We consider a network of identical nonlinear oscillators, coupled through linear nearest neighbor coupling. Each agent has an attractor and we are interested in stabilizing the synchronous attractor in the network. In this talk:

(i) We derive necessary conditions for synchronization with symmetric coupling;

(ii) when these conditions are not met, we show that it is always possible to choose an appropriate non symmetric nearest neighbor coupling of the agents so that the synchronous solution is asimptotically stable;

(iii) we show that this guarantee of stability comes without having to let the coupling strength be too large.

Our construction, based on the Master Stability Function, requires the computation of Lyapunov exponents of a non autonomous linear system and the solution of a suitable inverse eigenvalue problem for the coupling matrix. Numerical implementations will be provided. This is a joint work with Luca Dieci and Alessandro Pugliese.