Stability of Simple Periodic Orbits and Chaos in a Fermi -- Pasta -- Ulam Lattice

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 (SPOs). In particular, we show that there is a relatively high $q$ mode $(q=2(N+1)\mathrm{/}3)$ 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 $N=5+3m,m=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}$ and whose first destabilization, $E_{2u}$, as the energy (or $\beta$) increases for {\it any} fixed $N$, practically {\it coincides} with the onset of a weak'' form of chaos preceding the break down 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 SPOs have different (positive) Lyapunov spectra in their vicinity. However, as the energy increases further (at fixed $N$), these spectra converge to {\it 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.