Isochronous systems are not rare

A (classical) dynamical system is called isochronous if it features an open (hence fully dimensional) region in its phase space in which all its solutions are completely periodic (i.e., periodic in all degrees of freedom) with the same fixed period (independent of the initial data, provided they are inside the isochrony region). When the isochrony region coincides with the entire phase-space one talks of entirely isochronous systems. A trick is presented associating to a dynamical system a modified system depending on a parameter so that when this parameter vanishes the original system is reproduced while when this parameter is positive the modified system is isochronous. This technique is applicable to large classes of dynamical systems, justifying the title of this talk. An analogous technique, even more widely applicable — for instance, to any translation-invariant (classical) many-body problem — transforms a real autonomous Hamiltonian system into an entirely isochronous real autonomous Hamiltonian system. The modified system is of course no more translation-invariant, but in its centre-of-mass frame it generally behaves quite similarly to the original system over times much shorter than the isochrony period $T$ (which may be chosen at will). Hence, when this technique is applied to a “realistic” many-body Hamiltonian yielding, in its centre of mass frame, chaotic motions with a natural time-scale much smaller than (the chosen) $T$, the corresponding modified Hamiltonian shall yield a chaotic behavior (implying statistical mechanics, thermodynamics with its second principle, etc.) for quite some time before the entirely isochronous character of the motion takes over hence the system returns to its initial state, to repeat the cycle over and over again. We moreover show that the quantized version of this modified Hamiltonian features an infinitely degenerate equispaced spectrum.