Resonance Interactions between Oscillators

Abstract

We shall consider one of the basic, but at the same time one of the most elementary, problems in the theory of nonlinear oscillations and waves, namely the interaction between three coupled oscillators with quadratic nonlinearity. In the absence of nonlinearity, as we know, a system of three coupled oscillators will undergo motions that are the simple superpositions of oscillations at the three normal frequencies (ω ₁, ω ₂,ω ₃). The equation of the system written in normal coordinates takes the form ${\ddot x_j} + w_j^2{x_j} = 0$ (j = 1, 2, 3). The presence of weak nonlinearity leads to a small right-hand side in the equation, i.e.,

$$ {\ddot x_j} + w_j^2{x_j} = \mu f\left( {{X_1},\,{X_2},\,{X_3}} \right) $$

((17.1))

, where µ ≪ 1. (17. 1) Two questions naturally arise: 1) why have we chosen three oscillators for analysis and 2) why have we limited ourselves to quadratic nonlinearity? These questions are related. In fact, if any of the quantities are nonlinear functions, e.g., of the potential field or potential (the nonlinearity, whatever it is, being weak), then this function may be expanded in the form of a power series of the potential.