  Understanding conditions for the synchronization of a network of
interconnected oscillators is a challenging problem. Typically, only sufficient
conditions are reported for the synchronization problem. Here, we adopted the
Lyapunov-Floquet theory and the Master Stability Function approach in order to
derive the synchronization conditions for a set of coupled nonlinear
oscillators. We found that the positivity of the coupling constant is a
necessary and sufficient condition for synchronizing linearly full-state
coupled identical oscillators. Moreover, in the case of partial state coupling,
the asymptotic convergence of volume in state space is ensured by a positive
coupling constant. The numerical calculation of the Master Stability Function
for a benchmark two-dimensional oscillator validates the synchronization
corresponding to the positive coupling. The results are illustrated using
numerical simulations and experimentation on benchmark oscillators.
