## Abstract

We formulate the periodic FPU problem with four alternating masses which is the simplest nontrivial version. The analysis involves normal form calculations to second order producing integrable normal forms with three timescales. In the case of large alternating mass the system is an example of dynamics with widely separated frequencies and three timescales. The presence of approximate integrals and the stability characteristics of the periodic solutions lead to weak interaction of the modes of the system.

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## References

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## Appendix

### Appendix

In the error estimates of the normal form analysis integral inequalities can be useful. We will use the *specific Gronwall lemma* formulated in [6], lemma 1.3.3.

### Lemma 5.1

Let \(\phi \) be a real-valued continuous (or piecewise continuous) functions on a real *t* interval \(I: t_0 \le t \le T\). Assume \(\phi (t) >0\) on *I* and \(\delta _1(\varepsilon ), \delta _2(\varepsilon )\) positive order functions (\(\varepsilon \) a small, positive parameter). If the inequality

holds on *I*, then

We apply the *specific Gronwall lemma* to obtain:

### Lemma 5.2

Consider the perturbation problem:

for \( I: t_0 \le t \le T, x \in D \subset {\mathbb {R}}^n\), \(\delta _1, \delta _2, \delta _3(\varepsilon )\) order functions with \(\delta _2(\varepsilon ) = o(\delta _1(\varepsilon ))\) as \(\varepsilon \rightarrow 0\) and continuous differentiability of the vector fields *f*,Â *R* on \(I \times D\); in particular we have \(||R(t, x)|| \le M, M>0\) for \(t \ge 0\). We neglect small terms to consider the solution of

and we approximate *y*(*t*) by a procedure (averaging) for which we know that \(||y(t)- \bar{y}(t)||= O(\delta _3(\varepsilon ))\) on the timescale \(1/ \delta _1(\varepsilon )\). Then we have on the timescale \(1/ {\delta _1(\varepsilon )}\) the estimate

### Proof

We formulate the equivalent integral equations for *x*(*t*),Â *y*(*t*):

Subtracting the two equations we have:

Using the Lipschitz continuity of *f* (Lipschitz constant *L*) and the estimate for *R* we have:

and with Lemma 5.1:

We conclude that *y*(*t*) approximates *x*(*t*) with error \(O( \frac{ \delta _2(\varepsilon )}{ \delta _1(\varepsilon )})\) on the timescale \(1/ \delta _1(\varepsilon )\). We conclude with the triangle inequality that

or

on the timescale \(1/ {\delta _1(\varepsilon )}\).

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Bruggeman, R., Verhulst, F. (2018). Dynamics of a Chain with Four Particles, Alternating Masses and Nearest-Neighbor Interaction. In: Belhaq, M. (eds) Recent Trends in Applied Nonlinear Mechanics and Physics. Springer Proceedings in Physics, vol 199. Springer, Cham. https://doi.org/10.1007/978-3-319-63937-6_5

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