Isochronous dynamical systems, the arrow of time and the definitions of “chaotic” versus “integrable” behaviours

by Francesco Calogero (Università di Roma “La Sapienza”)

Monday 24 May 2010, 13:15 → 14:15 Europe/Stockholm

FA32

Any (autonomous) dynamical system can be extended or modified, obtaining thereby a new (autonomous) dynamical system involving a constant T---the value of which can be freely assigned---and featuring the following two properties: (i) all solutions of the new model are isochronous (completely periodic in all their degrees of freedom with the assigned period T); (ii) starting from generic initial data, the time evolution of the new dynamical system over time intervals of order is essentially identical to that of the original dynamical system, up to a constant rescaling of time and of corrections of order . These findings entail that, in some sense, “isochronous systems are not rare” and moreover that such systems may feature an “extremely complicated “ time-evolution. They are also valid in the context of Hamiltonian dynamics; they are in particular applicable to the most general many-body problem (provided it is, overall, translation-invariant), entailing remarkable observations about statistical mechanics, thermodynamics and the issue of the “arrow of time” for macroscopic physics. Since completely periodic systems are maximally superintegrable (possessing the maximal number of functionally independent constants of motion compatible with the time evolution not being frozen), these findings also entail that any (Hamiltonian) dynamics can be embedded into a superintegrable (Hamiltonian) dynamics; and again, that “integrable (indeed, superintegrable) Hamiltonian systems are not rare” and that such systems may feature an “extremely complicated “ time-evolution. All these findings have been obtained together with François Leyvraz. Some of them are reported in a recent monograph (F. Calogero, Isochronous systems, Oxford University Press, 2008); others are more recent, see references listed below. An even more recent finding demonstrates how to modify an arbitrary (autonomous) dynamical system so that the (also autonomous) modified system is isochronous (with an arbitrarily assigned period T) yet its dynamics for an arbitrary fraction (of course, less than unity) of its (periodic) time evolution is exactly identical to that of the original system (F. Calogero and F. Leyvraz, “Isochronous systems, the arrow of time, and the definition of deterministic chaos”, submitted to Lett. Math. Phys., January 2010). These findings suggest the need to invent new definitions --- associated with a finite time scale --- of the “chaotic” versus “integrable” behaviours of dynamical systems (all current definitions refer instead to the behaviour over infinite time).