Coupled Hopf-bifurcations: Persistent examples of n-quasiperiodicity determined by families of 3-jets
Geometric methods in dynamics (I) : Volume in honor of Jacob Palis, Astérisque, no. 286 (2003), pp. 223-229.
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     title = {Coupled {Hopf-bifurcations:} {Persistent} examples of $n$-quasiperiodicity determined by families of 3-jets},
     booktitle = {Geometric methods in dynamics (I) : Volume in honor of Jacob Palis},
     editor = {de Melo, Wellington and Viana, Marcelo and Yoccoz, Jean-Christophe},
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Broer, Henk. Coupled Hopf-bifurcations: Persistent examples of $n$-quasiperiodicity determined by families of 3-jets, dans Geometric methods in dynamics (I) : Volume in honor of Jacob Palis, Astérisque, no. 286 (2003), pp. 223-229. http://www.numdam.org/item/AST_2003__286__223_0/

[1] C. Baesens, J. Guckenheimer, S. Kim and R. S. Mackay, Three coupled oscillators: Mode-locking, global bifurcation and toroidal chaos, Physica D, (1991), 387-475. | DOI | MR | Zbl

[2] B. L. J. Braaksma & H. W. Broer, On a quasi-periodic Hopf bifurcation, Ann. Institut Henri Poincaré, Analyse non linéaire, 4, no.2 (1987), 115-168. | DOI | EuDML | Numdam | MR | Zbl

[3] B. L. J. Braaksma, H. W. Broer and G. B. Huitema, Towards a Quasiperiodic Bifurcation Theory, Mem. AMS, 83(421) (1990), 83-175. | Zbl

[4] H. W. Broer, Formal normal forms for vector fields and some consequences for bifurcations in the volume preserving case. In: D. Rand and L.S. Young (eds.), Dynamical Systems and Turbulence, Warwick 1980 LNM 898 (1981), 75-89, Springer-Verlag. | MR | Zbl

[5] H. W. Broer, KAM-Theory: Multi-Periodicity in conservative and dissipative systems, Nieuw Arch. Wish. 14(1), (1996), 1-15. | MR | Zbl

[6] H. W. Broer, G. B. Huitema and F. Takens, Unfoldings of Quasi-Periodic Tori, Mem. AMS, 83(421) (1990), 1-82. | Zbl

[7] H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-periodic tori in families of dynamical systems: order amidst chaos, LNM 1645, Springer Verlag 1996. | Zbl

[8] H. W. Broer & F. Takens, Mixed spectrum and rotational symmetry, Arch. Rational Mech. An. 124 (1993), 13-42. | DOI | MR | Zbl

[9] H. W. Broer, F. Takens and F. O. O. Wagener, Integrable and non-integrable deformations of the skew Hopf bifurcation, Reg. and Chaot. Dyn. 4 (1999), 17-43. | DOI | MR | Zbl

[10] H. W. Broer and F. O. O. Wagener, Quasi-periodic stability of subfamilies of an unfolded skew-Hopf bifurcation, Arch. Rational Mech. An. 152 (2000), 283-326. | DOI | MR | Zbl

[11] S-N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press 1994. | MR | Zbl

[12] B. Fiedler and P. Poláčik, Complicated dynamics of scalar reaction diffusion equations with a nonlocal term, Proc. Roy. Soc. Edinburgh Sect. A, (Mathematical and Physical Sciences) 115(1-2) (1990), 167-192. | DOI | MR | Zbl

[13] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Appl. Math. Sciences 42, Springer Verlag 1983. | DOI | MR | Zbl

[14] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, LNM 583, Springer Verlag 1977. | MR | Zbl

[15] E. Hopf, A mathematical example displaying features of turbulence. Commun. Appl. Math. 1 (1948), 303-322. | DOI | MR | Zbl

[16] A. Jorba and J. Villanueva, On the Normal Behaviour of Partially Elliptic Lower-Dimensional Tori of Hamiltonian Systems, Nonlinearity 10 (1997), 783-822. | DOI | MR | Zbl

[17] J. K. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136-176. | DOI | EuDML | MR | Zbl

[18] S. E. Newhouse, D. Ruelle and F. Takens, Occurrence of strange Axiom A attractors near quasiperiodic flows on 𝕋 m ,m3, Commun. Math. Phys. 64 (1978), 35-40. | DOI | MR | Zbl

[19] P. Poláčik and V. Sosovička, Stable periodic solutions of a spatially homogeneous non­local reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, (Mathematical and Physical Sciences) 126(4) (1996), 867-884. | DOI | MR | Zbl

[20] D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys. 20 (1971), 167-192; | DOI | MR | Zbl

D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys. 23 (1971), 343-4. | DOI | Numdam | MR | Zbl

[21] C. Simó, Perturbations of translations in the two-dimensional torus: the case near resonance, Proceedings VI CEDYA, Universidad de Zaragoza, 1984.

[22] F. Takens, Singularities of Vector Fields, Commun. Math. Phys. 20, 1970, 167-192, | Zbl

F. Takens, Singularities of Vector Fields, Commun. Math. Phys. 23, (1971), 343-344. | Zbl

F. Takens, Singularities of Vector Fields, Publ. I.H.E.S. 43 (1974), 47-100. | DOI | EuDML | Numdam | MR | Zbl

[23] F. Takens and F. O. O. Wagener, Resonances in skew and reducible quasi-periodic Hopf bifurcations, Nonlinearity 13 (2000), 377-396. | DOI | MR | Zbl

[24] A. Vanderbauwhede, Centre Manifolds, Normal Forms and Elementary Bifurcations, Dynamics Reported 2 (1989), 89-170. | DOI | MR | Zbl

[25] F. O. O. Wagener, On the skew Hopf bifurcation. PhD thesis, University of Groningen, 1998. | Zbl

[26] H. Zoladek, Bifurcation of a certain family of planar vector fields tangent to axes, Journ. Diff. Eqns. 67 (1987), 1-55. | DOI | MR | Zbl