We investigate the connection between local and global dynamics in the Fermi-Pasta-Ulam (FPU) $\beta$ model from the point of view of stability of its simplest periodic orbits (SPO’s). In particular, we show that there is a relatively high-$q$ mode $[q=2(N+1)∕3]$ of the linear lattice, having one particle fixed every two oppositely moving ones (called SPO2 here), which can be exactly continued to the nonlinear case for $N=5+3m$, $m=0,1,2,\dots$, and whose first destabilization $E_{2u}$, as the energy (or $\beta$) increases for any fixed $N$, practically coincides with the onset of a “weak” form of chaos preceding the breakdown of FPU recurrences, as predicted recently in a similar study of the continuation of a very low $(q=3)$ mode of the corresponding linear chain. This energy threshold per particle behaves like $\frac{E_{2u}}{N}\propto N^{-2}$. We also follow exactly the properties of another SPO [with $q=(N+1)∕2$] in which fixed and moving particles are interchanged (called SPO1 here) and which destabilizes at higher energies than SPO2, since $\frac{E_{1u}}{N}\propto N^{-1}$. We find that, immediately after their first destabilization, these SPO’s have different (positive) Lyapunov spectra in their vicinity. However, as the energy increases further (at fixed $N$), these spectra converge to the same exponentially decreasing function, thus providing strong evidence that the chaotic regions around SPO1 and SPO2 have “merged” and large-scale chaos has spread throughout the lattice. Since these results hold for $N$ arbitrarily large, they suggest a direct approach by which one can use local stability analysis of SPO’s to estimate the energy threshold at which a transition to ergodicity occurs and thermodynamic properties such as Kolmogorov-Sinai entropies per particle can be computed for similar one-dimensional lattices.

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