Simulation and analysis of Kuramoto-Sakaguchi populations interacting through complex networks

This repository contains the code accompaining the paper From chimeras to extensive chaos in networks of heterogeneous Kuramoto oscillator populations by Pol Floriach, Jordi Garcia-Ojalvo, and Pau Clusella.

Two different set of codes are provided:

1. Some auto-07p files and instructions to obtain the main bifurcations displayied in the paper for a two-population model (Fig. 2). These is all provided in the auto folder. See the TUTORIAL.md in that folder.
2. The Julia code to reproduce most of the simulations in the paper. These functions are provided in the KuramotoPopulationNetwork.jl module in the src directory. In the following we show how to use the functions in that module to simulate the system using some examples.

Using the KuramotoPopulationNetwork module

Open a Julia REPL in the directory, activate the project and load/install all required dependencies:

] activate .
] instantiate
using KuramotoPopulationNetwork

Apart from these packages, to run these examples we also need three other packages. Load (or install) them with:

using Graphs, Plots, ColorSchemes

Then we can use Graphs.jl to generate a ring network with nearest neighbours:

N = 128;
g_ring = watts_strogatz(N, 2, 0.0);

Now we can reproduce of Fig. 5(a) in the paper:

t,R,ϕ,u0 = simulation(;K=7.0,p=0.9,α=1.2,gr=g_ring,trans=5e2,tmax=4e2,ic="homogeneous");
n = size(R,1)
range = n-10000:1:n;
heatmap(1:128,t[range],ϕ[range,:],c=:cyclic_mrybm_35_75_c68_n256)
heatmap(1:128,t[range],R[range,:],c=:linear_bmy_10_95_c78_n256)

We can compute and plot the full Lyapunov spectra for this case (this can take some minutes):

λ, = simulationLE(;K=7,p=0.9,α=1.2,gr=g_ring,trans=2e2,tmax=1e3,nLE=2*size(g_ring,1),ic=u0);
plot(λ)

With this data we can compute the attractor dimension and the dynamical entropy:

dimKY(λ)
entropy(λ)

Reproduction of Fig. 5(b) in the paper:

@time t,R,ϕ,u0 = simulation(;K=15.0,p=0.5,α=1.2,gr=g_ring,trans=5e2,tmax=4e2,ic="homogeneous");
n = size(R,1)
range = n-10000:1:n;
heatmap(1:128,t[range],ϕ[range,:],c=:cyclic_mrybm_35_75_c68_n256)
heatmap(1:128,t[range],R[range,:],c=:linear_bmy_10_95_c78_n256)

Simulation on the slow regime that appears for p=0.5. Notice we change tmax and dt:

t,R,ϕ,u0 = simulation(;K=20.0,p=0.5,α=1.2,gr=graph,trans=0e3,tmax=40e3,dt=1e-1,ic="homogeneous");
n = size(R,1)
range = n-1000:10:n;
heatmap(1:128,t[range],ϕ[range,:],c=:cyclic_mrybm_35_75_c68_n256)
heatmap(1:128,t[range],R[range,:],c=:linear_bmy_10_95_c78_n256)

Let's compute now the LE for a simulation of the ER network:

n = 128;
avk = 10;
g_er = erdos_renyi(n, avk*1.0/n);

t,R,ϕ,u0 = simulation(;K=14.0,p=0.9,α=1.2,gr=g_er,trans=5e2,tmax=4e2,ic="homogeneous");
λ, = simulationLE(;K=14,p=0.9,α=1.2,gr=g_er,trans=2e2,tmax=1e3,nLE=2*size(g_er,1),ic=u0);

and plot the results:

n = size(R,1)
trange = n-10000:1:n;
heatmap(1:128,t[trange],ϕ[trange,:],c=:cyclic_mrybm_35_75_c68_n256)
heatmap(1:128,t[trange],R[trange,:],c=:linear_bmy_10_95_c78_n256)
plot(λ)