Periodic solutions of a system of complex ODEs. II. Higher periods

In a previous paper the real evolution of the system of ODEs [GRAPHICS] z(n) drop z(n)(t), (z)over dot(n) drop dz(n)(t)/dt, n = 1,...,N is discussed in C-N, namely the N dependent variables z(n), as well as the N(N-1) (arbitrary!) "coupling constants" g(nm), are considered to be complex numbers, while the independent variable t (time) is real. In that context it was proven that there exists, in the phase space of the initial data z(n)(0), (z)over dot(n)(0), an open domain having infinite measure, such that all trajectories emerging from it are completely periodic with period 2pi, z(n)(t+2pi)=z(n)(t). In this paper we investigate, both by analytical techniques and via the display of numerical simulations, the remaining solutions, and in particular we show that there exist many-emerging out of sets of initial data having nonvanishing measures in the phase space of such data-that are also completely periodic but with periods which are integer multiples of 2pi. We also elucidate the mechanism that yields nonperiodic solutions, including those characterized by a "chaotic" behavior, namely those associated, in the context of the initial-value problem, with a sensitive dependence on the initial data.