Resonance tongues in the quasi-periodic hill-Schrödinger equation with three frequencies
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Cita com:
hdl:2117/11094
Tipus de documentReport de recerca
Data publicació2010-07
Condicions d'accésAccés obert
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Abstract
In this paper we investigate numerically the following Hill’s equation
x00 + (a + bq(t))x = 0 where q(t) = cos t + cosp2t + cosp3t is a quasiperiodic
forcing with three rationally independent frequencies. It appears,also,
as the eigenvalue equation of a Schr¨odinger operator with quasi-periodic potential.
Massive numerical computations were performed for the rotation number
and the Lyapunov exponent in order to detect open and collapsed gaps, resonance
tongues. Our results show that the quasi-periodic case with three
independent frequencies is very different not only from the periodic analogs,
but also from the case of two frequencies. Indeed, for large values of b the
spectrum contains open intervals at the bottom. From a dynamical point of
view we numerically give evidence of the existence of open intervals of a, for
large b where the system is nonuniformly hyperbolic: the system does not have
an exponential dichotomy but the Lyapunov exponent is positive. In contrast
with the region with zero Lyapunov exponents, both the rotation number and
the Lyapunov exponent do not seem to have square root behavior at endpoints
of gaps. The rate of convergence to the rotation number and the Lyapunov
exponent in the nonuniformly hyperbolic case is also seen to be different from
the reducible case.
Forma part[prepr2010087PuiS]
URL repositori externhttp://www.ma1.upc.edu/~jpuig/preprints/puig-simo_10_2.pdf
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