Abstract
The paper deals with the problem of the existence of a normal form for a nearly-integrable real-analytic Hamiltonian with aperiodically time-dependent perturbation decaying (slowly) in time. In particular, in the case of an isochronous integrable part, the system can be cast in an exact normal form, regardless of the properties of the frequency vector. The general case is treated by a suitable adaptation of the finite order normalization techniques usually used for Nekhoroshev arguments. The key point is that the so called “geometric part” is not necessary in this case. As a consequence, no hypotheses on the integrable part are required, apart from analyticity. The work, based on two different perturbative approaches developed by Giorgilli et al., is a generalisation of the techniques used by the same authors to treat more specific aperiodically time-dependent problems.
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Notes
It is easy to see that any attempt to consider the limit \(r \rightarrow \infty \) would imply the degeneration into a trivial problem, (i.e. in which the allowed perturbation size reduces to zero, see also Giorgilli and Galgani 1985, formula (46), P. 105).
We stress that this hypothesis is usually not satisfied in the case of periodic or quasi-periodic time dependence.
In particular, if \(g \in {\mathfrak {C}}_{\rho ,\sigma }\) then \(|g_k|_{\rho } \le \left\| g \right\| _{\rho ,\sigma } \exp (-|k|\sigma )\) for all \(t \in {\mathbb {R}}^+\).
See bound (54).
Recall (2), then use the inequality \(\sum _{k \in {\mathbb {Z}}^n} \exp (-\delta |k|{\hat{\sigma }}) \le (e \delta ^{-1} {\hat{\sigma }}^{-1})^{2n}\). Its variant \(\sum _{k \in {\mathbb {Z}}^n} (1+|\omega ||k|)\exp (-\delta |k|{\hat{\sigma }}) \le C_{\omega } (e \delta ^{-1} {\hat{\sigma }}^{-1})^{2n}\) is used to obtain the second of (13).
Use the inequality \( \ln (1-x) \ge -2 x \ln 2\), valid for all \(x \in [0,1/2]\).
The reason for using \(n C_{\omega }\) in the definition of \(\Theta \) will be clear in the proof of Lemma 4.6.
A similar (and even stronger) phenomenon could have been noticed in the original setting. Namely, suppose by induction that \(\left\| F^{(j)} \right\| _{(\rho _j,\sigma _j)} \le \epsilon _j \exp (-a_j t)\). By Lemma 4.2 and (19), one finds that \(\left\| F^{(j+1)} \right\| _{(\rho _{j+1},\sigma _{j+1})} \le \epsilon _{j+1} \exp (- 2 a_j t)\) and so on. This leads to a remarkable rate of decay (\(a_j=2^{j}a\)) but not to a substantial improvement of the estimates and of the threshold (15) of \(\varepsilon _a\), as these are uniform in j.
Namely, those terms of the diagram which Fourier harmonics belong to \(\Lambda _s\).
Obviously, \(\Omega (\rho ) \equiv 0\) for all \(\rho \) in the case of an isochronous system, so that (32) would impose no restrictions on \(\rho \).
Use the inequality \(\sum _{|k| \ge (s-1)N} \exp (-\delta |k|\sigma ) \le \exp (-N \delta n (s-1) \sigma ) (\sum _{m=0}^{+\infty }\exp (-\delta m \sigma ))^n \le (2/\delta )^n\), where in this case \(\delta :=d_{s+\frac{1}{2}}-d_s= d/(2r)\).
From a “computational” point of view, first compute \(\theta _1\) then proceed with \(\beta _s,\theta _s\) for all \(s=2,\ldots ,r.\)
Clearly (49) holds for \(s \le r\) if \(y(r)\le \exp (r-1)\) for all \(r \ge 3\) (let it be directly checked for \(r=1,2\)). Hence set \(r=n+1\) and prove that \(y(r)_{r=n+1} \le \exp (n)\) for all \(n \ge 2\), conclusion that is immediate as one can find that \(y(n) \le n+1 + 3 e^n/(4n)\).
The use of (12) with \(\delta =0\) would have given \((s/j)^2\) instead of (s / j), producing in this way a troublesome factorial in the estimates.
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Acknowledgments
The first author is grateful to Proff. D. Bambusi, L. Biasco, A. Giorgilli and T. Penati for very useful discussions on a preliminary version of this paper. This research was supported by ONR Grant No. N00014-01-1-0769 and MINECO: ICMAT Severo Ochoa project SEV-2011-0087.
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Fortunati, A., Wiggins, S. Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations. Celest Mech Dyn Astr 125, 247–262 (2016). https://doi.org/10.1007/s10569-016-9684-1
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DOI: https://doi.org/10.1007/s10569-016-9684-1