Towards a theory of chaos explained as travel on Riemann surfaces

Abstract

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.

A Solvable N-body Problem of Goldfish Type Featuring N2 Arbitrary Coupling Constants

“…To do so one must investigate the time evolution of each coordinate over the Riemann surface associated with the configuration of the zeros of the polynomial (2.2). This is not a trivial endeavour, as demonstrated by various papers where this phenomenology has been studied in considerable detail [8,12,[22][23][24][25].…”

Three New Classes of Solvable N-Body Problems of Goldfish Type with Many Arbitrary Coupling Constants

“…However, there are lots of solutions that are completely periodic with periods which are integer submultiples of T MAX . The detailed identification of these solutions and their periods is a nontrivial matter, as shown, for instance, by the discussion of this phenomenology in the paper [11]-that treats the "periodic goldfish model" (for this terminology, see [8]), which is in fact characterized by the same equations of motions (11a), but with all coupling constants vanishing, g m " 0-and by the detailed investigation of the structure of the Riemann surfaces associated with other analogous many-body models [21][22][23][24][25]. Proposition 2.2.…”

“…If a time-dependent polynomial P N pz; tq , of degree N in z, is time-periodic with periodT, P N`z ; t`T˘" P N pz; tq, the unordered set z ptq of its N zeros z n ptq is of course periodic with the same periodT, z`t`T˘" z ptq (since after a period the polynomial is unchanged); however, due to the possibility that these zeros, as it were, "exchange their places" over their time evolution, the period of each individual zero z n ptq, considered as a continuous function of time, may be a positive integer multiple ofT; although of course that multiple cannot exceed the number N! of permutations of the N elements of the unordered set z ptq (for a detailed discussion of this phenomenology in analogous many-body contexts see [11,[21][22][23][24][25]). …”

Asymptotically isochronous systems

“…In this case the model without F is generally not integrable, yet (if considered in the complex , namely without restricting the dependent variables z n -nor, for that matter, the coupling constants g 2 nm -to be real) it still does feature an open, hence fully dimensional, region in its phase space where all solutions are completely periodic with the same period T , see (4a) [18,24]; while in other regions of its phase space it might also be periodic but with periodsT = pT where the numbers p are integers but might be very large, or it might even display an aperiodic, quite complicated (in some sense chaotic) behavior [25] (for recent progress in the understanding of this phenomenology see [26][27][28][29]). It then stands to reason that the solutions of the generalized model (14) with (14a) replaced by (25) (and of course x in (14b) replaced by z) shall again approach asymptotically solutionsincluding, from open regions of initial data, completely periodic ones -of the model (25) without F : entailing a remarkable, and quite rich, phenomenology.…”