Nonlinear harmonic oscillators. (English) Zbl 1116.34319
This paper deals with the existence of assembles of an arbitrary number of complex oscillators, or equivalently of an arbitrary even number of real oscillators, characterized by Newtonian equations of motion (“acceleration equal force”) with one-body velocity-dependent linear forces and many-body velocity-independent cubic forms, all the nonsingular solutions of which are isochronous. As for the singular solutions, as usual they emerge, in the context of the initial-value problem, from a closed domain in phase space having lower dimensionality.
Reviewer: Messoud A. Efendiev (Berlin)
MSC:
34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
34C25 | Periodic solutions to ordinary differential equations |
37C27 | Periodic orbits of vector fields and flows |
37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
70K42 | Equilibria and periodic trajectories for nonlinear problems in mechanics |