Institute of Mathematics for Industry

Kyushu University

Hayato Chiba chiba@imi.kyushu-u.ac.jp

Feb/14/2014

A Spectral Theory of

Linear Operators on a Gelfand Triplet

and

its Application to Coupled Oscillators

Contents

・ Synchronization –Kuramoto model-

・Idea

・ Spectral theory on Gelfand triplets

・ Bifurcation

Kuramoto model (1975)

synchronization occurs.

Kuramoto model (1975)

Order parameter :

synchro de-synchro

Open problem.

Kuramoto conjecture (1984):

: distribution function for

If is even and unimodal, then

Infinite- dimensional Kuramoto model

Evolution eq. for : prob. dens.

func. for

continuous limit

Fourier coefficients:

Evolution equations on

Trivial solution (de-synchro state)

, multiplication operator

, projection

Spectrum of

・conti. spec:

・eigenvalues: roots of

which exist iff .

Stability? Bifurcations?

Let’s reformulate in terms of the Gelfand triplet.

Contents

・ Synchronization –Kuramoto model-

・Idea

・ Spectral theory on Gelfand triplets

・ Bifurcation

continuous spectrum

Resolvent. , analytic w.r.t. on

Eq. ,

Sol.

Resolvent. , analytic w.r.t. on

The resolvent diverges on as a function in . . .

But it may converge in a larger space.

e.g. Gaussian. (as a usual func.)

(Dirac delta.)

Idea: Consider the multiplication operator on

Resolvent.

When , However, the inner product may exist!

(if and are continuous )

Spectrum.

(if and are holomorphic )

The anayltic continuation

exists only when and are holomorphic.

: some class of holo. functions on .

: the dual space (the set of conti. linear functionals on )

The mapping defines a linear functional on . The mapping defines a linear map from into . The resolvent has an analytic continuation from the lower half plane to the upper half plane as an operator from into .

Gelfand triplet.

spectrum singularities of the resolvent (on ).

generalized spectrum singularities of an analytic

continuation of the resolvent (on a Riemann surface).

The first Riemann sheet The second Riemann sheet

Through the Laplace inversion formula,

the generalized spectrum induces an exponential

decay of a solution.

Contents

・ Synchronization –Kuramoto model-

・Idea

・ Spectral theory on Gelfand triplets

・ Bifurcation

Infinite- dimensional Kuramoto model

Evolution eq. for : prob. dens.

func. for

continuous limit

Stability? Bifurcations?

Let’s reformulate in terms of the Gelfand triplet.

A spectral theory on a Gelfand triplet.

For the Kuramoto model, is an inductive limit of a certain series of Banach spaces of holomorphic functions near the real axis. The dual space is a Frechet Montel space.

A spectral theory on a Gelfand triplet.

Define by

is an analytic continuation of in the generalized sense.

Eigen-problem

Since the analy. conti. of in the dual sp. is …

Def. If the equation

has a nonzero solution , is called the generalized eigenvalue and is called the generalized eigenfunction associated with .

is given by

Since the analy. conti. of in the dual sp. is …

Define the generalized resolvent by

: isolated generalized eigenvalue.

Define the generalized Riesz projection by

Main theorems. : the set of gene. eigenvalues.

(i)

(ii)

(iii) is holomorphic on

(iv)

For the Kuramoto model, is countable. They lie on the second Riemann sheet of the resolvent.

Spectrum in -sense Generalized spectrum

continuous spectrum

Semigroup in -sense. Semigroup on the dual sp.

Theorem. (spectral decomposition)

For any ,

Theorem. (completeness)

(i) A system of generalized eigenfunctions is complete

( ).

(ii) linearly independent :

(iii) the decomposition is unique.

The next purpose is a bifurcation at .

When , all gener. eigenvalues lie on the left

half plane, which proves the stability of the de-synchro state.

Contents

・ Synchronization –Kuramoto model-

・Idea

・ Spectral theory on Gelfand triplets

・ Bifurcation

On , the center subspace is

infinite-dim because

Generalized center subspace:

Existence of a finite-dim center mfd. on .

For the Kuramoto model, this is one-dimensional.

(continuous model)

inclusion.

Dynamics on a center manifold.

Kuramoto conjecture is proved.

References H.Chiba, A proof of the Kuramoto conjecture for a bifurcation structure of the infinite dimensional Kuramoto model, Ergo. Theo. Dyn. Syst, (2013) H. Chiba, A spectral theory of linear operators on rigged Hilbert spaces under certain analyticity conditions. (arXiv:1107.5858) H.Chiba, I.Nishikawa, Center manifold reduction for a large population of globally coupled phase oscillators, Chaos, 21, 043103, (2011)

(continuous model)

Dynamics on a center manifold.