A dynamical system is called isochronous if it features in its phase space an open, fully-dimensional region where all its solutions are periodic in all its degrees of freedom with the same, fixed period. Recently a simple transformation has been introduced, applicable to quite a large class of dynamical systems, that yields autonomous systems which are isochronous. This justifies the notion that isochronous systems are not rare. In this book the procedure to manufacture isochronous systems is reviewed, and many examples of such systems are provided. Examples include many-body problems characterized by Newtonian equations of motion in spaces of one or more dimensions, Hamiltonian systems, and also nonlinear evolution equations (PDEs). The book shall be of interest to students and researchers working on dynamical systems, including integrable and nonintegrable models, with a finite or infinite number of degrees of freedom. It might be used as a basic textbook, or as backup material for an undergraduate or graduate course.
Francesco Calogero, Department of Physics, University of Rome "La Sapienza"
1. Introduction ; 2. Isochronous systems are not rare ; 3. A single ODE of arbitrary order ; 4. Systems of ODEs: many-body problems, nonlinear harmonic oscillators ; 5. Isochronous Hamiltonian systems are not rare ; 6. Asymptotically isochronous systems ; 7. Isochronous PDEs ; 8. Outlook ; Appendix A: Some useful identities ; Appendix B: Two proofs ; Appendix C: Diophantine findings and conjectures