5,184 research outputs found
Synchronisation in Coupled Sine Circle Maps
We study the spatially synchronized and temporally periodic solutions of a
1-d lattice of coupled sine circle maps. We carry out an analytic stability
analysis of this spatially synchronized and temporally periodic case and obtain
the stability matrix in a neat block diagonal form. We find spatially
synchronized behaviour over a substantial range of parameter space. We have
also extended the analysis to higher spatial periods with similar results.
Numerical simulations for various temporal periods of the synchronized
solution, reveal that the entire structure of the Arnold tongues and the
devil's staircase seen in the case of the single circle map can also be
observed for the synchronized coupled sine circle map lattice. Our formalism
should be useful in the study of spatially periodic behaviour in other coupled
map lattices.Comment: uuencoded, 1 rextex file 14 pages, 3 postscript figure
Stable synchronised states of coupled Tchebyscheff maps
Coupled Tchebyscheff maps have recently been introduced to explain parameters
in the standard model of particle physics, using the stochastic quantisation of
Parisi and Wu. This paper studies dynamical properties of these maps, finding
analytic expressions for a number of periodic states and determining their
linear stability. Numerical evidence is given for nonlinear stability of these
states, and also the presence of exponentially slow dynamics for some ranges of
the parameter. These results indicate that a theory of particle physics based
on coupled map lattices must specify strong physical arguments for any choice
of initial conditions, and explain how stochastic quantisation is obtained in
the many stable parameter regions.Comment: 18 pages, postscript figures incorporated into the text;
acknowledgements adde
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
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
Chimera states in coupled sine-circle map lattices
Systems of coupled oscillators have been seen to exhibit chimera states, i.e.
states where the system splits into two groups where one group is phase locked
and the other is phase randomized. In this work, we report the existence of
chimera states in a system of two interacting populations of sine circle maps.
This system also exhibits the clustered chimera behavior seen earlier in delay
coupled systems. Rich spatio-temporal behavior is seen in different regimes of
the phase diagram.We carry out a detailed analysis of the stability regimes and
map out the phase diagram using numerical and analytic techniques.Comment: 10 pages, 5 picture
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