time. Essays on Dynamical Systems. Economic Process and Related Topics, Springer-Verlag, 1980. | MR | Zbl

[Sy] J.L. Synge, On the Motion of Three Vortices, Can. Journal of Math., Vol. 1, 1949, pp. 257-270. | MR | Zbl

[Z] S.L. Ziglin, Non Integrability of a Problem on the Motion of Four Point Vortices, Soviet. Math. Dokl., Vol. 21, 1, 1980, pp. 296-299. | Zbl