Volume 9, Issue 4, November 2001, Pages 483 - 516
Periodic Solutions of a System of Complex ODEs. II. Higher Periods
- F CALOGERO, M SOMMACAL
- Corresponding Author
- F CALOGERO
Available online November 2001.
- https://doi.org/10.2991/jnmp.2002.9.4.9How to use a DOI?
- 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.
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Cite this article
TY - JOUR AU - F Calogero AU - M Sommacal PY - 2001/11 DA - 2001/11 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 - 1402-9251 UR - https://doi.org/10.2991/jnmp.2002.9.4.9 DO - https://doi.org/10.2991/jnmp.2002.9.4.9 ID - CALOGERO2001/11 ER -