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We give a simple proof of the existence of response solutions in somequasi-periodically forced systems with a degenerate fixed points. The samequestions were answered in \cite{ss18} using two versions of KAM theory.  Our method is based on reformulating the existence of response solutions as afixed point problem in appropriate spaces of smooth functions. By algebraicmanipulations, the fixed point problem is transformed into a contraction.  Compared to the KAM method, the present method does not incur a loss ofregularity. That is, the solutions we obtain have the same regularity as theforcing. Moreover, the method here applies when problems are only finitelydifferentiable. It also weakens slightly the non-degeneracy conditions. Sincethe method is based on the contraction mapping principle, we also obtainautomatically smooth dependence on parameters and, when studying complexversions of the problem we discover the new phenomenon of monodromy. We alsopresent results for higher dimensional systems, but for higher dimensionalsystems, the concept of degenerate fixed points is much more subtle than in onedimensional systems.  To illustrate the power of the method, we also consider two problems notstudied in \cite{ss18}: the forcing with zero average and second orderoscillators. We show that in the zero average forcing case, the solutions arequalitatively different, but the second order oscillators is remarkablysimilar.

Response solutions to the quasi-periodically forced systems with  degenerate equilibrium: A simple proof of a result of W. Si and J. Si. and  extensions