This thesis is concerned with degenerate weakly coupled systems of Hamilton-Jacobi equations, imposed on flat torus, using both PDE and dynamical methods. The PDE approach relies essentially on control and viscosity solutions tools. Our main contribution is the construction of an algorithm through which we can get a critical solution to the system as limit of monotonic sequence of subsolutions and we also adapt the algorithm to non compact setting. Moreover, we get a characterization of isolated points of the Aubry set and establish semi-concavity type estimates for critical subsolution. A crucial step in our work is to reduce our analysis from systems into either scalar Eikonal equations or discounted ones. Whereas, in the dynamical approach we use the random frame introduced by H.Mitake, A.Siconolfi, H.V. Tran, and N. Yamada to provide a cycle condition characterizing the points of Aubry set. This generalizes a property already known in the scalar case.
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