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I'm reading the original article about the Fermi-Pasta-Ulam-Tsingou (FPUT) problem and I have some problems about the conclusion. Here the behavior of the system as was reported in the article: $$x_i=(x_{i+1}+x_{i-1}-2x_i)+\alpha[(x_{i+1}-x_i)^2-(x_i-x_{i-1})^2]$$ for example for the quadratic perturbation, $i=1,2,...,64$.

The paradox arises from the bizarre behavior of the system which presents the predominance of one of the modes, i.e. one of the modes has more energy than all the others put together. Only the first few modes exchanges energy between each others and do this regularly. This were the FPUT original observations. At the end of the article there is a reference to the Frobenius and Perron theorem which is summarily recalled

Let $A$ be a matrix with positive elements. Consider the linear transformation of the $n$-dimensional space defined by this matrix. One can assert that if $\bar{x}$ is any vector with all its components positive, and if $A$ is applied repeatedly to this vector, the directions of the vectors $\bar{x}$, $A\bar{x}, A^2\bar{x},..., A^i\bar{x}$, will approach that of a fixed vector $\bar{x}_0$ in such that a way that $A\bar{x}=\lambda\bar{x}_0$. This eigenvector is unique among all vectors with all their components non-negative. [...]

This above is quoted by the FPUT's article, which continues:

[...] Let $Q$ be a transformation of a $n$-dimensional space which is non-linear but is still rather simple algebriacally (let us say, quadratic in all the coordinates). Consider any vector $\bar{x}$ and the iterates of the transformation $Q$ acting on the vector $\bar{x}$. In general, there will be no question of convergence of these vectors $Q^n\bar{x}$ to a fixed direction. But a weaker statement is perhaps true. The directions of the vectors $Q^n\bar{x}$ sweep out certain cones $C_{\alpha}$ or solid angles in space in such a fashion that the time averages, i.e., the time spent by $Q^n\bar{x}$ in $C_{\alpha}$, exist for $n\to\infty$. These time averages may depend on the initial values, given $C_{\alpha}$. In other words, the space of all direction divides into a finite number of regions $R_i$, such that for vector $\bar{x}$ taken from any one of these regions the percentage of time spent by images $\bar{x}$ under the $Q^n$ are the same in any $C_{\alpha}$.

Now I can't understand the meaning of these $C_{\alpha}$ I think that if the system goes into a "cyclic behavior" for what is about energy exchange I can somehow think to the modes like vectors in a space and I can think the system in approaching these vectors one by one cyclically, so I can read the $C_{\alpha}$ like a formalization of this idea, is it correct?

Another question is: what does it means, in this context, to iterate the application of a transformation ($A$ or $Q$) on a vector? Could it represent somehow the evolution of the system?

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  • $\begingroup$ pter26, The English is fine, though I did correct some typos. $\endgroup$ – stafusa Mar 12 at 8:15

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