34,126 research outputs found
Stability Properties of 1-Dimensional Hamiltonian Lattices with Non-analytic Potentials
We investigate the local and global dynamics of two 1-Dimensional (1D)
Hamiltonian lattices whose inter-particle forces are derived from non-analytic
potentials. In particular, we study the dynamics of a model governed by a
"graphene-type" force law and one inspired by Hollomon's law describing
"work-hardening" effects in certain elastic materials. Our main aim is to show
that, although similarities with the analytic case exist, some of the local and
global stability properties of non-analytic potentials are very different than
those encountered in systems with polynomial interactions, as in the case of 1D
Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Our approach is to study the motion
in the neighborhood of simple periodic orbits representing continuations of
normal modes of the corresponding linear system, as the number of particles
and the total energy are increased. We find that the graphene-type model is
remarkably stable up to escape energy levels where breakdown is expected, while
the Hollomon lattice never breaks, yet is unstable at low energies and only
attains stability at energies where the harmonic force becomes dominant. We
suggest that, since our results hold for large , it would be interesting to
study analogous phenomena in the continuum limit where 1D lattices become
strings.Comment: Accepted for publication in the International Journal of Bifurcation
and Chao
Iterated maps for clarinet-like systems
The dynamical equations of clarinet-like systems are known to be reducible to
a non-linear iterated map within reasonable approximations. This leads to time
oscillations that are represented by square signals, analogous to the Raman
regime for string instruments. In this article, we study in more detail the
properties of the corresponding non-linear iterations, with emphasis on the
geometrical constructions that can be used to classify the various solutions
(for instance with or without reed beating) as well as on the periodicity
windows that occur within the chaotic region. In particular, we find a regime
where period tripling occurs and examine the conditions for intermittency. We
also show that, while the direct observation of the iteration function does not
reveal much on the oscillation regime of the instrument, the graph of the high
order iterates directly gives visible information on the oscillation regime
(characterization of the number of period doubligs, chaotic behaviour, etc.)
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