J. H. Lowenstein, G. Poggiaspalla, F. Vivaldi
Sticky orbits in a kicked-oscillator model
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ABSTRACT. We study a 4-fold symmetric kicked-oscillator map with sawtooth kick
function. For the values of the kick amplitude $\lambda=2\cos(2\pi
p/q)$ with rational $p/q$, the dynamics is known to be pseudochaotic,
with no stochastic web of non-zero Lebesgue measure.
We show that this system can be represented as a piecewise affine map
of the unit square ---the so-called local map--- driving a lattice map.
We develop a framework for the study of long-time behaviour of the
orbits, in the case in which the local map features exact scaling.
We apply this method to several quadratic irrational values of
$\lambda$, for which the local map possesses a full Legesgue
measure of periodic orbits; these are promoted to either periodic
orbits or accelerator modes of the kicked-oscillator map.
By constrast, the aperiodic orbits of the local map can generate
various asymptotic behaviours. For some parameter values the orbits
remain bounded, while others have excursions which grow logarithmically
or as a power of the time. In the power-law case, we derive rigorous
criteria for asymptotic scaling, governed by the largest eigenvalue of
a recursion matrix. We illustrate the various behaviours by performing
exact calculations with algebraic numbers; the hierarchical nature of
the symbolic dynamics allows us to sample extremely long orbits with
high efficiency, i.e., uniformly on a logarithmic time scale.