On the asymptotic approximations of the solutions of a system of two non-linearly coupled harmonic oscillators

A procedure, closely related to the averaging method, is given for the construction of asymptotic approximations for the solutions of the Lagrangian system

$\text{d}\text{x}{}_1\text{d}\text{t}\text{=x}{}_2\text{,}\text{d}\text{x}{}_2\text{d}\text{t}\text{=?x}{}_1\text{+ε}\text{∑}\text{i, j=1}\text{4}\text{α}{}_{\mathrm{ij}}\text{x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{j}}\text{,}\text{d}\text{x}{}_3\text{d}\text{t}\text{=x}{}_4\text{,}\text{d}\text{x}{}_4\text{d}\text{t}\text{=?4(1+δ)x}{}_3\text{+ε}\text{∑}\text{i, j=1}\text{4}\text{β}{}_{\mathrm{ij}}\text{x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{j}}\text{, x}{}_1\text{(0)=a}{}_0\text{, x}{}_2\text{(0)=0, x}{}_3\text{(0)=b}{}_0\text{, x}{}_4\text{(0)=0,}$

where ε > 0 and δ > 0 are small parameters, a_ij = a_ji and β_ij = β_ji are real constants. The solutions can be represented as follows:

$\text{x}{}_1\text{=a}\text{cos}\text{(t+ξ), x}{}_2\text{=a}\text{sin}\text{(t+ξ), x}{}_3\text{=b}\text{cos}\text{(2t+ψ), x}{}_4\text{=2b}\text{sin}\text{(2t+ψ).}$

The asymptotic approximations (in a certain well-defined sense) of a, b, ξ and ψ are given. It is shown that a rapid variation of the phase, ψ(t), will take place if b → 0 for the case that δ ≈ ε or δ ? ε. In addition for the case δ ≈ ε the phenomenon of bifurcation will be discussed.