Periodic solutions of a system of complex ODEs

We consider the real evolution of the system of ODEs [GRAPHICS] in C-N, namely we assume the N dependent variables zn. as well as the N(N - 1) (arbitrary!) "coupling constants" g(nm), to be complex numbers, while the independent variable t ("time") is real. In this context we prove 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). However, completely real initial data z(n)(0), <(z)over dot>(n)(0) are generally not included in this domain. There also hold analogous results for more general systems of ODEs. such as that which obtains by replacing the right-hand sides of the above equations of motion with arbitrary analytic functions Fn((z) under bar) satisfying the scaling property Fn(lambda(z) under bar) = lambda(-3)F(n)((z) under bar). (C) 2002 Elsevier Science B.V. All rights reserved.