A Note on an Existence of Conditionally Periodic Oscillation in a one-dimensional Anharmonic Lattice

Abstract

The oscillations of a one-dimensional anharmonic lattice is investigated sufficiently near the equilibrium state, which is described by the equation d²yᵢ/dt²＝y₊₁-2yᵢ+yᵢ₋₁+α{(yᵢ₊₁－yᵢ)³－(yᵢ－yᵢ₋₁)³}, －∞＜t＜+∞, where i＝1, 2, ・・・ n, α＝constant, boundary conditions are y₀＝yₙ₊₁＝O and initial conditions yᵢ(0), dyᵢ(O)/dt are given. A large part of the oscillations starting from initial data sufficiently near the origin of the phase space of the system is proved to be a conditionally periodic motion under the incommensurable condition on linear frequencies mentioned later.