Approximate synchronization can be achieved even for heterogeneous Van der Pol oscillators

It may be surprising to see that the heterogeneous oscillators achieve synchronization. It is approximate since exact, or asymptotic synchronization is simply not possible due to their heterogeneity. In spite of parametric differences in their dynamics, strong coupling can make them oscillate in the “same” frequency! In this result, each agent communicates only one variable with neighboring agents. A key to success is the choice of this single variable to communicate. A simulation result is posted here, and detailed analysis will appear in the future. $\dot{x} = a a \dot{y} = b b$

Number of agents
N = 4;
Coupling parameter
a = 1; b = 1;
Parameters for each
c1 = 0.9; w1 = 0.9;
c2 = 1.2; w2 = 3.2;
c3 = 0.8; w3 = 2.8;
c4 = 1.1; w4 = 4.1;
Graph Laplacian
L = [1, -1, 0, 0; -1, 2, -1, 0; 0, -1, 2, -1; 0, 0, -1, 1];

Simulation results:

Uncoupled case (k=0):

Coupled case (k=10):

A document containing the details of simulation: simVP.pdf