197 research outputs found
Multi-cluster dynamics in coupled phase oscillator networks
In this paper we examine robust clustering behaviour with multiple nontrivial
clusters for identically and globally coupled phase oscillators. These systems
are such that the dynamics is completely determined by the number of
oscillators N and a single scalar function (the coupling
function). Previous work has shown that (a) any clustering can stably appear
via choice of a suitable coupling function and (b) open sets of coupling
functions can generate heteroclinic network attractors between cluster states
of saddle type, though there seem to be no examples where saddles with more
than two nontrivial clusters are involved. In this work we clarify the
relationship between the coupling function and the dynamics. We focus on cases
where the clusters are inequivalent in the sense of not being related by a
temporal symmetry, and demonstrate that there are coupling functions that give
robust heteroclinic networks between periodic states involving three or more
nontrivial clusters. We consider an example for N=6 oscillators where the
clustering is into three inequivalent clusters. We also discuss some aspects of
the bifurcation structure for periodic multi-cluster states and show that the
transverse stability of inequivalent clusters can, to a large extent, be varied
independently of the tangential stability
Discrete Breathers
Nonlinear classical Hamiltonian lattices exhibit generic solutions in the
form of discrete breathers. These solutions are time-periodic and (typically
exponentially) localized in space. The lattices exhibit discrete translational
symmetry. Discrete breathers are not confined to certain lattice dimensions.
Necessary ingredients for their occurence are the existence of upper bounds on
the phonon spectrum (of small fluctuations around the groundstate) of the
system as well as the nonlinearity in the differential equations. We will
present existence proofs, formulate necessary existence conditions, and discuss
structural stability of discrete breathers. The following results will be also
discussed: the creation of breathers through tangent bifurcation of band edge
plane waves; dynamical stability; details of the spatial decay; numerical
methods of obtaining breathers; interaction of breathers with phonons and
electrons; movability; influence of the lattice dimension on discrete breather
properties; quantum lattices - quantum breathers. Finally we will formulate a
new conceptual aproach capable of predicting whether discrete breather exist
for a given system or not, without actually solving for the breather. We
discuss potential applications in lattice dynamics of solids (especially
molecular crystals), selective bond excitations in large molecules, dynamical
properties of coupled arrays of Josephson junctions, and localization of
electromagnetic waves in photonic crystals with nonlinear response.Comment: 62 pages, LaTeX, 14 ps figures. Physics Reports, to be published; see
also at http://www.mpipks-dresden.mpg.de/~flach/html/preprints.htm
Coupled Oscillator Model for Nonlinear Gravitational Perturbations
Motivated by the gravity/fluid correspondence, we introduce a new method for
characterizing nonlinear gravitational interactions. Namely we map the
nonlinear perturbative form of the Einstein equation to the equations of motion
of a collection of nonlinearly-coupled harmonic oscillators. These oscillators
correspond to the quasinormal or normal modes of the background spacetime. We
demonstrate the mechanics and the utility of this formalism within the context
of perturbed asymptotically anti-de Sitter black brane spacetimes. We confirm
in this case that the boundary fluid dynamics are equivalent to those of the
hydrodynamic quasinormal modes of the bulk spacetime. We expect this formalism
to remain valid in more general spacetimes, including those without a fluid
dual. In other words, although borne out of the gravity/fluid correspondence,
the formalism is fully independent and it has a much wider range of
applicability. In particular, as this formalism inspires an especially
transparent physical intuition, we expect its introduction to simplify the
often highly technical analytical exploration of nonlinear gravitational
dynamics.Comment: 17 pages, 3 figures. Minor fix to match published versio
Fundamentals and applications of spatial dissipative solitons in photonic devices : [Chapter 6]
We review the properties of optical spatial dissipative solitons (SDS). These are stable, self‐localized optical excitations sitting on a uniform, or quasi‐uniform, background in a dissipative environment like a nonlinear optical cavity. Indeed, in optics they are often termed “cavity solitons.” We discuss their dynamics and interactions in both ideal and imperfect systems, making comparison with experiments. SDS in lasers offer important advantages for applications. We review candidate schemes and the tremendous recent progress in semiconductor‐based cavity soliton lasers. We examine SDS in periodic structures, and we show how SDS can be quantitatively related to the locking of fronts. We conclude with an assessment of potential applications of SDS in photonics, arguing that best use of their particular features is made by exploiting their mobility, for example in all‐optical delay lines
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