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We investigate the connection between local and global dynamics in the Fermi-- Pasta -- Ulam (FPU) $\beta$ -- model from the point of view of stability ofits simplest periodic orbits (SPOs). In particular, we show that there is arelatively high $q$ mode $(q=2(N+1)/{3})$ of the linear lattice, having oneparticle fixed every two oppositely moving ones (called SPO2 here), which canbe exactly continued to the nonlinear case for $N=5+3m, m=0,1,2,...$ and whosefirst destabilization, $E_{2u}$, as the energy (or $\beta$) increases for {\itany} fixed $N$, practically {\it coincides} with the onset of a ``weak'' formof chaos preceding the break down of FPU recurrences, as predicted recently ina similar study of the continuation of a very low ($q=3$) mode of thecorresponding linear chain. This energy threshold per particle behaves like$\frac{E_{2u}}{N}\propto N^{-2}$. We also follow exactly the properties ofanother SPO (with $q=(N+1)/{2}$) in which fixed and moving particles areinterchanged (called SPO1 here) and which destabilizes at higher energies thanSPO2, since $\frac{E_{1u}}{N}\propto N^{-1}$. We find that, immediately aftertheir first destabilization, these SPOs have different (positive) Lyapunovspectra in their vicinity. However, as the energy increases further (at fixed$N$), these spectra converge to {\it the same} exponentially decreasingfunction, thus providing strong evidence that the chaotic regions around SPO1and SPO2 have ``merged'' and large scale chaos has spread throughout thelattice.Comment: Physical Review E, 18 pages, 6 figure

Stability of simple periodic orbits and chaos in a Fermi-Pasta-Ulam lattice