RESPONSE SOLUTIONS TO HARMONIC OSCILLATORS BEYOND MULTI-DIMENSIONAL BRJUNO FREQUENCY

This paper focuses on the quasi-periodically forced nonlinear harmonic oscillators <(x)over dot> + lambda(2)x = epsilon f(omega t, x), where lambda is an element of O, a closed interval not containing zero, the forcing term f is real analytic, and the frequency vector omega is an element of R-d (d >= 2) is beyond Brjuno frequency, which we call as Liouvillean frequency. For the given class of the frequency omega is an element of R-d, which will be given later, we prove the existence of real analytic response solutions (the response solution is the quasi-periodic solution with the same frequency as the forcing) for the above equation. The proof is based on a modified KAM (Kolmogorov-Arnold-Moser) theorem for finite-dimensional harmonic oscillator systems with Liouvillean frequency.

Automated summary

This paper focuses on quasi-periodically forced nonlinear harmonic oscillators and proves the existence of real analytic response solutions for a given class of frequencies, based on a modified KAM theorem.