Domains of analyticity for response solutions in strongly dissipative forced systems

We study the ordinary differential equation εx ̈ + x ̇ + εg(x) = εf (ωt), where g and f are real-analytic functions, with f quasi-periodic in t with frequency vector ω. If c0 ∈ R is such that g(c0) equals the average of f and g′(c0) ̸= 0, under very mild assumptions on ω there exists a quasi-periodic solution close to c0 with frequency vector ω. We show that such a solution depends analytically on ε in a domain of the complex plane tangent more than quadratically to the imaginary axis at the origin.