334 research outputs found
Homoclinic points of 2-D and 4-D maps via the Parametrization Method
An interesting problem in solid state physics is to compute discrete breather
solutions in coupled 1--dimensional Hamiltonian particle chains
and investigate the richness of their interactions. One way to do this is to
compute the homoclinic intersections of invariant manifolds of a saddle point
located at the origin of a class of --dimensional invertible
maps. In this paper we apply the parametrization method to express these
manifolds analytically as series expansions and compute their intersections
numerically to high precision. We first carry out this procedure for a
2--dimensional (2--D) family of generalized Henon maps (=1), prove
the existence of a hyperbolic set in the non-dissipative case and show that it
is directly connected to the existence of a homoclinic orbit at the origin.
Introducing dissipation we demonstrate that a homoclinic tangency occurs beyond
which the homoclinic intersection disappears. Proceeding to , we
use the same approach to determine the homoclinic intersections of the
invariant manifolds of a saddle point at the origin of a 4--D map consisting of
two coupled 2--D cubic H\'enon maps. In dependence of the coupling the
homoclinic intersection is determined, which ceases to exist once a certain
amount of dissipation is present. We discuss an application of our results to
the study of discrete breathers in two linearly coupled 1--dimensional particle
chains with nearest--neighbor interactions and a Klein--Gordon on site
potential.Comment: 24 pages, 10 figures, videos can be found at
https://comp-phys.tu-dresden.de/supp
The Discrete Nonlinear Schr\"odinger equation - 20 Years on
We review work on the Discrete Nonlinear Schr\"odinger (DNLS) equation over
the last two decades.Comment: 24 pages, 1 figure, Proceedings of the conference on "Localization
and Energy Transfer in Nonlinear Systems", June 17-21, 2002, San Lorenzo de
El Escorial, Madrid, Spain; to be published by World Scientifi
Artificial graphene as a tunable Dirac material
Artificial honeycomb lattices offer a tunable platform to study massless
Dirac quasiparticles and their topological and correlated phases. Here we
review recent progress in the design and fabrication of such synthetic
structures focusing on nanopatterning of two-dimensional electron gases in
semiconductors, molecule-by-molecule assembly by scanning probe methods, and
optical trapping of ultracold atoms in crystals of light. We also discuss
photonic crystals with Dirac cone dispersion and topologically protected edge
states. We emphasize how the interplay between single-particle band structure
engineering and cooperative effects leads to spectacular manifestations in
tunneling and optical spectroscopies.Comment: Review article, 14 pages, 5 figures, 112 Reference
Coupled Maps with Growth and Death: An Approach to Cell Differentiation
An extension of coupled maps is given which allows for the growth of the
number of elements, and is inspired by the cell differentiation problem. The
growth of elements is made possible first by clustering the phases, and then by
differentiating roles. The former leads to the time sharing of resources, while
the latter leads to the separation of roles for the growth. The mechanism of
the differentiation of elements is studied. An extension to a model with
several internal phase variables is given, which shows differentiation of
internal states. The relevance of interacting dynamics with internal states
(``intra-inter" dynamics) to biological problems is discussed with an emphasis
on heterogeneity by clustering, macroscopic robustness by partial
synchronization and recursivity with the selection of initial conditions and
digitalization.Comment: LatexText,figures are not included. submitted to PhysicaD
(1995,revised 1996 May
Scale invariant distribution functions in quantum systems with few degrees of freedom
Scale invariance usually occurs in extended systems where correlation
functions decay algebraically in space and/or time. Here we introduce a new
type of scale invariance, occurring in the distribution functions of physical
observables. At equilibrium these functions decay over a typical scale set by
the temperature, but they can become scale invariant in a sudden quantum
quench. We exemplify this effect through the analysis of linear and non-linear
quantum oscillators. We find that their distribution functions generically
diverge logarithmically close to the stable points of the classical dynamics.
Our study opens the possibility to address integrability and its breaking in
distribution functions, with immediate applications to matter-wave
interferometers.Comment: 8+10 pages. Scipost Submissio
Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems
We study numerically statistical distributions of sums of chaotic orbit
coordinates, viewed as independent random variables, in weakly chaotic regimes
of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam
(FPU-) oscillator chains with different boundary conditions and numbers
of particles and a microplasma of identical ions confined in a Penning trap and
repelled by mutual Coulomb interactions. For the FPU systems we show that, when
chaos is limited within "small size" phase space regions, statistical
distributions of sums of chaotic variables are well approximated for
surprisingly long times (typically up to ) by a -Gaussian
() distribution and tend to a Gaussian () for longer times, as the
orbits eventually enter into "large size" chaotic domains. However, in
agreement with other studies, we find in certain cases that the -Gaussian is
not the only possible distribution that can fit the data, as our sums may be
better approximated by a different so-called "crossover" function attributed to
finite-size effects. In the case of the microplasma Hamiltonian, we make use of
these -Gaussian distributions to identify two energy regimes of "weak
chaos"-one where the system melts and one where it transforms from liquid to a
gas state-by observing where the -index of the distribution increases
significantly above the value of strong chaos.Comment: 32 pages, 13 figures, Submitted for publication to Physica
Scale invariant distribution functions in quantum systems with few degrees of freedom
Scale invariance usually occurs in extended systems where correlation
functions decay algebraically in space and/or time. Here we introduce a new
type of scale invariance, occurring in the distribution functions of physical
observables. At equilibrium these functions decay over a typical scale set by
the temperature, but they can become scale invariant in a sudden quantum
quench. We exemplify this effect through the analysis of linear and non-linear
quantum oscillators. We find that their distribution functions generically
diverge logarithmically close to the stable points of the classical dynamics.
Our study opens the possibility to address integrability and its breaking in
distribution functions, with immediate applications to matter-wave
interferometers.Comment: 8+10 pages. Scipost Submissio
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