Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A proof of the Kuramoto conjecture for a bifurcation structure of the
  infinite dimensional Kuramoto model   The Kuramoto model is a system of ordinary differential equations for
describing synchronization phenomena defined as a coupled phase oscillators. In
this paper, a bifurcation structure of the infinite dimensional Kuramoto model
is investigated. For a certain non-selfadjoint linear operator, which defines a
linear part of the Kuramoto model, the spectral theory on a space of
generalized functions is developed with the aid of a rigged Hilbert space to
avoid a continuous spectrum on the imaginary axis. Although the linear operator
has an unbounded continuous spectrum on a Hilbert space, it is shown that it
admits a spectral decomposition consisting of a countable number of
eigenfunctions on a space of generalized functions. The semigroup generated by
the linear operator is calculated by using the spectral decomposition to prove
the linear stability of a steady state of the system. The center manifold
theory is also developed on a space of generalized functions. It is proved that
there exists a finite dimensional center manifold on a space of generalized
functions, while a center manifold on a Hilbert space is of infinite
dimensional because of the continuous spectrum on the imaginary axis. The
results are applied to the stability and bifurcation theory of the Kuramoto
model to obtain a bifurcation diagram conjectured by Kuramoto. If the coupling
strength $K$ between oscillators is smaller than some threshold $K_c$, the
de-synchronous state proves to be asymptotically stable, and if $K$ exceeds
$K_c$, a nontrivial solution, which corresponds to the synchronization,
bifurcates from the de-synchronous state.

12
The categories of this paper are: nucl-th
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph