Resonance modes of near-conservative nonlinear systems: PMM vol. 38, n≗ 3, 1974, pp. 459–464

Abstract

The Rauscher method is used to construct the steady-state resonance solutions of near-conservative nonautonomous multi-dimensional systems. It is assumed that the generating system has an analytic potential and admits of normal oscillations with rectilinear trajectories in configuration space. As is well known, the forced oscillations of systems with one degree of freedom in the resonance region are close to the natural oscillations of unperturbed conservative systems [1]. We present the possibility of generalizing this result to the multi-dimensional case, using the concept of normal forms of oscillations of conservative nonlinear systems [2, 3]. By selecting special types of external actions it was shown in [4] that the resonance modes possess the properties of the normal oscillations of conservative systems. For sufficiently general types of external periodic perturbations of quasi-linear systems close to Liapunov systems, Malkin [5] has exhaustively studied the periodic modes.