63 research outputs found
The degenerate C. Neumann system I: symmetry reduction and convexity
The C. Neumann system describes a particle on the sphere S^n under the
influence of a potential that is a quadratic form. We study the case that the
quadratic form has l+1 distinct eigenvalues with multiplicity. Each group of
m_\sigma equal eigenvalues gives rise to an O(m_\sigma)-symmetry in
configuration space. The combined symmetry group G is a direct product of l+1
such factors, and its cotangent lift has an Ad^*-equivariant Momentum mapping.
Regular reduction leads to the Rosochatius system on S^l, which has the same
form as the Neumann system albeit for an additional effective potential.
To understand how the reduced systems fit together we use singular reduction
to construct an embedding of the reduced Poisson space T^*{S^n}/G into
R^{3l+3}$. The global geometry is described, in particular the bundle structure
that appears as a result of the superintegrability of the system. We show how
the reduced Neumann system separates in elliptical-spherical co-ordinates. We
derive the action variables and frequencies as complete hyperelliptic integrals
of genus l. Finally we prove a convexity result for the image of the Casimir
mapping restricted to the energy surface.Comment: 36 page
Quasi-periodic stability of normally resonant tori
We study quasi-periodic tori under a normal-internal resonance, possibly with
multiple eigenvalues. Two non-degeneracy conditions play a role. The first of
these generalizes invertibility of the Floquet matrix and prevents drift of the
lower dimensional torus. The second condition involves a Kolmogorov-like
variation of the internal frequencies and simultaneously versality of the
Floquet matrix unfolding. We focus on the reversible setting, but our results
carry over to the Hamiltonian and dissipative contexts
Normal-normal resonances in a double Hopf bifurcation
We investigate the stability loss of invariant n-dimensional quasi-periodic
tori during a double Hopf bifurcation, where at bifurcation the two normal
frequencies are in normal-normal resonance. Invariants are used to analyse the
normal form approximations in a unified manner. The corresponding dynamics form
a skeleton for the dynamics of the original system. Here both normal
hyperbolicity and KAM theory are being used.Comment: 22 pages, 6 figure
Normal resonances in a double Hopf bifurcation
We introduce a framework to systematically investigate the resonant double Hopf bifurcation. We use the basic invariants of the ensuing T1-action to analyse the approximating normal form truncations in a unified manner. In this way we obtain a global description of the parameter space and thus find the organising resonance droplet, which is the present analogue of the resonant gap. The dynamics of the normal form yields a skeleton for the dynamics of the original system. In the ensuing perturbation theory both normal hyperbolicity (centre manifold theory) and KAM theory are being used
Bifurcations and Monodromy of the Axially Symmetric 1:1:-2 Resonance
We consider integrable Hamiltonian systems in three degrees of freedom near
an elliptic equilibrium in 1:1:-2 resonance. The integrability originates from
averaging along the periodic motion of the quadratic part and an imposed
rotational symmetry about the vertical axis. Introducing a detuning parameter
we find a rich bifurcation diagram, containing three parabolas of Hamiltonian
Hopf bifurcations that join at the origin. We describe the monodromy of the
resulting ramified 3-torus bundle as variation of the detuning parameter lets
the system pass through 1:1:-2 resonance
The 1:2:4 resonance in a particle chain
We consider four masses in a circular configuration with nearest-neighbour
interaction, generalizing the spatially periodic Fermi--Pasta--Ulam-chain where
all masses are equal. We identify the mass ratios that produce the
~resonance --- the normal form in general is non-integrable already
at cubic order. Taking two of the four masses equal allows to retain a discrete
symmetry of the fully symmetric Fermi--Pasta--Ulam-chain and yields an
integrable normal form approximation. The latter is also true if the cubic
terms of the potential vanish. We put these cases in context and analyse the
resulting dynamics, including a detuning of the ~resonance within
the particle chain.Comment: 1 Latex file, 6 figure
Dynamical stability of quasi-periodic response solutions in planar conservative systems
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