Let's.

Just

say.

Thank you Flavio for the introduction
I don't believe Google scores.

It's great it's great to be this
NG physics Georgia Tech again.

I think this is my fourth
time in a school of physics

my first time was in December of
one nine hundred ninety four and

I got to see Joe Ford So
I'm really grateful that that happened and

of course I'm very honored to.

Be giving this Joseph Ford a lecture.

I've had just so much fun today.

I know a lot of people here and it's just
so much fun to see everybody again and

talk about what they're doing and
science and so forth so

I start my clock so I don't run over.

I wanted to start with by
talk let me go back and

start by acknowledging my coworkers
Annette Taylor who was at the University

of Leeds but now it's a Sheffield Mark
tensely it's with W.V.U.

and our students Hong Kong Mon and
Simba Russia.

And of course the National Science
Foundation so in the last.

I guess six or
seven years we have focused on

discrete coupled couple
discrete oscillators.

And.

And that's what I'm going to tell you
about today kind of an overview of

coupled oscillator work the story starts.

With.

Actually refereed this paper
by Sylvia to Monte from

some problem Sorenson from Copenhagen and
their students and

that's a wonderful paper in
which they studied yeast.

In a flow reactor and
were able to do some fantastic

experiments that I'll tell you about but
actually the yeast system

you know in earnest goes back
forty years now to Aldridge and

Pai who found that if you
have a solution of yeast and

you dilute that solution say with
nutrient broth first of all when it's

before you dilute it it's
oscillating synchronously and

you can follow the oscillations
by looking at in A.D.H.

because that's in vault
in the glycolysis And so

you can see these things oscillating
synchronously synchronously as you diluted

you reach a critical concentration
a critical number density where

these oscillations stop being
synchronous and the and

the signal goes flat so after Origin
pi the yeast system became very

you know paired to magic system for
studying so all signaling because

it's obvious that if there synchronized
they're talking to each other but

it's not understood was this
transition from synchrony

to unsynchronised behavior and
it could occur in two possible ways

these are these this is
a figure from this periscope.

HEGAN paper and this is one way it could
occur say you're oscillating fine here and

at time zero you dilute the medium so

that the number density goes down what
could happen is all of the oscillators

could become Do you think when ised and
because they're all out of sync

the global signal showing
in dark black is flat OK so

that this is like a classic
synchronization transition I've call

it occur moto synchronization transition
because she could curve motor was

the first one that really figured out this
critical critical coupling strength for

the transition but there could be another
possibility so you oscillating fine right

here all in synchrony you dilute it
below some threshold concentration and

because the enter cellular
signaling molecules change.

Each oscillator changes state and
goes to the stationary state and

say if it saves spiral a focus station or
state spirals in and

so so it's a completely different type of.

Mechanism and
it was not known how this worked the.

Parrots Copenhagen study
showed pretty definitively for

their conditions it was this this so
called quantum sensing type

mechanism the reason it's called quantum
sensing is because there's a branch

in physics that I'm sorry
in biology that studies

bacteria populations and
when some bacteria get above

a critical density then they
switch states completely and

one example is this viral
fissure I that live in squid and

they become chemical luminescent
above a critical number density.

Another example is
the pseudomonas organ also which

suddenly becomes a biofilm producer which
can be important for human health and

so this is a big topic big
area in biology because it's

very important that how these
bacteria behave this yeast system is

you know it's a different system but
but it provided a lot of insights

into this Quorum sensing transmission and
phenomenon why is it called core

because you need to have enough you
know a high enough number density so

it's like you have to have a core arm in
the faculty meeting to you know to vote.

OK So when we saw this I said.

We should do this with these
micro oscillators and because.

It offers an opportunity to get some
fundamental insights into into this

transition and so our experiment is
very simple I'll tell you about these

micro oscillators in the next slide No I'm

sorry.

Well I left out one of my so.

I'll tell you about that Mike Ross later
is right now these are very simple to make

their cattle and exchange resin beads that
are about two hundred microns in diameter

you load on the catalyst on to this be.

The catalyst to the blue stuff Jabotinsky
a solitary reaction and then you put this.

Into a catalyst free solution and
you have an autonomy little micro

oscillator and
in this case you have oscillator.

Can communicate with the other
oscillators through the solution now

this blue software potentially
reaction puts out two.

Critical species into
the solution run is the activator

so-called promise acid and
one is the inhibitor and

so they communicate with each
other through the solution and

here you can see the picture

we monitor this system both with
an electrode electro chemical Ali And

we also monitor it optically and
you can see some of these

beads are kind of bluish gray and
some are red when they're blue they just

reach the peak of their oscillation when
they're red they're at the minimum and so

here's a typical oscillation it's
red red even those is black and

white suddenly turns blue red red and
so on so.

So that's a very simple experiment
we're able to freeze frame

even though we're starting at three
hundred to six hundred R.P.M.

this this system using a shutter
speed of four tenths of a millisecond

one thing about this micro oscillator
system is that allows you to have

extremely large populations of
oscillators like in a two by two Q.

that you can have one hundred thousand
oscillators communicating and so

it offers a very nice
opportunity to study commune.

Later communication.

OK so here's an example.

Of what happens at a low exchange rate
this is when we stir at three hundred

rpm and
we step up the density the number density

of oscillators by this red line and
you can see that.

But there's a flat out noisy
signal at a low number down city

you start to see some oscillations then
start seeing real oscillations and

going it goes into full blown oscillations
by the time you step the number density

up to the maximum if you look
at the electro chemical signal

you don't have zero at this
kind of noisy fly zone but

some finite signal and
then you kind of a smooth increase in

the amplitude of these oscillations
these are actually the electro chemical

oscillations if you look at the the.

Optic with the reconstruction.

Of the oscillation from the camera
you're looking at these red and

blue particles the blue particles
transmit light more and so

you can also talk about the maximum in
the oscillation of the optical signal and

you see that it starts here nonzero
at about zero point two and

then goes up gradually and
the period is just kind of flat.

Here we see an example of the experiment.

It's a little bit hard these are bubbles
these That's a bead stuck to a bubble that

we omit from our image analysis but
there are blue ones that are going by and

red ones and that's on the other
side you see oscillations occurring.

Now one can.

Simulate the system we you know they're
pretty good mammals for the B.Z.

reaction to Soft have a tense cue reaction
we use a model that I'm going to show

you later called the Z B K E model from
Jabotinsky and Epstein from brain.

Dies and the reason we use it is because
it allows a variation of the it shows

variation of the period and
our experimental system has a distribution

in frequency and so we need to
build that into our system and so

this is what we simulate here
is our global flat signal.

Given by this these blue beads that and

the catalyst is this oxidized blue
catalyst here are the individual

oscillators that are out of phase so
they sum up to the flat mean and

then you have this system becoming
more and more synchronized

the frequency being sharper and
sharper until you have synchronization

this is you know your classic cool motor
like synchronization transmission and

you can also look in order parameter
this is a particular order parameters.

Chena moto order parameter
which basically just looks at

the phases over the oscillators and.

Gives you an idea of the order depending
on whether they're all in the same

face then your parameters close to one it
is one if they're in the same phase or

if they're all in different phases
the order parameter is zero and

this is the time average.

OK so
now we can go look at the system in a.

Higher.

Exchange rate and
this is six hundred R.P.M.

these are again explain experiments and
we're going to step up the density like so

and you can see that it's
we have a flat signal and

you know kind of flat once it settles
down and then suddenly full blown

oscillations and so
in the electro chemical C..

No it's a zero until you have large
up to docile ations basically

from the reconstruction of the optical
signal zero to full blown and

then we have this all.

Looks like what looks like period
lengthening it's probably a saddle loop or

a saddle node bifurcation we still
have yet to completely sort that out.

Here is the.

The movie you saw that's when.

They want it high density just also
lated the one at low density has no

oxidized loss leaders This means that
all of them are in the steady state.

OK So so
this is this is really a very important

set of experimental observations.

Wait just a minute I think
that we're going to see this

thing off late once more.

Here is the simulation and

you know at the steady state
zero frequency no frequency and

then we start getting oscillations here
and there basically all at the same

frequency this should have more to occur
more to order parameter really shouldn't

isn't a political to the steady state so
it really shouldn't be used there but but

this is the quantum sensing type
transmission versus the current

moto synchronization type transmission so.

And this is the model I'm not going to.

Bore you with the chemistry behind it but

basically here is our activator
Roman SAS and our inhibitor bromide and

here's our catalyst so it's a three
variable model the activator and

the inhibitor go to the solution and

they affect their neighbor their
effect their oscillators it's a glue

it's a mean field coupling and
the inhibitor.

Oxidized metal catalyst is.

Immobilized on to this little
cat on an exchange beat.

This just shows for the solution you have
an exchange on into the solution and

back onto the particle maybe another
particle and this is the Z B K.

Model.

OK So actually we were really.

Thrilled that we got both this

quantum sensing type transition and
the current moto.

Like transition in this chemical system so
if we look at a high exchange rate

say three were several seconds we go along
in the steady state and then suddenly it's

like a first order transition we
suddenly have full blown amplitude

full amplitude oscillations this is the
amplitude Max amplitude of the oscillation

on the other hand if we take a small
exchange rate point three or

so it's just this gradual
synchronization I say crime or

to like because her moto studied phase
off the leaders and this is anything but

a face off later if we
look at the other way and

her number density constant
say eighty two hundred

then we see this true this line like so so

this is kind of an increasing career MOTOS
synchronization on the other hand if we.

Look at forty two hundred we have the.

Transition and then suddenly the the.

Quantum sensing transition and

I show that again here these are the true.

Transitions this is Quorum sensing and
this is Coronato and now this.

One where we hold number density constant
this is increasing synchronization

like term auto and so it would be going.

This is exchange rate here and so
we're stepping up exchange rate and

then suddenly we fall off this
like first order transition

here there are experiments and we have
this noisy signal because it's all out of

phase also later and
then it starts getting in phase and

then suddenly it goes to the steady state.

Like what you can you can look at I'm
sorry that this goes the opposite

direction but these are the individual
oscillators with the oxidized metal

catalyst all out of phase giving you
a flat signal oscillations in phase and

then popping down to the to the steady
state and the corresponding

frequency distributions so I guess.

What I'd like to.

Leave you with on this
in this study is that.

This kind of tells us you
know if you believe that.

You saw that oscillating because you've
stopped catabolism you're looking at

the solitary glycolysis if you
believe it's an oscillator then

it should also show all type transitions
if the parameter ranges accessible and

so even though the Copenhagen study
showed that it was this type of

transition this quantum sensing transition
you should probably also see this and

you should probably also see this in all
of these corm sensing transitions you

probably also see chromo transitions if
the parameter range is accessible and

so I guess.

The bottom line for.

For me anyway is that I I think
that studying this chemical system

where your state variables are just
simple concentrations your.

Governing equations or so
are well established kinetics and

transport and then the experiments
are very straightforward allows you

to do experiments to give insights into
biological behavior that's very similar

and I think that this is a case
where we really can say that we have

provided some insights that
weren't weren't available before.

So we can also look at this system
as a spatial temporal system

in some sense this is more
like the actually bacteria

experiment the last one was very
much like the yeast system but

that's more like a bacteria
experiment because they grow more or

less two dimensional usually on surfaces
and I'll show you the experiments and

our interpretation behind these behaviors.

So we start with in this
says system we don't

store the solution we
simply use these same.

Loaded catalyst loaded beads and
we mix up the solution so

that it's in the excitable
steady state all right and

this is actually a movie but it's not
very interesting because it does nothing.

It's in the study state.

But if we now also bring up these B.

before in this setting we
always only only look at

these oscillators that
are two at least three

to three diameters apart because we don't
want to look at the diffuse of coupling.

In the steady state but then we bring them
up basically in contact and they have.

A diffusive coupling and so
we bring up fifty of them and

nothing happens and we deliberately I
think it's was we thought it was kind of

nice to make it look a little bit like
the bacteria colony you don't need them in

a perfect way and so you bring up
sixty nothing happens you bring up

seventy and then finally you bring up.

Eighty eight beads and absolutely nothing
is happening you bring up the eighty ninth

and you suddenly see.

The thing comes alive and
not only that but

you notice that there's
not some salsa later and

here causing this to happen these waves
originate from completely different

areas this desist in is actually just.

Now become a solitary system collectively.

The you know you can see
fairly good phase wave.

Synchronization here although
the center moves around a lot

we can also find spiral particles
I couldn't show an example

of when this the onset.

When when all when all these were just in
the steady state but now you know I think

who do we have I don't know how many
particles we have but but wait but

it's the same story you bring up the last
particle and you start getting a spiral

behavior and there's this transition you
know twenty percent of the time we would

see spirals and eighty percent of
the time we would see target patterns.

The.

Their face there's very little face
synchronization here because this way.

Moves all over all over the place and
I'll see if I can get this going again.

And so the wave tip is because
it's two hundred you had a genius

to get nice phase wave
dynamics we can we can

take a look at this system in the.

Matter all and I'm sorry before we get
to the model these are experiments

where we're majoring is the probability
that you see wave behavior as a function

of the group size this is not number
density This is two dimensional group and

so you can see that were plotting
probability as a function of the.

Size of the group as well
as this concentration.

Brought me to a reactant that
it's at high bromates more active

low brain it's less active so
you see these curves.

After every each of these points
represents about eight experiments and

there's about three hundred
experiments in this plot and

after every single experiment my
students I want young Hong separated

the beads to be at least three two or
three diameters apart and

not we didn't find one
a solitary particle and so so so

it was very good evidence that
it is truly a collective.

Behavior here is our modeling.

I think this one is thirteen by thirteen
by three I think we also did twenty five

twenty two by twenty two by four but
we basically put a model layer of loaded.

Catalyst particles in the middle and

then where there it's empty
that is the solution where

where these chemicals the activator and
the inhibitor can actually react.

Here is what the.

Activator looks like and
the inhibitor the activator is.

Let's see which is or
yes the activators the blue one and

the inhibitor Bromide is the red one but

when we look at this probability for
wave activity as a function of group size

you can see that it doesn't look
very much like our experiment

it just you know it's a step function
at some value of the group size and

in addition we always had wave activity
starting from the center of the group.

But when we add some heterogeneity and
basically what we

were what we do here is
yes here is the mean we

make this system heterogeneous in
this catalyst concentration and

so you can see that as we go to.

Increasing concentration of the reactant
row mate the system it does look quite

a bit like the experiments in addition we
can look at this behavior as a function

of the heterogeneity just the standard
deviation of this distribution and

you can see that below what
three point seven or so

times in minus four molar we get wave
activity virtually all the time and above.

Stand a standard deviation of say six
times in one of four we don't see any so

this this result was kind of interesting
that they had originated actually

suppresses the wave activity rather than
supports it we have some speculation about

this amplitude death and so forth but
I think it's pretty speculative.

OK Well let's now take a look
at another couple DOS later.

System.

Let's go back to the Stood system like
the Quorum sensing in the car moto

synchronization system but let's change
the chemistry a bit to a new catalyst.

And what we see we focus on the current
synchronization transition and

instead of seeing the smooth
increase in synchronization we see

phase clusters and
basically save face clusters or

groups of oscillators in
the population that also late

out of phase with another group and
so the basic.

Global manifestation of this is
that your signal that you see

looks like a complex oscillation because
it's a composite of these clusters and

so we decided to.

Take a look at this and.

Yes and so here is a movie now this
catalyst is different colored it's.

Harder to see but with image analysis
you can do still do a good job and so

this is these are the oscillations
with no clusters and

in the next movie I show also Asians
that are about half the amplitude and

twice the frequency.

That we would like to think
our face clusters and

but in fact you kind of if you're doing
experiments you have to wonder whether

you're watching some kind of period
doubling or whatever you know you can't

you have to think about how to prove that
you're really looking at phase clusters so

in fact the phase clusters
were we started with the.

Two phase clusters and
after a stirring part of a.

And it went to the one cluster system and

on the left hand side you can see you can
kind of see with your eye that this has

two oscillations there's an amplitude
Max and there's an amplitude Max But

whereas over here it's harder to see but
there's really only one solution so

we take these pieces and
read cut them out and we look at them and

you know this is what we saw and
before that the.

Small amplitude oscillations are about
half the size and twice the frequency

of the large salacious but how do we
know that their face clusters and

the way that we know at least the way
that we add evidence to thinking we have

is to do the reconstruction
of these images and

we basically each dot here corresponds
to one of these panels and

what it tells us is that this small
oscillations has only fifty percent.

Of the oscillators.

Reaching the maximum.

And then later fifty percent or
so do it again

whereas in the large oscillations almost
one hundred percent reach the maximum so

that tells us there face clusters
Well it isn't proof positive but

we still can do some modeling and

be convinced ourselves pretty well so
we take the same model.

And we look remember I would
like to emphasize that

these phase clusters
are in a two by two queue

that of one hundred thousand
OSS later nonetheless.

The collection the population of
oscillators still finds a way to partition

itself into oscillating half of
the oscillators oscillate and

then the other half
they also letters offs.

Wait PIO to face so here is our.

Unsynchronized system this
is a four hundred oscillator

simulation this is a one
hundred sample and

there's no spatial meaning to this
grid this is just a sample OK And

this is the global signal it's flat.

And if we look at the phase this is just
a phase there's a linear extrapolation

of the phases from peak to peak you
see that they're all spread out and

here is our period distribution and

here are three time series for.

Three different oscillators twenty six
thirty and twenty three natural periods.

OK We go to.

A higher number density we when we were
at four hundred now we're going to double

the number density day eight hundred
here is our sample of one hundred and

now you're looking now what you
see is this time series and

this is really kind of you know back in
the olden days when we were doing stirred

tech reactors this sort of mixed mode
oscillations almost the same but but

they're not they're clusters
face clusters of four

clusters and you can see them
they're not synchronized here but

most of them are with a pretty.

Sharp frequency period distribution.

OK We advanced the number
density a little bit more and

now we see what looks kind of
like to cluster system but

in fact it's more complicated we
should control that because of

these kind of randomly varying
amplitudes and in fact

what we're seeing is a two Questor system
but with a switcher and these which are.

So we call them switchers are these core
clusters that run into the back of one

group the front of the group believes runs
into the back of the other group and so

you keep getting this very complex
dynamics look at the the period

distribution now and these individual time
traces of the oscillators are very complex

so there's a lot of
complexity in these phase.

Phase groups.

Phase and so here here we show
the overall picture here's

unsynchronized we go to four four
cluster three cluster two cluster and

all synchronized in a one cluster state so
I see I need to hurry up a little.

So we'll turn to another
couple doth later experiment.

Back in two thousand and two Kermit on but

talk talk published this paper about.

Coupled oscillators that they coupled all
come back to this in a little bit but

this they coupled in what they call
non-local coupling where you have strong

coupling nearest neighbors weaker coupling
as you go further out and weaker yet

if you go further out with this
kind of coupling arrangement and

then in two thousand and four.

Current moto show that
you can also get spiral.

Manifestation of this same behavior so
what is it it is the coexistence of

a sub population that is synchronized
with another sub population

that is unsynchronized in a population
of oscillators that are identical and

identically coupled so
this seems like impossible.

You know why should the Democrats
lawyers that are identical.

Coupled split into two populations
one synchronized one on Surprised so

strong in two thousand and
four an Abrams Stanley Abrams.

Coined this state because of
course mirrors or this Greek

mythological creature that are in Congress
you know should not exist together kind of

like synchronized and unsynchronized
oscillators and of course spoke at.

Many.

Studies on this state so we thought
we should look at these because.

Then I need sperm and examples I should
mention the Roy who is just here I think

not long ago he also decided to look
at these systems in Maryland and

we kind of work together on on
making sure that we both got.

Way to say what we found and.

And so.

I thought I don't think I need
to spend too much time on this

basically the way we couple the also
leaders in the system is that we look

at each oscillator we major
The transmitted light intensity and

then we major an oscillator it's coupled
to its transmitted light intensity we take

the difference of this light intensity
the two light intensity is multiplied

by a coupling constant and then shine
light at four hundred sixty nanometers

that causes this chemical reaction to
occur and generate the auto catalyst and

that's that's basic basically it in
a nutshell this is just the B.Z.

the blue Soft have intense chemistry and
the auto catalyst H.B.R.

O two is the fundamental
thing that drives it.

OK here is a model by Danny Abrams and
Steve Strogatz and another.

Students that.

As opposed to having coupling
that fast off from strong to weak

kind of exponentially they wanted to
take this simplest version of this

namely strong coupling for group one.

Oscillators toss leaders from
group one strong coupling for

oscillators toss players and group two but
weaker coupling for Also leaders and

group want to group to and vice versa so
this kind of like a two step

non-local coupling strong up
of strong coupling of your

local group weaker coupling
to a distant group and

the reason they did that is because
they could solve it analytically but

that's what we decided to start on and and

this shows our experiment.

We basically monitor as I said each of
these last letters with a camera and

then we take the difference
of the oscillator

eye that we're looking at and
then we get the mean.

Intensity for all of the oscillators
in the system we use a time delay

here that we can talk about later
if anyone is interested and

then we have a this is
the end trial coupling and

this is the in group coupling and

that's basically the Abrams
the Abrams model and

here's what we find.

You know a low.

A particular I think that we're
doing this as a function.

Of delay time but
this is the simplest behavior

that we see we have forty oscillators
it's in there in Group A.

Group B. And here is Group A always
synchronized and Group B.

also late you know Pi out of phase more or
less.

Then we can also have face clusters
Here's group a here are also leaders

into clusters in Group B. you can see
they're not there not to like one hundred

twenty degrees out of phase because
this is a relaxation oscillator and

this is the kind here this is
group a Also synchronized and

this is group B. unsynchronized And then
finally we found this rather odd behavior

that is not still not characterized very
well that we called semi synchronized and

we think now that it's basically trance
in phase clusters bunching up and

but but I should qualify this
experimental observation in

the sense that we indeed have all
twenty all slayers here Adenike

we coupled to each other in the twenty all
Slater's over here coupled to them and

vice versa but the oscillators are not
identical the oscillators have a.

Frequency distribution so
it's not it's not exactly the mathematical

curiosity of that term
OTOH discovered and.

Many other theory theoretical
works have been carried out on but

it's still quite a dramatic example
because they're done a quick coupled.

There are some other interesting
things even though these are red and

blue there are these through the Group B.
And here we see Group A As

always thank Here is a long
lived this is the mean signal

of these unsynchronized toss letters
this is the kind near the collapses

after some you know twenty minutes and
becomes.

Phase synchronized with Group A And
so they have a for.

Night life time and
here we see the life time

as a function of the standard
deviation of the frequency

distribution and
you can see that there looks like there

is an asymptotic asymptotically
stable came nearer but

you never can tell by you know
with numerics but it fails off and

I think one thing that this says is that
heterogeneity in this case actually makes

it less likely to see a kind of
mirror state rather than more likely.

OK So this is the kind of thing more than

a mathematical curiosity and
you know what what might it be good for

well some people have suggested
that it could be relevant to

uni hemispheric sleep I think
Steve Strogatz suggested this.

You know at uni hemispheric sleep occurs
in mammals that need to come up for

air because they've got to get some sleep
but they also have to know how to come up

for air also there are lots of
these migratory birds need to get

some rest and
also many animals of prey and

and so this seal is keeping one
eye out open for this shark and so

one hemisphere of the brain
is in this very slow wave.

Sleep.

Mode and you have minimal Assoc Colin
release the other hemisphere

is like the Awake mode and
with maximal as a tool Colin released.

It's probably OK fact some way
it's not the greatest paradigm

because I think that we know that.

Synchronized behaviors a little bit
pathological in the brain and probably.

Completely and synchronise behavior
doesn't exist and so but but

it does uni hemispheric sleep in the sense
that you have two states that really

shouldn't co-exist I think in that
in that way it's of fine metaphor.

So now let's look at this
original chrome auto.

System you know where where this coupling
strength falls off exponentially

because you have this coupling constant K.
with this Kappa in

an exponent in the exponent and
basically this is a D. K. constant So

at high cap the system decays rapidly and

the coupling radius is very small for
low Kappa it goes very far and

you have a long range coupling So
this cap is a very critical parameter

we're going to do experiments just
exactly as I described for the two

groups system except now we're going to
have forty oscillators in a ring so that

we have periodic boundary conditions and
don't have to worry about the boundary.

And so this is actually
a simulation of homogeneous B.Z.

oscillators which we can't do in the lab
yet but we can do on the computer and

this is what you saw when you
the opening slide and I really

like it because to me it's a much more
intuitive way of thinking about chi Mirah

then then all the other descriptions
basically you have a random distribution

of phase distribution and what happens
is you have accidental correlations and

you can probably see some here and
they start firing together these.

Oscillators are close in phase when
they fire together they there twice is

powerful you know two is
twice as strong as one

in the terms of the signal and
then three four etc.

Until it finally reaches
an asymptotic state like this and so

I think that's a much more intuitive
way of thinking of the kind Mirah.

Here is the heterogeneous oscillator
that is has a heterogeneity that's

very much like the heterogeneity we
would have in experiments this is

the initial random phase distribution in
this is the final state of course this

could be in the middle because
it's periodic boundary conditions.

Here is the system we just looked
at these are the phases so

here are the synchronized
phases here is a.

Little cool order parameter that's
like the Carmel to order parameter but

you just a sample.

Off letters to the right and to the left
with any phase with any sampling radius

and you can see for me I find these.

Little spatulas of trance in
order I think more interesting

than the disorder because they're they're
trying to they're telling us you know

they're telling us things and I think that
will get to that this is doesn't look

like a very dramatic slide but it's
very important it shows the probability

of finding a completely do synchronize
state I can mirror state or

completely synchronized state as
a function of this DK constant

that determines the length
of the range of coupling so

at a high Kappa we have
all the synchronized

oscillators at a low Kapil we have
mostly synchronized oscillators but

at point three and point four We
have both these synchronized and

synchronized here's
the surprise come here and

synchronized this is very
important if you're studying

a system with heterogeneous
oscillators because if you are.

Studying heterogeneous oscillators and
you don't see coexistence with

a synchronized state you can't be sure
that you're seeing a real crime here or

you could be looking at
a distribution that's want to ride and

the tails are on synchronized you know
it's not in the in trainmen zone and

you think you have a kind of Europe but
you don't and

so it's very important
to see this coexistence.

These are experimental results in our
experiment we have these phase waves

that I think are also
quite interesting here

are here is this what we just saw
this phase wave here are the faces

of the second eyes heart and the phase
which right here and you can see in

this local order parameter high order
the face waves don't show up with high

order because the order parameter isn't
meant to show high water there but

it collapses into an anti phase
oscillation right here and

so and these are just the period
at two different times and

we'll take a better look
at period in a moment.

Here are more of these phase waves.

They're kind of right they have

the theorists have studied a lot of splays
states these are not really splay States

because they don't go from
zero to two PIII even.

Phase differences but but
you can see that this

this phase wave goes here and this one
comes down here through the periodic

boundary to here and so
they are that is order.

There we also see phase clusters here
are two phase clusters in phase Here's one

out of phase these are experiments
here and the simulations here.

OK Here here we take a look at period and

we could look at frequency as well but
here is.

Here is are our zero to two Pif A's and

our synchronized region this is
heterogeneous oscillators and

you can see that the period
starts bouncing around.

The region of synchronization meets
the region of an synchronization

there's kind of this frustration and
that's what that's what really makes

the kind mirror in my eye now if
you're look at the theory papers and

all of phase oscillators that's this abuse
beautiful smooth curve of the frequency

on follow up but of course we don't we
don't have that in the chemical system

here we look at the homogeneous by the way
we're running the five times faster so

we can accumulate periods over here and

you can see that you know some of these
unsynchronized also later as that were due

to the initial random phase distribution
really don't do much they kind of

keep the same phase it's only the ones
that are kind of in this frustrated zone

that really have this dynamics that
reminds you of chaotic Diane am except

OK I'm just about ready to
finish up here we see also

clusters you know here's a one class or
state that spontaneously gives a to

sequester state through Questers and
friend clusters finally.

We can see.

A system that has one clusters two
clusters through clusters four and

then when you get to the five
cluster state it starts drifting so

this is all unchartered
territory we are trying

to wrap up into some sort of paper and

you one can as I mentioned before
Carmel told showed in two thousand and

four that you can see the kind
mirror in two dimensions and

then two dimensions this is this is
local local order parameter the.

Sure of the.

Springtime euro is made up of
unsynchronized off Slater's

the spiral wave of course is synchronized
in the sense that it's phase

wave synchronized And so
what you see is this this is

the local minimum in the order
parameter and you can see this.

Spiral wave kind of wandering
around with broad me and.

Dynamics you very this delay and

you see there a different behavior
this crazy looping behavior and

you also see that this spiral
start looking differently and

you would never see this in a direct
action shouldn't say never because

maybe you have seen it but but it would be
unusual to see in a continuous medium and

so you this big loops.

And you see that here here there's
the Brownian Here are the loops and

here's the difference in the spirals but

you find that on changing this TOWIE
even further you have instability

in this spiral core and

this is what we're trying to
work on now and you can see that

you know I you can see these
fleeting patches of order and

you know that this is that it to
have one of these regions break

apart you have to have synchronization
going through the metal and so

I think there's a lot of tremendous amount
to learn on these spiral kind Mira's and

all stopped by just letting this
movie play this is simply the order

parameter of the last slide speeded up so
thank you for your attention.

Would you just.

Explain.

It.

All right it's right there each one.

This is this is for
the spontaneous wave for storm something.

We have not done it in one day.

That that could that would be
interesting to do it in one day but

we've only done it in two D..

Our our modeling kind of indicates
that the these Fontayne and

cooperative behavior that
you see is due to basically

it's the last rate of the catalyst it is

decreases with increasing systems size and
so the one D.

might be might be problematic but
it might be interesting.

Three D. I'm sure yeah yeah I like that.

That's right I think it would
yeah you could do I think three

D..

With respect.

But.

This is.

The busiest one hundred times easier
to work with than any other solitary

reaction that from my experience you
know it's just it's just so easy I'm.

And you can make you know you make an
opening you can make it open system very

easily because it takes a very minimal
flow of reactants because the reaction

is so slow and
like brigs Rosher has bubbles and

and it's yeah and so to be easy it's
really it's a it's it's just so

easy compared to other
a solitary reactions.

Well just think you know OK.