14,416 research outputs found
Lam\'e polynomials, hyperelliptic reductions and Lam\'e band structure
The band structure of the Lam\'e equation, viewed as a one-dimensional
Schr\"odinger equation with a periodic potential, is studied. At integer values
of the degree parameter l, the dispersion relation is reduced to the l=1
dispersion relation, and a previously published l=2 dispersion relation is
shown to be partially incorrect. The Hermite-Krichever Ansatz, which expresses
Lam\'e equation solutions in terms of l=1 solutions, is the chief tool. It is
based on a projection from a genus-l hyperelliptic curve, which parametrizes
solutions, to an elliptic curve. A general formula for this covering is
derived, and is used to reduce certain hyperelliptic integrals to elliptic
ones. Degeneracies between band edges, which can occur if the Lam\'e equation
parameters take complex values, are investigated. If the Lam\'e equation is
viewed as a differential equation on an elliptic curve, a formula is
conjectured for the number of points in elliptic moduli space (elliptic curve
parameter space) at which degeneracies occur. Tables of spectral polynomials
and Lam\'e polynomials, i.e., band edge solutions, are given. A table in the
older literature is corrected.Comment: 38 pages, 1 figure; final revision
Vanishing Twist in the Hamiltonian Hopf Bifurcation
The Hamiltonian Hopf bifurcation has an integrable normal form that describes
the passage of the eigenvalues of an equilibrium through the 1: -1 resonance.
At the bifurcation the pure imaginary eigenvalues of the elliptic equilibrium
turn into a complex quadruplet of eigenvalues and the equilibrium becomes a
linearly unstable focus-focus point. We explicitly calculate the frequency map
of the integrable normal form, in particular we obtain the rotation number as a
function on the image of the energy-momentum map in the case where the fibres
are compact. We prove that the isoenergetic non-degeneracy condition of the KAM
theorem is violated on a curve passing through the focus-focus point in the
image of the energy-momentum map. This is equivalent to the vanishing of twist
in a Poincar\'e map for each energy near that of the focus-focus point. In
addition we show that in a family of periodic orbits (the non-linear normal
modes) the twist also vanishes. These results imply the existence of all the
unusual dynamical phenomena associated to non-twist maps near the Hamiltonian
Hopf bifurcation.Comment: 18 pages, 4 figure
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