Isochronous systems. (English) Zbl 1091.34023

Calogero, Francesco

Mladenov, Ivaïlo M.(ed.) et al., Proceedings of the 6th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 3–10, 2004. Sofia: Bulgarian Academy of Sciences (ISBN 954-84952-9-5/pbk). 11-61 (2005).

From the text: Recently – via a simple trick, amounting essentially to a change of independent (and possibly as well as of dependent) variables – the possibility has been noted to modify a quite general evolution system so that the modified system possesses a lot of completely periodic, indeed isochronous, solutions. Generally, these isochronous solutions emerge out of an open domain of initial data having full dimensionality in the space of initial data. And many of the isochronous systems obtained in this manner seem rather interesting. In this paper, these developments are reviewed, mainly in the context of dynamical systems (systems of ODEs – in particular, systems interpretable as many-body problems), and some specific examples are discussed in detail, including an analysis of the transition (to motions with higher periods, or aperiodic, or perhaps chaotic) occurring when the initial data get outside the region producing isochronous motions. The applicability of this approach in the context of nonlinear evolution PDEs is also outlined. This review paper covers the material presented at the conference via four lectures organized as follows: 1. Overview: isochronous systems are not rare; 2. The “goldfish”: theory and simulations; 3. Novel technique to identify solvable dynamical systems and a solvable extension of the goldfish many-body problem; 4. Isochronous PDEs.
For the entire collection see [Zbl 1066.53003].

Reviewer: Iliya Iliev (Sofia)

MSC:

34C25	Periodic solutions to ordinary differential equations
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70F10	\(n\)-body problems
70H12	Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics
70H06	Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations

Keywords:

isochronous systems; N-body problem; superintegrable systems; Hamiltonian function; Diophantine relations