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Periodic solutions of a system of complex ODEs. II. Higher periods | | F. Calogero
; M. Sommacal
; | | Date: |
18 Jun 2003 | | Journal: | J. Nonlinear Math. Phys., volume9, no.4 (2002) 483-516 | | Subject: | Exactly Solvable and Integrable Systems | nlin.SI | | Abstract: | In a previous paper the extit{real} evolution of the system of ODEs ddot{z}_{n} + z_{n}=sumlimits_{m = 1, m
e 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 extit{complex} numbers, while the independent variable t (``time’’) is extit{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 extit{infinite} measure, such that extit{all} trajectories emerging from it are extit{completely periodic} with period 2pi, z_{n} (t + 2pi) = 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 extit{completely periodic} but with periods which are extit{integer multiples} of 2pi. We also elucidate the mechanism that yields extit{nonperiodic} solutions, including those characterized by a ``chaotic’’ behavior, namely those associated, in the context of the initial-value problem, with a extit{sensitive dependence} on the initial data. | | Source: | arXiv, nlin.SI/0306034 | | Services: | Forum | Review | PDF | Favorites | |
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