Connectance and stability of nonlinear symplectic systems

We have revisited the problem of the transition from ordered to chaotic motion for increasing number of degrees of freedom in nonlinear symplectic maps. Following the pioneer work of Froeschlé (Phys. Rev. A 18, 277–281, 1978) we investigate such systems as a function of the number of couplings among the equations of motion, i.e. as a function of a parameter called connectance since the seminal paper of Gardner and Ashby (Nature 228, 784, 1970) about linear systems. We compare two different models showing that in the nonlinear case the connectance has to be intended as the fraction of explicit dynamical couplings among degrees of freedom, rather than the fraction of non-zero elements in a given matrix. The chaoticity increases then with the connectance until the system is fully coupled.