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kuramoto.m
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kuramoto.m
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% We focus on closing the gap and come-up with analytical modelling by analyzing
% at the mesoscopic level what parameters shape (de-re)synchronization of the
% parallel programs based on a system of Kuramoto-type coupled phase oscillators.
function Hout = kuramoto(n,kappa,beta)
if nargin == 1
preset = n;
else
preset = 0;
end
if nargin == 0
n = 5;
kappa = rand;
beta = .660*kappa;
elseif nargin < 3
n = []; % Number of oscillators. real scalars
kappa = []; % Coupling coefficient. real scalars
beta = []; % Half-width of omega interval.
else % nargin == 3
% n,kappa,beta are input
end
width = []; % Standard deviation of display ring.
step = []; % Display step size.
pot = []; % Graph potential or order parameter.
uni = []; % Uniform or normal distribution of omega.
quit = []; % 0 <= t <= quit.
H = []; % Optional output
wanth = nargout > 0;
delta = [];
init_vars(preset)
% omega and theta are real column vectors of length n
point = []; flag = []; animal = []; titl = [];
theta = []; omega = []; colors = []; r = [];
rotate = []; txt = []; ax2 = []; psi = [];
% topology_ij the elements of the adjacenct/connective topology matrix of size NxN
% topology_ij = 1 if there is a connection between ossilator i and j, or topology_ij = 0, otherwise
% periodicity/non-periodicity can make this symmetric / asymmetric.
topology = zeros(n);
% all-to-all
% topology = ones(n);
% 1. Direct short distance communication
% uni--directional non-periodic
% topology(2:n+1:end) = 1; % lower first subdiaginal
% no propogation as sudden hit by boundary
% d=1;
% topology(d*n+1:n+1:end) = 1; % upper first subdiagonal
%uni--directional periodic
% d=1;
% topology(d*n+1:n+1:end) = 1; % upper first and lower (n-1)th subdiagonal
% topology(n,1) = 1;
% bi-directional non-periodic % both upper and lower 1st subdiagonal
% topology(2:n+1:end) = 1;
% d=1;
% topology(d*n+1:n+1:end) = 1;
% bi-directional periodic % both upper and lower 1st and (n-1)th subdiagonals
% topology(2:n+1:end) = 1;
% d=1;
% topology(d*n+1:n+1:end) = 1;
% d=n-1;
% topology(d*n+1:n+1:end) = 1;
% topology(n,1) = 1;
%2. Indirect long distance communication
% bi-directional non-periodic next-to-next ossilator only (d=+-2) % both upper and lower 2nd subdiagonal
% d=2;
% topology(d*n+1:n+1:end) = 1;
% topology(3:n+1:end) = 1;
% topology(1,n) = 0;
% bi-directional periodic next-to-next ossilator only (d=+-2) % both upper and lower 2nd and (n-2)th subdiagonals
% d=2;
% topology(d*n+1:n+1:end) = 1;
% topology(3:n+1:end) = 1;
% topology(1,n) = 0;
% d=n-2;
% topology(d*n+1:n+1:end) = 1;
% topology(n-1,1) = 1;
% topology(n,2) = 1;
for i = 1:n
for j = 1:n
L = j-i;
d = 1; % next oscillator communication only
%d = 2; % next-to-next oscillator communication only
%if L == d % uni--directional non-periodic: lower dth subdiagonal
%if -L == d % uni--directional non-periodic: upper dth subdiagonal
%if (L == d) | (L == d-1) % uni--directional non-periodic compact: all upper subdiagonals till d (here d=2)
%if (-L == d) | (L == n-d) % uni--directional periodic: upper dth and lower (n-d)th subdiagonals
if abs(L) == d % bi-directional non-periodic: both upper and lower dth subdiagonals
%if (abs(L) == d) | (abs(L) == n-d) % bi-directional periodic: both upper and lower dth and (n-d)th subdiagonals
topology(j,i) = 1;
end
end
end
% rng(0)
stop = 0;
while ~stop
init_controls
init_figure
init_plot
shg
t = 0;
loop = 1;
flag = 0;
while loop
s = max(step,.01);
options = odeset('outputfcn',@outputfcn, ...
'maxstep',s,'initialstep',s);
if isinf(quit)
tspan = t:s:t+delta; % s=timestep, delta=number of timesteps
else
tspan = t:s:t+quit; % infinite run i.e., number of timesteps=inf
end
[t,theta] = ode45(@ode,tspan,theta,options);
hold on;
t = t(end);
theta = theta(end,:)';
end
end
if wanth
Hout = H;
end
close(gcf)
%-------------------------------------------------------------
%% Kuramoto's system of odes
function theta_dot = ode(~,theta)
% The nonlinear term is the gradient of the potential, partial(v)/partial(theta).
theta_dot = omega - kappa/n*gradv(theta);
% noise injection for certain time
% if t < 700 %t = 0:s/100:5000
% %random noise with a standard deviation of 2
% theta_dot = theta_dot + (5*theta_dot/100).*rand(n,1); %right way
% % theta_dot = theta_dot + 0.2*sqrt(t).*rand(n,1);
% % theta_dot = theta_dot + 500*sin(2*pi*0.1*t) .* rand(n,1); % dependent on time step
% % Gaussian Pulse
% %theta_dot = theta_dot + 1/(4*sqrt(2*pi*0.01))*(exp(-t.^2/(2*0.01)));
% %theta_dot = theta_dot + 2*sin(2*pi*15*t) + 2.5*gallery('normaldata',size(t),4)
% end
end
% Cases:
% 1. sync/unstable desync/phase-locked stable desyn
% potential: sin / tanh
% 2. sync/unstable desync (never get phase-locked stable desync system)
% cos/-cos/sec/-sec/-sech
% 5. sync/phase-locked stable desync
% sinh
% 3. unstable desync/phase-locked stable desync system
% -sin, tan, -tan, -tanh, sech
% 4. interaction with strongest possible force, i.e., immediate speed-up or slow-down
% depending on direction (modify frequency of ossilator by a strong non differential step)
% sign, kroneckerDelta, dirac, sign(sin), heaviside, square, rectpuls,
% rectangularPulse, tripuls, sawtooth, triangularPulse, ceil, floor
% x = square(t) generates a square wave with period 2π for the elements of the time array t.
% square is similar to the sine function but creates a square wave with values of –1 and 1.
% x = square(t,duty) generates a square wave with specified duty cycle.
% The duty cycle (0-100) is the percent of the signal period in which the square wave is positive.
% tanh(round(theta-theta'))): lock-phased stable desync (bottleneck
% applications with WF_amplitude=round_factor)
% intitial injection of -3 on second ossilotor -> pull third ossilator by -0.5 (second ossilator=-2.5)
% -> pull forth ossilator by -0.5 (second ossilator=-1.5)
% -> pull fifth ossilator by -0.5 (second ossilator=-1)
% second ossilator=-0.5)
% tanh(theta-theta'): sync (decay of idle waves in applications)
% intitial injection of -3 on second ossilotor -> pull all ossilators by
% different forces (more stongly nearest ossilator and more weakly distant neighbour)
% discrete functions like sign / square wave etc. there is no interation b/w
% ossilators without decay
function g = gradv(theta)
% theta-theta' is a matrix with elements theta(j)-theta(k).
% The sum is by colums and produces a column vector.
x=theta-theta';
% random unsuccessful tries
% x=abs(x);
% x=sign(x);
% x=round(x);
% x=eq(x,zeros(n)); % let the ossilators interact that has zero phase-differences
% engineered wavefront by using unitstep with everything shifted to the right by 0.5
% interaction length for phases (= initial_injection/n)
% x(x>=-0.2) = 0;
% g = dot(topology,tanh(x),2);
% or
% f = @(t) sin(t).*(t<-0.5); %t = linspace(-4*pi,4*pi); plot(t, f(t)) axis([4*pi 4*pi -1.5 1.5])
% g = dot(topology,f(x),2);
alpha=1.5;
sigma=(pi/alpha)*1.5;
f = @(t) -sin(alpha*t).*(abs(t)<sigma) + 1.*(t>sigma) -1.*(t<-sigma);
% Half-Wave Rectified Sine Function
%f = @(t) sin(t).*(sin(t)>=0);
%t = linspace(-4*pi,4*pi); plot(t, f(t)) axis([4*pi 4*pi -1.5 1.5])
g = dot(topology,f(x),2);
% original global potential / interaction matrix (any non-zero value change frequencies)
% g = sum(sin(x),2);
% modified with topology
%g = dot(topology,-sin(2*x),2);
% modified for scalable programs with tanh()
%g = dot(topology,tanh(x),2);
% modified for botlleneck programs with -tanh()* gaussian
% gaussian = @(t,amp,mu,sigma) amp*exp(-(((t-mu).^2)/(2*sigma.^2)));
% g = dot(topology, tanh(x) * gaussian(t,1,0,40), 2); %t, amp= 1 / 1/(4*0.1*sqrt(2*pi)), mu=0, sigma=0.1
%expo = @(t,interaction_length) exp(-interaction_length * t);
%g = dot(topology, tanh(x) * expo(x,0.9), 2); %t, amp= 1 / 1/(4*0.1*sqrt(2*pi)), mu=0, sigma=0.1
end
% function v = potential(theta)
% v = 0;
% for k = 1:n
% j = k+1:n;
% v = v + sum(sin((theta(j)-theta(k))/2).^2);
% end
% v = (4/n^2)*v;
% end
%%
%-------------------------------------------------------------
function status = outputfcn(t,theta,odeflag)
% Called after each successful step of the ode solver.
if isequal(odeflag,'init') && wanth
% assert(t(1) == 0)
H.t = 0;
H.theta = theta;
H.pot = potential(theta);
H.gradv = gradv(theta);
H.psi = psi;
H.order = 0-0;
end
if isempty(odeflag) % Not 'init' or 'last'.
fileID = fopen('open_d1_d2.txt','a+'); %'w'
for j = 1:length(t)
if loop == 0
break
end
% Order parameter.
z = 1/n*sum(exp(1i*theta(:,j)));
psi = 0;
if get(rotate,'value') == 1
% Rotating frame of reference.
psi = angle(z);
z = abs(z);
end
for k = 1:n
set(point(k), ...
'xdata',r(k)*cos(theta(k,j)-psi), ...
'ydata',r(k)*sin(theta(k,j)-psi))
end
% Length of arrow is order parameter.
arrow(0,z);
if pot
% Potential
% v = potential(theta(:,j));
% set(titl,'string', ...
% ['potential (' sprintf('%6.3f',v) ')'])
% addpoints(animal,t(j),v)
v=sum(dot(topology,tanh(abs(theta(:,j)-theta(:,j)')),2));
%v = sum(abs(gradv(theta))); % sum(abs(dot(topology,tanh(theta-theta'),2)))
g = dot(topology,theta(:,j)-theta(:,j)',2)
%if g(1,1) > 0.004
if abs(v) > 0.004
format long
fprintf(fileID,'%6.3f %6.3f %6.3f \n',t,g(1,1),v);
%fprintf(fileID,'%6.3f %6.3f \n',t,v);
%save('potential','t','v');
end
set(titl,'string', ...
['timeline of phase differences, potential (=' sprintf('%6.3f',v) ')'])
for k = 1:n
addpoints(animal,t(j),g(k))
end
else
% Order parameter
set(titl,'string', ...
['order parameter (' sprintf('%6.3f',abs(z)) ')'])
addpoints(animal,t(j),abs(z))
end
if wanth
nt = length(H.t) + 1;
H.t(nt) = t(j);
H.theta(:,nt) = theta(:,j);
H.pot(nt) = potential(theta(:,j));
H.gradv(:,nt) = gradv(theta(:,j));
H.psi(nt) = psi;
H.order(nt) = abs(z);
end
if isinf(quit), qt = delta; else, qt = quit; end
if rem(t(j),qt) < 1 && t(j) > 10 || t(j) >= quit
clearpoints(animal)
set(ax2,'xlim',[t(j)-1 t(j)+qt])
if t(j) >= quit
loop = 0;
stop = 1;
end
end
end
fclose(fileID);
end
status = flag + stop;
drawnow limitrate
end
function init_vars(preset)
width = 0; % Standard deviation of display ring.
step = 0.1; % Display step size.
pot = true; % Graph potential or order parameter.
uni = true; % Uniform distribution of omega.
quit = inf; % 0 <= t <= quit.
%delta = 200*(1+double(preset==3));
delta = 100;
if wanth
quit = delta;
end
%-------------------------------------------------------------
%% presets
if preset > 0
n = 5;
end
switch preset
case 0 % strong coupling
kappa = 1; %d_avg/Tcomp+Tcomm
beta = 0;
step = 0.1;
n = 8;
case 1 % free ossilations
kappa = 0;
beta = 0;
step = 1;
case 2 % strong coupling
kappa = .75;
beta = 0;
step = 1;
case 3
kappa = .36; % strong coupling compare to frequency spread of oscillators
beta = .24;
step = 0.5;
case 4
kappa = .36; % weak coupling compare to frequency spread of oscillators
beta = .23;
step = 0.5;
case 5
uni = false; %100 oscillators
n = 100;
kappa = .10;
beta = .05;
step = 0.5;
otherwise
display(['unknown preset:' int2str(preset)])
scream
end
%-------------------------------------------------------------
%%
if wanth
H.n = n;
H.kappa = kappa;
H.beta = beta;
H.t = [];
H.theta = [];
H.pot = [];
H.gradv = [];
H.psi = [];
H.order = [];
end
loop = 0;
end
function txtval(v,vmin,vmax,fmt,cb,k)
% Slider with text.
txt(k) = uicontrol('style','text', ...
'string',sprintf(fmt,v), ...
'units','normal', ...
'position',[.04 0.92-.10*k .18 .05], ...
'background',[.94 .94 .94], ...
'fontsize',get(0,'defaultuicontrolfontsize')+2, ...
'horiz','left', ...
'background','w');
uicontrol('style','slider', ...
'units','normal', ...
'position',[.04 0.88-.10*k .18 .05], ...
'min',vmin, ...
'max',vmax, ...
'value',v, ...
'callback',cb);
end
function t = toggle(str,v,cb,k,lr)
% Toggle switches.
switch lr
case 'l' % left side
x = .04;
dx = .18;
y = .34;
dy = .07;
case 'r' % right side
x = .90;
dx = .08;
y = .98;
dy = .08;
end
t = uicontrol('style','toggle', ...
'units','normal', ...
'position',[x y-k*dy dx dy-.02], ...
'string',str, ...
'value',v, ...
'callback',cb);
end
function init_controls
% Initialize buttons, sliders and toggles.
shg
% Preset buttons
uicontrol('style','text', ...
'string','preset', ...
'units','normal', ...
'horiz','left', ...
'background','w', ...
'position', [.04 .92 .10 .04])
for k = 1:5
uicontrol('style','radio', ...
'units','normal', ...
'position',[.04*k .88 .04 .04], ...
'background','w', ...
'value',double(k == preset), ...
'callback',@radiocb)
end
% Sliders
txt = zeros(5,1);
txtval(n,1,100,'n = %3d',@ncb,1);
txtval(kappa,0,1.0,'kappa = %5.3f',@kappacb,2);
txtval(beta,0,1.0,'beta = %5.3f',@betacb,3);
txtval(step,0,1,'step = %5.3f',@stepcb,4);
txtval(width,0,0.2,'width = %5.3f',@widthcb,5);
% Toggles
toggle('restart',0,@restartcb,1,'l');
rotate = toggle('rotate',1,[],2,'l');
if uni
toggle('uniform / random',0,@unicb,3,'l');
else
toggle('random / uniform',1,@unicb,3,'l');
end
if pot
toggle('potential / order',1,@potcb,4,'l');
else
toggle('order / potential',0,@potcb,4,'l');
end
toggle('exit',0,@stopcb,1,'r');
toggle('help',0,@helpcb,2,'r');
toggle('blog',0,@blogcb,3,'r');
flag = 0;
stop = 0;
end
function init_figure(~)
% Initialize figure window.
set(gcf,'menubar','none','numbertitle','off', ...
'name','kuramoto','color','white')
ax1 = axes('position',[.30 .34 .60 .60]);
circle = exp((0:.01:1)*2*pi*1i);
line(real(circle),imag(circle),'color',grey)
axis(1.2*[-1 1 -1 1]) % change circle radius for figure 1
axis square
set(gca,'xtick',[],'ytick',[])
box on
ax2 = axes('position',[.3 .07 .6 .2]); % change dimentions for potential figure 2
%animal = animatedline('linewidth',2, ...
%'color',cyan);
animal = animatedline('lineStyle','none','Marker','o','MarkerSize',1,'color',cyan);
if isinf(quit)
%axis([0 delta 0 1.2]) % change axis of potential figure 2
axis([0 delta -2*pi 2*pi])
else
%axis([0 quit 0 5.2])
axis([0 quit -2*pi 2*pi])
end
titl = title('');
box on
axes(ax1)
end
%-------------------------------------------------------------
%% Initial condition
function init_plot
% Initialize plot.
% Oscillators initially uniform around circle.
%theta = zeros(n,1); % lock-step symmetry
%theta = (1:n)'/n*2*pi; % uniform distribution on all ossilators in [0,2π]
%theta = 2*pi.*rand(n,1) -pi; % uniform distribution on all ossilators in [-π,π]
theta = 2*pi*rand(n,1); % random distribution on all ossilators in [0,2π]
%theta = double((1:n)' <1.1) * 4; %-3*pi/2; % disturbance on a single ossilator
omega = omegas(uni);
% Plot radii.
r = ones(n,1) + width*randn(n,1);
% Parula, blue is fast, yellow is slow.
colors = flipud(parula(ceil(n)));
point = zeros(n,1);
for k = 1:n
point(k) = line(r(k)*cos(theta(k)),r(k)*sin(theta(k)), ...
'linestyle','none', ...
'marker','o', ...
'markersize',6, ...
'markeredgecolor','k', ...
'markerfacecolor',colors(mod(k-1,length(colors))+1,:));
end
title(sprintf('kappa = %5.3f, beta = %5.3f',kappa,beta))
end
function omega = omegas(uni)
% Intrinsic freqencies in interval of width beta centered at 1.
if uni
omega = ones(n,1) + beta*(-1:2/(n-1):1)';
else
omega = ones(n,1) + beta*(2*rand(n,1)-1);
end
end
%%
function ncb(arg,~)
% Number of oscillators.
n = round(get(arg,'value'));
set(arg,'value',n)
set(txt(1),'string',sprintf('n = %d',n))
loop = 0;
end
function kappacb(arg,~)
% Coupling parameter.
kappa = get(arg,'value');
set(txt(2),'string',sprintf('kappa = %5.3f',kappa))
end
function betacb(arg,~)
% Spread of intrinsic frequencies.
beta = get(arg,'value');
set(txt(3),'string',sprintf('beta = %5.3f',beta))
omega = omegas(uni);
end
function radiocb(arg,~)
pos = get(arg,'position');
preset = pos(1)/.04;
init_vars(preset)
end
function stepcb(arg,~)
% Step size for display.
step = get(arg,'value');
set(txt(4),'string',sprintf('step = %5.3f',step))
flag = 1;
end
function widthcb(arg,~)
% Standard deviation of random radii.
oldw = width;
width = get(arg,'value');
set(txt(5),'string',sprintf('width = %5.3f',width))
r = (r - 1)*width/(oldw+realmin) + 1;
flag = 1;
end
function restartcb(~,~)
% Restart with current parameters.
clearpoints(animal)
drawnow
loop = 0;
%flag = 1;
end
function unicb(arg,~)
uni = get(arg,'value') == 1;
if uni
set(arg,'string','uniform / random');
else
set(arg,'string','random / uniform');
end
omega = omegas(uni);
end
function potcb(arg,~)
pot = get(arg,'value') == 1;
if pot
set(arg,'string','potential / order')
else
set(arg,'string','order / potential')
end
end
function stopcb(~,~)
loop = 0;
stop = 1;
end
function helpcb(~,~)
doc('kuramoto')
toggle('help',0,@helpcb,2,'r');
end
function blogcb(~,~)
web(['https://blogs.mathworks.com/cleve/2019/10/30' ...
'/stability-of-kuramoto-oscillators'])
toggle('blog',0,@blogcb,3,'r');
end
function arrow(z0,z1)
delete(findobj('tag','arrow_shaft'))
delete(findobj('tag','arrow_head'))
rho = angle(z1-z0);
x = real(z1);
y = imag(z1);
u = [0 -.08 -.05 -.08 0];
v = [0 -.05 0 +.05 0];
s = u;
u = u*cos(rho) - v*sin(rho) + x;
v = s*sin(rho) + v*cos(rho) + y;
line([real(z0) real(z1)],[imag(z0) imag(z1)], ...
'linewidth',1.5, ...
'color',cyan, ...
'tag','arrow_shaft');
patch(u,v,cyan, ...
'tag','arrow_head');
end
function c = grey
c = [.8 .8 .8];
end
function c = cyan
c = [0 .6 .6];
end
end