Stochastic transition in the unequal-mass Toda lattice

Abstract

In this paper, the results of various computer experiments performed on the two-particle, unequal-mass Toda lattice are discussed. These experiments indicate that this system exhibits a transition from near-integrable to macroscopically observable stochastic behavior at a critical value of the total energy ${E}_{c}$ which depends on the mass ratio of the two particles ($\frac{{m}_{2}}{{m}_{1}}$). This transition appears to occur for every mass ratio in the open interval $0<\frac{{m}_{2}}{{m}_{1}}<1$. Details of trajectory behavior at selected energies and mass ratios are exposed through presentation of a number of Poincar\'e surfaces of section. The transition discussed herein will likely prove to be significant for both mathematics and physics.