Towards a theory of chaos explained as travel on Riemann surfaces

Abstract

We investigate the dynamics defined by the following set of three coupled first-order ODEs: (z) over dot (n) + i omega z(n) = g(n+2)/z(n) - z(n+1) + g(n+1)/z(n) - z(n+2) It is shown that the system can be reduced to quadratures which can be expressed in terms of elementary functions. Despite the integrable character of the model, the general solution is a multiple-valued function of time (considered as a complex variable), and we investigate the position and nature of its branch points. In the semi-symmetric case (g(1) = g(2) not equal g(3)), for rational values of the coupling constants the system is isochronous and explicit formulae for the period of the solutions can be given. For irrational values, the motions are confined but feature aperiodic motion with sensitive dependence on initial conditions. The system shows a rich dynamical behaviour that can be understood in quantitative detail since a global description of the Riemann surface associated with the solutions can be achieved. The details of the description of the Riemann surface are postponed to a forthcoming publication. This toy model is meant to provide a paradigmatic first step towards understanding a certain novel kind of chaotic behaviour.