Periodic Solutions of a System of Complex ODEs. II. Higher Periods

AbstractIn a previous paper the realevolution of the system of ODEsis discussed in CN, namely the Ndependent variables zn, as well as the N(N−1) (arbitrary!) “coupling constants” gnm, are considered to be complexnumbers, 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 zn(0), żn(0), an open domain having infinitemeasure, such that alltrajectories emerging from it are completely periodicwith period 2π, zn(t+2π)=zn(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 periodicbut with periods which are integer multiplesof 2π. We also elucidate the mechanism that yields nonperiodicsolutions, including those characterized by a “chaotic” behavior, namely those associated, in the context of the initial-value problem, with a sensitive dependenceon the initial data.