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

DOI

10.2991/jnmp.2002.9.4.9How to use a DOI?

Abstract

In a previous paper the real evolution of the system of ODEs ¨zn + zn = N m=1, m=n gnm(zn - zm) -3 , zn zn(t), zn dzn(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 2, zn(t + 2) = zn(t). In this paper we investigate, both by analytcal 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 2. We also elcidate 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.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - F. Calogero
AU  - M. Sommacal
PY  - 2002
DA  - 2002/11/01
TI  - Periodic Solutions of a System of Complex ODEs. II. Higher Periods
JO  - Journal of Nonlinear Mathematical Physics
SP  - 483
EP  - 516
VL  - 9
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2002.9.4.9
DO  - 10.2991/jnmp.2002.9.4.9
ID  - Calogero2002
ER  -