User peter cudmore - MathOverflow

Answer by Peter Cudmore for Calculate the inverse of a matrix

2013-04-22T00:54:25Z

I think this is more appropriate for mathSE, however;

Recall that a linear differential equation can have fixed points as nodes, saddles and centers.

Your question requires the fixed point to be a node of some variety, so the real parts of the eigenvalues of $A$ must be the same sign.

If they all have negative real parts, the node is stable and you will get the solution you are after.

If they all have positive real parts, simply reverse the direction of time and solve as you figured in your comment above.

If they are of alternating sign or have zero real part, this approach will no longer work.

Lower bounds of laplace transform of characteristic functions

2013-02-22T02:39:13Z

Cross-posted on maths.stackexchange

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability distribution and $\mu$ is some complex variable with strictly positive real part $\mu_r>0$.

It is easy enough to prove that this function is bounded above, $|f(\mu)| \le \frac{1}{\mu_r}$.

However, numerics suggest that for 'nice' distributions (symmetric, and non-increasing on $[0,\infty)$), the integral is bounded below by $|f(\mu_r)|\le|f(\mu)|$.

So my questions are:

1) Are there established results on finding lower bounds of the Laplace transform of a characteristic function. (My trawling of google scholar and the like haven't produced anything. It seems like such an elementary problem that surely someone has considered it before.)

2) If not what general techniques are used to look for lower bounds on integrals such as this?

Background: The integral comes from investigating how non-identical frequency distribution of linear oscillators a particular collective behavior problem. This has occurred before in such problems and has been treated asymptotically for strictly real $\mu$ (ref. 1) and commented on for complex $\mu$ (appendix ref. 2). Due to the parameter regions we are interested in (around $\Re{\mu} =1$) the approximations made previously no longer hold.

Many Thanks, Pete.

References:

Ref. 1 - RE. Mirollo, SH Strogatz. Amplitude Death in Limit Cycle Oscillators. http://www.springerlink.com/index/ln051rp502550471.pdf

Ref. 2 - PC. Matthews, RE. Mirollo, SH Strogatz. Dynamics of a large system of coupled nonlinear oscillators. http://www.sciencedirect.com/science/article/pii/016727899190129W

edit: (typo) corrected to non-increasing

Answer by Peter Cudmore for Synchronization frequency of the Kuramoto model for coupling matrix with constant rows

2011-07-12T16:59:14Z

As far as literature goes:

"From Kuramoto to Crawford..." - S Strogatz: http://omnis.if.ufrj.br/~monica/ACMSM2008/StrogatzPD2000.pdf

"Synchronization in Complex Networks" - Arenas et al http://arxiv.org/PS_cache/arxiv/pdf/0805/0805.2976v3.pdf

However,

The special case you mention can be reduced to the mean field kuramoto model by introducing a complex order parameter. $ re^{i \Psi} = \frac{1}{n}\sum\limits_{j=1}^n e^{i\theta_j} $

So $\dot\theta_i = \omega_i + K_ir\sin(\Psi - \theta_i)$

Also, if you let $\omega_i \rightarrow \Omega + \Delta\omega_i$ and $\theta_i \rightarrow \theta_i + \Omega t $ where $\Omega$ is your mean frequency

you get

$\dot\theta_i = \Delta\omega_i + K_ir\sin(\Psi - \theta_i)$

with the mean of all $\Delta\omega_i$ are zero and most importantly, $\dot\theta_i = 0$ now corresponds to that oscillator running at your mean frequency.

Now if $K_i >\frac{2}{\pi g(0)} \forall i$ with $g$ the p.d.f for your $\Delta\omega_i$'s, then you'll definately get a large proportion of oscillators synching to your mean frequency. (Strogatz' paper covers this derivation for constant $K$)

How many is a tough question, depending on your p.d.f. For general $K_i$ i think Arenas' paper should provide a starting point.

Eigenvalues of the sum of a diagonal and a unit matrix

2011-06-16T10:03:40Z

I'm trying to find information on the eigenvalues of an $n \times n$ matrix A such that

$A = D + J$

Where $D$ is some complex valued diagonal matrix, and $J$ is an matrix consisting of all $1$'s.
When $D$ has identical values, the problem is equivalent to finding the eigenvalues of $J$.

So my question is this:
If $D$ has non-identical values (specifically, non-identical imaginary components), is there an elementary way to compute the eigenvalues of $A$ ?

The problem comes from linearising about the origin of a system of $n$ near identical coupled resonators. $D$ relates to the behaviour of each resonator, $J$ relates to the coupling process.

Comment by Peter Cudmore

2013-03-20T00:14:15Z

This looks strikingly familiar to the Poincare Linearization theorem and Normal Forms in the theory of ODEs.

Comment by Peter Cudmore

2011-06-16T13:11:05Z

Thanks very much, this was exactly what i was looking for.