Averaging method for quasi-periodic response solutions

TL;DR

This work develops an averaging method for obtaining quasi-periodic response solutions in real analytic quasi-periodic systems with Diophantine frequencies. By transforming the system to an averaged form and applying a rigorous, infinite-step KAM scheme, the authors construct a Cantorian parameter set on which a quasi-periodic solution exists and converges to the averaged equilibrium as the perturbation parameter vanishes. Crucially, hyperbolicity is not required; purely imaginary eigenvalues of the averaged linearization are allowed, and the approach extends to second-order and certain degenerate cases. Measure estimates show near-full parameter availability as the perturbation size shrinks, making the results broadly applicable to non-hyperbolic quasi-periodic dynamics in perturbed systems.

Abstract

In this paper, we present an averaging method for obtaining quasi-periodic response solutions in perturbed real analytic quasi-periodic systems with Diophantine frequency vectors. Assuming that the averaged system possesses a non-degenerate equilibrium and the eigenvalues of the linearized matrix are pairwise distinct, we show that the original system admits a quasi-periodic response solution for the parameter belonging to a Cantorian set. The proof is based on the KAM techniques, and this averaging method can be extended to the second-order systems. It is worth mentioning that our results do not require the equilibrium point to be hyperbolic, which means that the eigenvalues of the linearized matrix of the averaging system may be purely imaginary.

Theorems & Definitions (8)

• proof
• proof
• proof
• proof
• proof
• proof : Proof of Theorem \ref{['theorem']}
• proof : Proof of Theorem \ref{['remark']}
• proof : Proof of Theorem \ref{['theorem1']}