function [X, outcome] =AndrzejakChaosMain(N, b, alphaD, Deltaalpha, epsilon, couplingonset, X0, WindowLength, NumberWindows, dt)
%
% This source code allows you to run the simulations presented in our paper
% [1] Andrzejak, Ralph Gregor, Giulia Ruzzene, and Irene Malvestio.
% Generalized synchronization between chimera states." Chaos; 27 (5): 053114. (2017).
%
% In general, we follow the notation of [1]. Only the auxiliary response is called Theta instead of Psi'
%
% Input
% N: Number of oscillators in each network (50 in [1])
% b: broadness of coupling kernel (18 in [1])
% alphaD: phase lag parameter of the driver network (1.46 in [1])
% Deltaalpha: difference between the phase lag parameter of the driver and response network. alphaR = alphaD+Deltaalpha (ranged from 0 to 0.08 in [1])
% epsilon: coupling strength (ranged from 0 to 1 in [1])
% couplingonset: number of sample points at which the coupling is turned on from zero to epsilon (100,0000 in [1])
% X0: initial conditions in a vector of size (3*N,1). (random intial conditions uniformly distributed between 0 and 2*pi in [1], using rand(150,1)*2*pi)
%
% NumberWindows: Total number of windows -of WindowLength samples each- used for simulation (250 in [1])
% WindowLength: Number of samples in each window. The last window is given as output X. (20,000 used in [1])
% dt sampling time used for the numerical integration (0.01 in [1])
% Taken these three parameters, we have 250 windows * 20,000 samples/window = 5,000,000 samples. Corresponding to 50,000 dimensionless time units
%
% Output
% X: phases of all three network in one joint matrix of size: (3N,WindowLength);
% In X: 1:N phases of driver; N+1:2*N phases of response; 2*N+1:3*N phases of auxiliary response.
% outcome: string explaining what happened for this realization
% The three networks (driver Phi, response Psi, and auxiliary response Theta)
% are generated using one set of differential equations. We here set the
% indices to address them:
Phi = 1:N;
Psi = N+1:2*N;
Theta = 2*N+1:3*N;
% Writing the phase lag parameters into a matrix of size (3*N,3*N).
% This matrix representation is used for computational efficiency.
alphaR = alphaD+Deltaalpha;
alphaV = zeros(3*N);
alphaV(Phi,Phi) = alphaD;
alphaV(Psi,Psi) = alphaR;
alphaV(Theta,Theta) = alphaR;
% Defining the coupling kernel matrix G
Rectangularwindow = zeros(N,1);
Rectangularwindow(1:b+1)=1;
Rectangularwindow(N-b+1:N)=1;
G = zeros(3*N,3*N);
for i = 1:round(N)
G(i,1:N)=circshift(Rectangularwindow,i-1);
G(i+N,Psi)=G(i,Phi);
G(i+2*N,Theta)=G(i,Phi);
end
G = G/2/b;
% Allocating the memory space for the phases of all three networks.
X = zeros(WindowLength,N*3);
% Setting the last state of the network to the initial conditions.
x = X0;
% Specifying the function which contains the differential equation of the
% entire network (driver & response & auxiliary response). Included below.
deriv = @AndrzejakChaosNetworkDifferentialEquation;
% default entry for outcome
outcome = 'No chimera collapsed. No synchronization occurred';
TotalSamples = WindowLength*NumberWindows;
for j =1:TotalSamples
% one-time activation of coupling
if j==couplingonset;
for i = 1:round(N)
G(i+N,i)=epsilon;
G(i+2*N,i)=epsilon;
end
end
% Runge-Kutta integration of order four
k1 = dt*feval(deriv,x,G,alphaV);
k2 = dt*feval(deriv,x+k1/2,G,alphaV);
k3 = dt*feval(deriv,x+k2/2,G,alphaV);
k4 = dt*feval(deriv,x+k3,G,alphaV);
x = x + k1/6+k2/3+k3/3+k4/6;
% wrapping the phases
x = mod(x,2*pi);
% Writing the phases to the output X using the modulus of Windowlength.
% In this way, always the last WindowLength samples are in X.
X(mod(j-1,WindowLength)+1,:)=x;
% What follows all serves to check for possible chimera collapses in one of the
% networks or onsets of synchronization. This is done only every 100th
% sample to reduce the computational load.
% Order parameter of driver network to check if its chimera has collapsed
Z = 1/N*abs(sum(exp(1i*x(Phi))));
if Z >0.9999
outcome='Driver network collapsed to fully synchronized state.';
break
end
% Order parameter of response network to check if its chimera has collapsed
Z = 1/N*abs(sum(exp(1i*x(Psi))));
if Z >0.9999
outcome='Response network collapsed to fully synchronized state.';
break
end
% Order parameter of auxiliary response network to check if its chimera has collapsed
Z = 1/N*abs(sum(exp(1i*x(Theta))));
if Z >0.9999
outcome='Auxiliary response network collapsed to fully synchronized state.';
break
end
deltaphipsi=mean(abs(sin((x(Phi)-x(Psi))/2)));
if deltaphipsi == 0
outcome='Identical synchronization between driver and response.';
break
end
deltaphitheta=mean(abs(sin((x(Phi)-x(Theta))/2)));
if deltaphitheta == 0
outcome='Identical synchronization between driver and auxiliary response.';
break
end
deltapsitheta=mean(abs(sin((x(Psi)-x(Theta))/2)));
if deltapsitheta == 0
outcome='Identical synchronization between response and auxiliary response => Generalized synchronization between driver and response.';
break
end
end
% Make sure that the last point is always at the right end of the window.
% See above how the data is written circularly into X.
X = circshift(X,-(mod(j,WindowLength)));
disp(outcome);
function dy = AndrzejakChaosNetworkDifferentialEquation(x,G,alpha)
diffm=bsxfun(@minus,x,x');
dy = -sum(sin(diffm+alpha).*G,2);