2003/06/17 by F. Calogero, M. Sommacal
Physics and Astronomy · #nlin.SI
published as J. Nonlinear Math. Phys., volume9, no.4 (2002) 483-516 · arxiv version is already official
arxiv created 2003/06/17 · arxiv updated 2009/11/30
In a previous paper the real evolution of the system of ODEs zn + zn=∑m = 1, m ≠ nN gnm(zn - zm) - 3, zn ≡ zn(t), zn ≡ \fracd zn(t)dt, n = 1,...,N is discussed in CN, namely the N dependent variables zn, as well as the N(N - 1) (arbitrary!) ``coupling constants'' gnm, 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 zn(0), zn (0), an open domain having infinite measure, such that all trajectories emerging from it are completely periodic with 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 periodic but with periods which are integer multiples of 2π. 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.