The transition from regular to irregular motions, explained as travel on Riemann surfaces

Abstract

Abstract. We introduce and discuss a simple Hamiltonian dynamical system, interpretable as a 3-body problem in the (complex ) plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on Riemann surfaces. The interest of this phenomenology -illustrating the onset in a deterministic context of irregular motions -is underlined by its generality, suggesting its eventual relevance to understand natural phenomena and experimental investigations. Here only some of our main findings are reported, without detailing their proofs: a more complete presentation will be published elsewhere.

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.…”

Dynamical systems on infinitely sheeted Riemann surfaces

“…Although it was proved that all singularities are finite order branch points, the presence of an infinite number of them could still produce aperiodic motion as the solution visits an infinite number of sheets. In the subsequent work [14], the Riemann surface of the solution was infinitely sheeted for generic values of the parameters, yet it could be described in full detail. Using these rather novel techniques, Calogero et al were able to derive analytic expressions for the periods and prove sensitive dependence on initial conditions.…”

“…Using these rather novel techniques, Calogero et al were able to derive analytic expressions for the periods and prove sensitive dependence on initial conditions. In [14] the projection of square-root branch points of the infinitely sheeted Riemann surface cover densely a circle on the complex plane for generic values of the parameters. As a further step on the way to understand the connection between chaotic behaviour and analytic structure, it was natural to extend the study to Riemann surfaces whose branch points projected onto C do not cover densely a one-dimensional curve in C, but the whole complex plane.…”