On a quasi-periodic Hopf bifurcation
Annales de l'I.H.P. Analyse non linéaire, Tome 4 (1987) no. 2, pp. 115-168.
@article{AIHPC_1987__4_2_115_0,
     author = {Braaksma, B. L. J. and Broer, H. W.},
     title = {On a quasi-periodic {Hopf} bifurcation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {115--168},
     publisher = {Gauthier-Villars},
     volume = {4},
     number = {2},
     year = {1987},
     mrnumber = {886930},
     zbl = {0614.34043},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_1987__4_2_115_0/}
}
TY  - JOUR
AU  - Braaksma, B. L. J.
AU  - Broer, H. W.
TI  - On a quasi-periodic Hopf bifurcation
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 1987
SP  - 115
EP  - 168
VL  - 4
IS  - 2
PB  - Gauthier-Villars
UR  - http://www.numdam.org/item/AIHPC_1987__4_2_115_0/
LA  - en
ID  - AIHPC_1987__4_2_115_0
ER  - 
%0 Journal Article
%A Braaksma, B. L. J.
%A Broer, H. W.
%T On a quasi-periodic Hopf bifurcation
%J Annales de l'I.H.P. Analyse non linéaire
%D 1987
%P 115-168
%V 4
%N 2
%I Gauthier-Villars
%U http://www.numdam.org/item/AIHPC_1987__4_2_115_0/
%G en
%F AIHPC_1987__4_2_115_0
Braaksma, B. L. J.; Broer, H. W. On a quasi-periodic Hopf bifurcation. Annales de l'I.H.P. Analyse non linéaire, Tome 4 (1987) no. 2, pp. 115-168. http://www.numdam.org/item/AIHPC_1987__4_2_115_0/

[Ar1] V.I. Arnol'D, Proof of a theorem by A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Uspehi Math. Nauk., t. 18, 1963, p. 13-40. Russ. Math. Surveys, t. 18, 5, 1963, p. 9-36. | MR | Zbl

[Ar2] V.I. Arnol'D, Geometrical Methods in the Theory of Ordinary Differential Equations. Springer Verlag, 1983. | MR | Zbl

[Be] E.G. Belaga, On the reducibility of a system of ordinary differential equations in a neighborhood of a conditionally periodic motion. Sov. Math. Dokl., t. 143, 2, 1962, p. 250-255. | MR | Zbl

[Bo] N.N. Bogoljubov, On quasi-periodic solutions in nonlinear problems of mechanics. Proceedings of the First Mathematical Summer School, Pt. I. Naukova Dumka, Kiev, 1964, p. 11-101. | MR

[BN] H. Bohr, O. Neugebauer, Ueber lineare Differentialgleichungen mit konstanten Koeffizienten und fast-periodischen rechter Seite. Nachr. Akad. Wiss. Göttingen, Math. phys. Kl., 1962, p. 8-22. | JFM

[BB] B.L.J. Braaksma, H.W. Broer, Quasi-periodic flow near a codimension one singularity of a divergence free vectorfield in dimension four. In Bifurcation, Théorie Ergodique et Applications, 22-26 juin 1981. Asterisque, t. 98-99, 1982, p. 74-142. | Numdam | MR | Zbl

[Br1] H.W. Broer, Formal normal form theorems for singularities of vector fields and some consequences for bifurcations in the volume preserving case. In Dynamical Systems and Turbulence, Warwick, 1980 (eds. D. Rand and L.-S. Young). Springer L. N. M., t. 898, 1981, p. 54-74. | MR | Zbl

[Br2] H.W. Broer, Quasi-periodicity in local bifurcation theory. In Bifurcation Theory, Mechanics and Physics (eds. C. P. Bruter, A. Aragnol, A. Lichnérowicz). Reidel, 1983, p. 177-208. | MR | Zbl

[Ca] J. Carr, Applications of Centre Manifold Theory. Springer Applied Math. Sciences, t. 35, 1981. | MR | Zbl

[Ch] A. Chenciner, Bifurcations de points fixes elliptiques, I. Courbes invariantes. Publ. Math. I. H. E. S., t. 61, 1985, p. 67-127. | Numdam | MR | Zbl

[CI] A. Chenciner, G. Iooss, Bifurcation de tores invariants. Arch. Rat. Mech. An., t. 69, 2, 1979, p. 109-198. | MR | Zbl

[CH] S.-N. Chow, J.K. Hale, Methods of Bifurcation Theory. Springer, 1982. | MR | Zbl

[Fl] D. Flockerzi, Generalized bifurcation of higher dimensional tori. J. Diff. Eq., t. 55, 1984, p. 346-367. | MR | Zbl

[Fr 1] M. Friedman, On Almost Periodic Solutions of Nonlinear Ordinary Differential Equations. Ph. D. Thesis New York University, 1965.

[Fr2] M. Friedman, Quasi-periodic solutions of nonlinear ordinary differential equations with small damping. Bull. Am. Math. Soc., t. 73, 1967, p. 460-464. | MR | Zbl

[Ha1] J.K. Hale, Integral manifolds of perturbed differential systems. Ann. Math., t. 73, 1961, p. 496-531. | MR | Zbl

[Ha2] J.K. Hale, Ordinary Differential Equations. Wiley and Sons, 1969. Krieger, 1980. | MR | Zbl

[Hau] F. Hausdorff, Set Theory (2nd ed.). Chelsea Publishing Company, 1962. | MR | Zbl

[HPS] M.W. Hirsch, C.C. Pugh, M. Shub, Invariant Manifolds. Springer L. N. M., t. 583, 1977. | MR | Zbl

[Hö] L. Hörmander, On the division of distributions by polynomials. Ark. Mat., t. 3, 1958, p. 555-568. | MR | Zbl

[Io] G. Iooss, Bifurcation of Maps and Applications. North Holland, 1979. | MR | Zbl

[Ma] I.G. Malkin, Some Problems in the Theory of Nonlinear Oscillations. State Publishing House, Moscow, 1956.

[MM] J.E. Marsden, M. Mccracken, The Hopf Bifurcation and its Applications. Springer, 1976. | MR | Zbl

[VdM] J.C. Van Der Meer, The Hamiltonian Hopf Bifurcation. Springer L. N. M., t. 1160, 1985. | MR | Zbl

[Mo1] J.K. Moser, Combination tones for Duffing's equation. Comm. Pure Appl. Math., t. 18, 1965, p. 167-181. | MR | Zbl

[Mo2] J.K. Moser, Convergent series expansions for quasi-periodic motions. Math. Ann., t. 169, 1967, p. 136-176. | MR | Zbl

[Po] H. Poincaré, Thèse. In Œuvres 1, 1879, LIX-CXXIX, Gauthier-Villars, 1928.

[Pö] J. Pöschel, Integrability of Hamiltonian systems on Cantor sets. Comm. Pure Appl. Math., t. 35, 1982, p. 653-696. | MR | Zbl

[Se] G.R. Sell, Bifurcation of higher dimensional tori. Arch. Rat. Mech. An., t. 69, 3, 1979, p. 199-230. | MR | Zbl

[Sij] J. Sijbrand, Studies in Nonlinear Stability and Bifurcation Theory, Ph. D. Thesis University of Utrecht, 1981.

[Stei] E. Stein, Singular Integrals and Differentiability-Properties of Functions. Princeton, 1970. | MR | Zbl

[Ster] S. Sternberg, On the structure of local homeomorphisms of Euclidean n-spaces, II. Am. J. of Math., t. 80, 1958, p. 623-631. | MR | Zbl

[Sto] J.J. Stoker, Nonlinear Vibrations. Interscience New York, 1950. | MR

[Str] S. J. Van Strien, Center manifolds are not C∞. Math. Z., t. 116, 1979, p. 143-145. | MR | Zbl

[Ta] F. Takens, Singularities of vector fields. Publ. Math. I. H. E. S., t. 43, 1974, p. 48-100. | Numdam | MR | Zbl

[Wh] H. Whitney, Analytic extensions of differentiable functions defined in closed sets. Trans. Am. Math. Soc., t. 36, 1934, p. 63-89. | JFM | MR | Zbl

[Ze] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems I. Comm. Pure Appl. Math., t. 28, 1975, p. 91-140. | MR | Zbl