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

Abstract

In a previous paper the \textit{real} evolution of the system of ODEs \ddot{z}_{n} + z_{n}=\sum\limits_{m = 1, m \ne n}^{N} g_{nm}{(z_{n} - z_{m})} ^{- 3}, z_{n} \equiv z_{n}(t), \qquad \dot {z}_{n} \equiv \frac{d z_{n}(t)}{dt}, \qquad 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 \textit{complex} numbers, while the independent variable t (``time'') is \textit{real}. In that context it was proven that there exists, in the phase space of the initial data z_{n}(0), \dot {z}_{n} (0), an open domain having \textit{infinite} measure, such that \textit{all} trajectories emerging from it are \textit{completely periodic} with period 2π, z_{n} (t + 2π) = 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 \textit{completely periodic} but with periods which are \textit{integer multiples} of 2π. We also elucidate the mechanism that yields \textit{nonperiodic} solutions, including those characterized by a ``chaotic'' behavior, namely those associated, in the context of the initial-value problem, with a \textit{sensitive dependence} on the initial data.