Synchronization Transitions in Systems of Coupled Phase Oscillators
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00:00
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Transcript: English(auto-generated)
00:01
Yes, so finally I have to introduce myself as a speaker. My name is Matthias Wolfram, I'm working at the Weierstrass Institute. And I'm giving this final talk in the in the network session here. And after that we have seen so many kind of really applied problems. I decided as a mathematician from an institute on applied mathematics, anyhow, I decided to go a little bit back to the basics.
00:29
And what I'm going to show you in the next half an hour is kind of a collection of some interesting results that we obtained during the last years, where kind of the intuition that you might have from synchronization theory might fail in some cases under some specific circumstances.
00:55
This is joint work with my colleague Oleg Omechenko, who is now at Potsdam University, Svetlana
01:00
Gurevich from Münster, Carlo Lein from New Zealand, and Effie Janczuk who is actually in the audience. And all that is based on very simple models, which are not really related to real life systems, but which anyhow I think can showcase interesting nonlinear phenomena
01:20
that one should be aware of when dealing with coupled oscillator and synchronization problems in general. Okay, so to be very basic, I will tell you, give you at the beginning a brief introduction, just to the classical Kuramoto model, probably most of you know that very well but anyhow let's have a look.
01:44
So this is Yoshiki Kuramoto, and this is his model that he wrote down in 1975. So these are n oscillators, which all have their own frequency omega sub k, and they are coupled via this sinusoidal interaction function, and it's
02:07
just a global coupling so I take some overall oscillators in this coupling term, and I do that without any coupling weights at the moment. Okay. And since there is a minus sign in front of the coupling, this interaction works attractive
02:26
between the faces, so all faces rotate with slightly different velocities, but the velocities all attract each other. And so this attractive all to all coupling finally overcomes, so to say, the inhomogeneity of the
02:44
frequencies, and at the critical coupling frames kc synchronization sets in this picture is just from the, from the thermodynamic limit which I will introduce later, and you see that the zero state here,
03:01
which is complete coherence just gets unstable and we get a branch of partially synchronized states. So this is the universal scenario for the onset of synchronization, people from statistical mechanics also call that the second order phase transition.
03:23
Here I showed you just in the first line that this model can actually be rewritten if you go to complex notation so I here I switched back from the electrical engineering notation to the mathematical notation. So i is again the imaginary unit and j is an integer index and not
03:45
less precise, using previous talk where I was covering the j was the imaginary unit. And we can see that this global coupling so principle network or notes are just connected with the same strength, can be
04:02
replaced by a company to meet fee. So, instead of the summation of all the company terms at first, some of these exponential of the faces extract these complex order parameters, and then a couple my system just to the complex order parameter. This trick
04:23
by Coromoto is actually the back, the starting point for a lot of nice analytical methods that you can use to treat this model, which you lose immediately when you go to more general interaction functions. If you take a general
04:43
interaction function instead, where this Coromoto's sinusoidal coupling, so to say only the first fully component that is functional function. If you go to these more general face interaction functions, then actually there is theoretical background that tells
05:06
you that any system of coupled oscillators can be written in this way in the limit of V coupling. And this is a kind of follows from, from, say, the segmental ceiling for, for, for limits. And
05:29
if you're over assumed that the frequencies are fast compared to the terms that come from the action, actually, ever agency really tells you that you can run this interaction function as a difference of the face.
05:50
So, now, let us come to the kind of non universal transitions to synchrony in the Coromoto Sakaguchi model. So
06:04
two years after this fundamental paper, Coromoto gets into this alpha parameter, which is this phase gap in the interaction function. And it turns out that all the nice machinery, all the energy classes work equally well, however, much more interesting dynamics can be observed.
06:28
When the phase gap parameter is immodal smaller than pi over two, then the coupling is still attractive, so you expect synchronization for large number coupling.
06:42
And the last general assumption that we make is always that the distribution g of omega from which we draw these unknown or inhomogeneous frequencies omega k, that this is just a unimodal distribution with some distinct maximum around which we expect the signal.
07:05
And then it turns out that this was actually not only for us surprising that there are certain distributions j, I will specify them later, which are unimodal and where this classical synchronization scenario, which I showed you here again in the top row of the figures,
07:27
changes and we see new and sort of unexpected behavior in the sense that, for example, if we, there are situations where if we increase the attractive coupling, anyhow the synchronization may decrease.
07:44
So here the whole problem, for example, goes down even to zero, such that after some interval of k where you have partial synchrony, incoherence regains stability and is the only stable solution.
08:03
But there are also interesting cases of coexistence. So in the middle figure you see an example where stable incoherence coexists with a stable partially synchronized state and in the lower row you see that also, you see that there probably you see in another example which I will show you later on,
08:32
where you also have the coexistence of two stable partially synchronized states. So how can we understand that? Let's briefly walk through the theory.
08:44
The first thing is, you'll think about the large number of oscillators, so you take the infinity. In this case, you can consider probability density, which just tells you that at a given time
09:02
moment t, how big is the probability to find an oscillator with natural frequency omega at a position theta. For this probability density, you can write down a continuity equation. In this continuity equation you have of course to plug in a velocity.
09:28
For this velocity, I just take the right hand side of the Coromoto equation that I showed before, so this second was written there. The only difference is that of course now for the mean field, I have now to plug in the continuum
09:45
version, so what was formerly a summation over all the n oscillators is now just an integral of the state. And of course there is an obvious normalization such that I get out the g if I integrate over theta.
10:06
As I told you, there are nice analytical methods available for these systems of Coromoto type and one of the most powerful tools is actually the Otto Antonsen reduction, which was proposed by Otto Antonsen in 2008, I think in 2009, two seminal papers in the Chaos Journal.
10:31
And this tells you essentially that this unknown density function f, for which I formulated the continuity equation
10:41
on the slide before, can be substantially simplified, namely one can just drop the dependence on the angle theta and represent this density profile with respect to the parameter theta just by a simple complex number. And here is kind of the formula that you, how that works, but I recommend you not to look at the formula but rather at the picture that I show here.
11:11
So the red graph, which is these distributions with respect to the angle theta, so the horizontal axis is just the angle theta
11:20
and f of theta, and if z equals zero, then we just have homogeneous distribution, so complete incoherence of the local example of oscillators.
11:42
If z is one, and one is z is one, then I get a delta function for the distribution with the angular position of that delta function, which is taken by the argument of z, and if z is somewhere between zero and the modulus one, so between zero and one, then I get such a distribution, which is something between homogeneous,
12:14
homogeneous distribution and the delta distribution. For the probabilists, there's recently also a nice theory by Dennis Goldobin
12:22
from Herren, who showed that this can be seen as a general principle of what he called circular cumulants. So, generalizing the concept of cumulants to distributions on the circle, very nice theory.
12:41
Okay, so here is the polynomial equation that means the dynamic equation for this quantity z. So instead of solving a dynamic equation for this probability density f that I showed before, I will now solve a much simpler equation for this local order parameter z.
13:01
So, for example, if I have this complex quantity, the angle zeta has disappeared, and this nice equation. I think I would not bother you with the details, just let me draw
13:21
your attention to the fact that these partially coherent states that we are interested in, that they are actually given by rotating solutions with a uniform rotation frequency capital omega and a fixed profile with respect to alpha, with respect to omega.
13:42
So this profile A of omega. So if we want to find the solution of this type, we have to find this collective frequency omega, and this profile depending on the natural frequency omega, this profile A. And it turns out this profile A has a universal solution, which is given
14:05
by that formula that originates just from solving this problem, which is not very difficult. And the main point is that it contains, so to say, two parameters, one parameter is this collective frequency capital
14:21
omega, and the other parameter is p, that kind of tells you how much the synchrony is in the solution. Okay, so you can plug all that in, and you get, in a sense, a version of Kuramoto's self-consistency equation in the kind of bit more advanced form in this first formula.
14:44
So this is an improper integral, but never mind. And just take that as kind of a bifurcation equation, which relates the solution parameters p and omega to the system parameters k, the coupling space, and alpha, the step of the regime.
15:02
And now use that as a starting point for doing bifurcation. There are two main points that you can immediately extract from that. The first thing is, if this p goes to zero, that means just you go along your solution branch to compute the coherence of p, the limit p tending to zero.
15:28
Actually, you see that in this formula, this is for singular with respect to p, so there's some technicalities behind it, but morally p tending to zero just is this optimization threshold values.
15:44
And this determinant, just using this partial derivative of h, can give you faults of the bifurcating branch of partial coherence states.
16:02
Yes, so there should be no top-bias and additions without a theorem. In the first slide, I just wrote down the linearization of the potential equation. This is a loose equation, so you just have to linearize the right-hand side. This right-hand side contains these integral operators,
16:25
so the linearization is not just, as it would be in an ODE system, just Jacobi matrices, but these are operators in bound spaces.
16:41
And one can find that this comes in a very specific form, that is, from local dynamics, you get a multiplication operator. Multiplication operators generate a continuous spectrum, essential spectrum, and there is a compact integral operator, which is a very nice behavior, saying that it comes with a coupling term.
17:03
And the main message is linearizations of that type can have two types of spectrum. They can have point spectrum, which is similar to matrix spectrum, and continuous spectrum, which is rather not similar to matrix spectrum, but maybe similar to things that some people with a physics background know from arms.
17:27
And there is a way to directly calculate the ODE point spectrum. I will not go into the details of this one. Instead, I show you an example. Maybe you should first show the graph on the right-hand side.
17:41
This is now a kind of alternative version of this classical onset of synchronization scenario with subcritical instability. So you see the black line at zero level, this is the statement, complete incoherence. Then you see at roughly 0.1, the sort of, say, Coromoto threshold, where
18:04
complete incoherence loses stability, and in principle there is an onset of partial coherence. But at that point, this branch of partial coherence takes by the case subcritical, and that means there is a range for existence of the complete incoherence and the partial synchronized state.
18:26
And these purple insect graphs just show the spectrum of the corresponding solutions. There are these lines, which are like on the axis, or like T-shaped in some cases.
18:42
This is the continuous spectrum, and there are the black dots, which is the point. So on the left-hand side, you see a two-parameter bifurcation diagram. So the right-hand side diagram was this variable in k on the horizontal axis, and fixed alpha at this value here indicated by the dashed line.
19:04
And so the left-hand side diagram gives you now the three bifurcation diagrams with two parameters. This red region, which just shows the single solution complete coherence. Same, and the blue curve shows you this fold, where the branch of partial coherence solution folds over from being same.
19:28
And that example was obtained just for kind of the most, starting from the most simple thing, which is the Bauschian frequency distribution, but truncating the tails.
19:44
Okay, so that means already the simple example of a truncated Bauschian is the non-standard simplification transition of this sub-critical time.
20:03
But only if you choose the alpha sufficiently large. If you are with an alpha close to zero, then you just single classifies. So here comes another example, which is particularly interesting, and this is just the superposition of two Gaussians with different widths.
20:33
So think about a mixed population. One part of the population has a rather, has frequencies with a rather strong inhomogeneity, which is the width sigma equals one.
20:48
And another part of the population actually has a very small inhomogeneity of the frequencies, and they are just coupled together in a global fashion.
21:01
And there you see now all these different scenarios that I already depicted on one of the first slides. Here you see the two-parameter multiplication diagram again with alpha and K, while this is just the onset of synchrony or increase in K for different choices now of alpha.
21:29
So here you see again this reddish region with a state of incoherence, and whenever a horizontal vertical line intersects twice, then you have this effect of incoherence, regaining stability for increasing coupling strengths.
21:49
And then you have this blue line of fault bifurcation, this is whenever the branch of partially coherent solutions forms, or whenever such a dashed line intersects a blue line, then I'm at such a fault line.
22:10
Here you see some qualitative explanations how one can understand this counterintuitive behavior that increasing coupling strengths leads to less coherence.
22:26
And so in the uppermost figure, you see kind of the normal scenario, and there, I don't know whether this is visible, but there's not only the black line, which is the global order parameter, but there is also sin lines which indicate the order parameter of the subpopulations.
22:48
And then you see that this is the same blue line at the corona threshold, which is relatively low K, because we have this subpopulation with only a very small amount of genuity, that they already synchronize and stay synchronized all over.
23:05
And then somewhere in the region where you expect the synchronization threshold for this other subpopulation, also the other order parameters start to grow in every single slice. This is for small alpha. However, for bigger alpha, you see the following. You see that whenever the blue population
23:27
has already reached a nice level of synchrony, and then the purple population kind of starts to increase in synchronization,
23:41
then it sort of pushes down the synchrony of the blue population. And this is because due to the alpha parameter, the frequency of the synchrony shifts with respect to the central frequency of the distribution.
24:02
So here you see these windows of synchronized frequencies. So all oscillators with natural frequencies in the blue window are synchronized, and they are getting more, but at the same time, the window gets shifted such that the peak of the blue population falls out of the window of synchrony.
24:24
And in this way, the synchronization of the purple guys suppresses, can effectively suppress the synchronization of the blue guys, and in this way, the synchronization may break down again with the increase.
24:42
I don't know how much time I do have, but I think I will briefly also tell you something about spatial redistricting. This is now the same system that I showed before, but now not with global coupling, but with coupling weights g, k, j, which are just
25:09
what some people call non-local coupling. So we think about the oscillator scheme in a 1-D array, and we have a distance-dependent coupling. So next and second neighbors are coupled to some oscillators, but only with a finite distance, and the coupling space somehow decays with distance.
25:33
And we have again inhomogeneous frequencies omega j. And what you see again is, if you just look at the global pool parameter, that for small alpha you would
25:48
have seen something that reminds you to discuss this equalization scenario, what is your bigger values of alpha, some strange things happen. Here you first see that somehow the maximum of u and the minimum of u with respect to
26:04
space does not coincide anymore, but only very slightly, and here you see that there are big discrepancies. But then you come back to a state which again looks more or less as it is for the classical scenario.
26:23
So, let me again skip the details and just tell you what happens. There is of course again the corona threshold, so the zero-solutional stability, but since this is now a spatially extended system, this
26:47
is not just the usual onset of serialization, but this is a kind of a purely light instability with the central weight number zero. You get the egghouse-like scenario where solutions with different weight numbers merge, they are depicted here, so this is just the space,
27:08
this is kind of the snapshot of the oscillators, and for the weight number zero, you have just this kind of homogeneously partially
27:21
symbolized state, and for the other weight numbers, you get these states which some people call twisted states. And again, there is an important influence of the alpha parameter, namely instead of the classical egghouse scenario that you just get when the spatially homogeneous solution bifurcates,
27:46
you get here the scenario where an intermediate regime of k's appears. So, in this loop, by the equation diagrams, you see again alpha versus k, so similar as
28:01
we had before, and you see that the blue region is where you have the same trivial solution, and the light blue region is where you have to stay with weight solutions and twisted solutions, but for large alphabets, this in-between region, and in this in-between region, you can see a kind of a non-trivial collective phenomena which can be either amplitude chaos,
28:29
which you see here, so this is an extensive chaos where not only the faces fluctuate, but over the amplitudes,
28:41
so you see amplitudes, while in the other two pictures, you see just the face chaos where the amplitude is. So, let me conclude with some general remarks. Of course, this is fundamental collective behavior in coupled oscillator systems, and there can be many interesting phenomena
29:03
observed already in these very simple parameter based oscillator systems, which are valid in the recoupling machine, general oscillators. If you look at the discrete medium, as in my last example,
29:21
this non-doctrine coupling can generate qualitative view and interesting collective dynamics. I have not talked today about chimera states, which are another example of qualitative view, new collective dynamics in such discrete media. And the properties of these face response functions, which in Kuramoto's case was just the
29:44
sine and in Kuramoto Sakabuchi's case was the sine with the alpha inside, are very important. And in particular, this alpha parameter can be actually crucial for the dynamics, and in our example is kind of the main ingredient to see this context phenomena.
30:03
Thank you for your attention. Yeah, I'm ready to take your question. Yes, I have a question. So when we introduce the auto pairing, they use the gradient structure of the coupling.
30:25
So it's surprising that the eigenvalues stay real. Any ideas? Yes. I've seen the same in the finite non-PDE version of the promoting model, I could never explain that to myself.
30:41
Yeah, well, they stay real until we reach the onset of spectrum, then we get these T-shaped spectra and we have also spectrum on the imaginary axis. Yes. You wouldn't expect it, right, that we throw in an alpha parameter that breaks the gradient structure, the eigenvalues stay real. Yeah, yeah.
31:05
Other questions? Sehi? If you have alpha, you observe the coexistence of two different partially synchronized states. Do you have some intuitive explanation for this, what kind of states we have or difficult?
31:27
Yes, I mean, the kind of explanation that I do have is just coming back to these windows of synchronization within the frequency distribution and that you can, so you either have a small
31:51
window, which is not so far shifted or a larger window which is further shifted due to the alpha. Right. And those could coexist.
32:03
But again, I mean, I think for me the main intuition is this suppression mechanism where kind of the, where you shift and the peak outside of your synchronization window and this actually destabilizes the main branch and gives rise to the second window.
32:24
So the frequency on which the system synchronizes differ from the main frequency. Right. That's interesting. Have you ever located the noise also in your system, especially if you deploy your effect mode and all your oscillators?
32:47
So here we just used the quench disorder in the natural frequency and did not include tiny endless noise. Some of the things can actually be done also from noisy systems.
33:07
Actually, it gets more difficult, but the phenomena are, I would say, mostly similar.
33:23
And there are more kind of numerical-based studies, but we also have goals like non-uniform frequencies. A question about Anfinsen.
33:41
So it somewhat parameterizes solutions with low complexity. Are you sure that you capture all solutions with that ansatz, or could it be that you're missing some? Yes. So the solution, the Anfinsen equation does not describe the full set of solutions to the continuity equation.
34:02
So this is an invariant manifold, so whenever you start, the Anfinsen manifold will stay there. And in most cases, it is locally stable. That means if you start in the neighborhood of this Anfinsen manifold, then sooner or later the solution will come to your standards.
34:25
So this is, that means it should capture all the interesting man.
34:44
If there are no further questions, then let me come back to my role as chairman again and close the session. Thank you all for coming and for coming.