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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

We give a simple proof of the existence of response solutions in some quasi-periodically forced systems with degenerate fixed points. The same questions were answered by Si and Si (2018 Nonlinearity 31 2361–18) using two versions of Kolmogorov–Arnold–Moser (KAM) theory. Our method is based on reformulating the existence of response solutions as a fixed point problem in appropriate spaces of smooth functions. By algebraic manipulations, the fixed point problem is transformed into a problem dealt with contraction mapping principle. Compared to the KAM method, the present method does not incur a loss of regularity. That is, the solutions we obtain have the same regularity as the forcing. Moreover, the method here applies when problems are only finitely differentiable. It also weakens slightly the non-resonance conditions on the forcing frequencies. Since the method is based on the contraction mapping principle, we also obtain automatically smooth dependence on parameters and, when studying complex versions of the problem we discover the new phenomenon of monodromy. We also present results for higher dimensional systems, but for higher dimensional systems, the concept of degenerate fixed points is much more subtle than in one dimensional systems. To illustrate the power of the method, we also consider two problems not studied by Si and Si (2018 Nonlinearity 31 2361–18): the forcing with zero average and second order oscillators. We show that in the zero average forcing case, the solutions are qualitatively different, but for the second order oscillators are remarkably similar.