11,266 research outputs found
Why must we work in the phase space?
We are going to prove that the phase-space description is fundamental both in
the classical and quantum physics. It is shown that many problems in
statistical mechanics, quantum mechanics, quasi-classical theory and in the
theory of integrable systems may be well-formulated only in the phase-space
language.Comment: 130 page
Basins of Attraction for Chimera States
Chimera states---curious symmetry-broken states in systems of identical
coupled oscillators---typically occur only for certain initial conditions. Here
we analyze their basins of attraction in a simple system comprised of two
populations. Using perturbative analysis and numerical simulation we evaluate
asymptotic states and associated destination maps, and demonstrate that basins
form a complex twisting structure in phase space. Understanding the basins'
precise nature may help in the development of control methods to switch between
chimera patterns, with possible technological and neural system applications.Comment: Please see Ancillary files for the 4 supplementary videos including
description (PDF
Quasiclassical and Quantum Systems of Angular Momentum. Part II. Quantum Mechanics on Lie Groups and Methods of Group Algebras
In Part I of this series we presented the general ideas of applying
group-algebraic methods for describing quantum systems. The treatment was there
very "ascetic" in that only the structure of a locally compact topological
group was used. Below we explicitly make use of the Lie group structure. Basing
on differential geometry enables one to introduce explicitly representation of
important physical quantities and formulate the general ideas of quasiclassical
representation and classical analogy
Bistable Chimera Attractors on a Triangular Network of Oscillator Populations
We study a triangular network of three populations of coupled phase
oscillators with identical frequencies. The populations interact nonlocally, in
the sense that all oscillators are coupled to one another, but more weakly to
those in neighboring populations than to those in their own population. This
triangular network is the simplest discretization of a continuous ring of
oscillators. Yet it displays an unexpectedly different behavior: in contrast to
the lone stable chimera observed in continuous rings of oscillators, we find
that this system exhibits \emph{two coexisting stable chimeras}. Both chimeras
are, as usual, born through a saddle node bifurcation. As the coupling becomes
increasingly local in nature they lose stability through a Hopf bifurcation,
giving rise to breathing chimeras, which in turn get destroyed through a
homoclinic bifurcation. Remarkably, one of the chimeras reemerges by a reversal
of this scenario as we further increase the locality of the coupling, until it
is annihilated through another saddle node bifurcation.Comment: 12 pages, 5 figure
On the Hyperbolicity of Lorenz Renormalization
We consider infinitely renormalizable Lorenz maps with real critical exponent
and combinatorial type which is monotone and satisfies a long return
condition. For these combinatorial types we prove the existence of periodic
points of the renormalization operator, and that each map in the limit set of
renormalization has an associated unstable manifold. An unstable manifold
defines a family of Lorenz maps and we prove that each infinitely
renormalizable combinatorial type (satisfying the above conditions) has a
unique representative within such a family. We also prove that each infinitely
renormalizable map has no wandering intervals and that the closure of the
forward orbits of its critical values is a Cantor attractor of measure zero.Comment: 63 pages; 10 figure
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