Survey on dissipative KAM theory including quasi-periodic bifurcation theory (H. Broer)

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

“…For a diffeomorphism (written as in (4)), this transition turns into a quasiperiodic Hopf bifurcation, where a circle attractor loses stability and a two-torus attractor branches off. This scenario has been described extensively by Broer et al [7,9,10,26] as a part of dissipative kam theory (also see [38] for a treatment specific of the HSN case). In this setting, resonances play a very strong role, since they involve a Cantor set of Diophantine conditions in the ω-direction.…”

“…Fourier methods [31,32] might be used to compute it more accurately. However, we observe that for a diffeomorphism at least a threedimensional parameter space is necessary to find a smooth submanifold parameterising a Diophantine family of invariant two-dimensional tori: as prescribed by dissipative kam theory [7,9,10,26], parameter sets where the frequency vector of the invariant two-torus is fixed to a constant value are discrete (zero-dimensional) in the (δ, µ)-plane. So even if one of the two frequencies is fixed to a Diophantine value, resonances of the other frequency (or of the whole frequency vector) are unavoidable as parameter vary smoothly in the (δ, µ)-plane.…”

“…Turning to our model map G (3), the above discussion is summarised in Figure 3: the left panel contains the bifurcation diagram of the diffeomorphism Y 1 γ,µ,ω , given by the time-1 map of vector field family (11), inside the three-dimensional parameter space (γ, µ, ω); in the right panel we sketch our expectations for model map G. A number of theoretical results can be obtained for the vector field family Y γ,µ,ω (11) by invoking standard perturbation theory (normal hyperbolicity [33,37]) and quasi-periodic bifurcation theory [7,9,10,26]. The surface HET of heteroclinic bifurcations of the diffeomorphism Y 1 γ,µ,ω turns into a region characterised by heteroclinic intersections of the polar saddlelike fixed points for the model map G [11,12,13,54].…”

“…The 1:5 resonance gap of the circle C is evidenced by the fuchsia strip at the right of the Lyapunov diagram. In the blue regions outside this gap, the dynamics on C is quasi-periodic and normal-internal resonances are also forbidden [7,9,10,26].…”

Resonance and Fractal Geometry

“…For overviews see [12,48,91]. The quasi-periodic bifurcations are inspired by the classical ones in which equilibria or periodic orbits are replaced by quasi-periodic tori.…”