Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2008-05-27
Journal of Physics A 42 (2009) 015205
Nonlinear Sciences
Exactly Solvable and Integrable Systems
30 pages
Scientific paper
10.1088/1751-8113/42/1/015205
This paper presents a more complete version than hitherto published of our explanation of a transition from regular to irregular motions and more generally of the nature of a certain kind of deterministic chaos. To this end we introduced a simple model analogous to a three-body problem in the plane, whose general solution is obtained via quadratures all performed in terms of elementary functions. For some values of the coupling constants the system is isochronous and explicit formulas for the period of the solutions can be given. For other values, the motions are confined but feature aperiodic (in some sense chaotic) motions. This rich phenomenology can be understood in remarkable, quantitative detail in terms of travel on a certain (circular) path on the Riemann surfaces defined by the solutions of a related model considered as functions of a complex time. This model is meant to provide a paradigmatic first step towards a somewhat novel understanding of a certain kind of chaotic phenomena.
Calogero Francesco
Gomez-Ullate David
Santini Paola
Sommacal Matteo
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