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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. (English) Zbl 1473.37075

The authors consider the following quasi-periodically forced systems: \[ \dot{x} = x^l+h(\omega t,x)+\varepsilon f(\omega t,x), \] where \(x\in\mathbb{R}\), \(l\in\mathbb{N}\) with \(l\geq2\), \(0<|\varepsilon|\ll1\) is a small real parameter, and \(\omega\) is a vector in \(\mathbf{\mathbb{R}}^d\) with \(d\in \mathbb{N}\). It is assumed that the function \(h\) vanishes in \(x\) to order higher than \(l\). The authors are interested in response solutions \(x(t)=a+V(\omega t)\). Introducing the linear operator \(\mathcal{L}_a=\omega\cdot\partial_{\theta}-la^{l-1}\), the existence of response solutions to the above system can be rewritten as \[ \begin{split} V(\theta)&=\mathcal{L}_a^{-1}\big(S(a,V(\theta))+h(\theta,a+V(\theta))+\varepsilon \tilde{f}(\theta,0)+\varepsilon g(\theta,a+V(\theta))\big)\\ &:=\mathcal{T}_a(V)(\theta), \end{split} \] which is a fixed point problem in appropriate spaces. Under suitable assumptions, the authors obtain the existence of response solutions via contraction mapping principle for analytic (Theorem 8) and finitely differentiable (Theorem 12) cases.
The conclusion is a generalization of the results in [W. Si and J. Si, Nonlinearity 31, No. 6, 2361–2418 (2018; Zbl 1401.37067)]. Compared to the KAM method used in [loc. cit.], the method in this paper does not assume a loss of regularity and applies when problems are only finitely differentiable. Furthermore, the assumptions on the order of vanishing and the non-resonance assumptions when \(l\) is even are weaker than in [loc. cit.]. Another point worth noting is that the method used in this paper allows to discuss complex values of the parameters, which leads to the new phenomenon of monodromy. The authors also consider some problems in higher dimensions, namely the case of zero-average forcing (Section 7) and degenerate oscillators (Section 8).

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
47H10 Fixed-point theorems
70K40 Forced motions for nonlinear problems in mechanics

Citations:

Zbl 1401.37067
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References:

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