Coupled Hopf-bifurcations : Persistent examples of n-quasiperiodicity determined by families of 3-jets

In this note examples are presented of vector fields depending on parameters and determined by the 3-jet, which display persistent occurrence of n-quasiperiodicity. In the parameter space this occurrence has relatively large measure. A leading example consists of weakly coupled Hopf bifurcations. This example, however, is extended to full generality in the space of all 3-jets.

[1]  F. Takens,et al.  Occurrence of strange AxiomA attractors near quasi periodic flows onTm,m≧3 , 1978 .

[2]  P. Polácik,et al.  Stable periodic solutions of a spatially homogeneous nonlocal reaction–diffusion equation , 1996, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[3]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[4]  Florian Wagener,et al.  Resonances in skew and reducible quasi-periodic Hopf bifurcations , 2000 .

[5]  G. B. Huitema Unfoldings of quasi-periodic tori , 1988 .

[6]  S. A. Robertson,et al.  NONLINEAR OSCILLATIONS, DYNAMICAL SYSTEMS, AND BIFURCATIONS OF VECTOR FIELDS (Applied Mathematical Sciences, 42) , 1984 .

[7]  David Ruelle,et al.  Occurrence of strange Axiom A attractors near quasiperiodic flows on $T^{m}$,$\,m\geq 3$ , 1979 .

[8]  H. Broer,et al.  Quasi-Periodic Stability of Subfamilies of an Unfolded Skew Hopf Bifurcation , 2000 .

[9]  Floris Takens,et al.  Singularities of vector fields , 1974 .

[10]  H. Broer,et al.  On a quasi-periodic Hopf bifurcation , 1987 .

[11]  J. Guckenheimer,et al.  Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos , 1991 .

[12]  H. Broer,et al.  Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case , 1981 .

[13]  André Vanderbauwhede,et al.  Centre Manifolds, Normal Forms and Elementary Bifurcations , 1989 .

[14]  F. Takens,et al.  Mixed spectra and rotational symmetry , 1993 .

[15]  B. Fiedler,et al.  Complicated dynamics of scalar reaction diffusion equations with a nonlocal term , 1990, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[16]  Henryk Żoł̨dek,et al.  Bifurcations of certain family of planar vector fields tangent to axes , 1987 .

[17]  F. Takens,et al.  Integrable and non-integrable deformations of the skew Hopf bifurcation , 1999 .

[18]  George Huitema,et al.  Quasi-Periodic Motions in Families of Dynamical Systems: Order amidst Chaos , 2002 .

[19]  Angel Jorbayx,et al.  On the normal behaviour of partially elliptic lower-dimensional tori of Hamiltonian systems , 1997 .

[20]  E. Hopf A mathematical example displaying features of turbulence , 1948 .

[21]  F. Takens,et al.  On the nature of turbulence , 1971 .

[22]  George Huitema,et al.  Toward a quasi-periodic bifurcation theory , 1990 .

[23]  Jürgen Moser,et al.  Convergent series expansions for quasi-periodic motions , 1967 .

[24]  S. Chow,et al.  Normal Forms and Bifurcation of Planar Vector Fields , 1994 .

[25]  George Huitema,et al.  Unfolding and bifurcations of quasi-periodic tori. II: Toward a quasi-periodic bifurcation theory , 1990 .