On the Phenomenon of Low-Frequency, Large-Amplitude Oscillations in a High-Dimensional Linear Dynamical System

This article considers a linear dynamic system that models a chain of coupled harmonic oscillators, under special boundary conditions that ensure a balanced energy flow from one end of the chain to the other. The energy conductivity of the chain is controlled by the parameter $\alpha$ of the system.
In a numerical experiment on this system, with a large number of oscillators and at certain values of $\alpha$, the phenomenon of low-frequency high-amplitude oscillations was discovered. The primary analysis showed that this phenomenon has much in common with self-oscillations in nonlinear systems. In both cases, periodic motion is created and maintained by an internal energy source that does not have the corresponding periodicity. In addition, the amplitude of the oscillations significantly exceeds the initial state amplitude. However, this phenomenon also has a fundamental difference from self-oscillations in that it is controlled by the oscillation synchronization mechanism in linear systems and not by the exponential instability suppression mechanism in nonlinear systems.
This article provides an explanation of the observed phenomenon on the basis of a complete analytical solution of the system. The solution is constructed in a standard way by reducing the dynamic problem to the problem of eigenvalues and eigenvectors for the system matrix. When solving, we use methods from the theory of orthogonal polynomials. In addition, we discuss two physical interpretations of the system. The connection between these interpretations and the system is established through the Schrödinger variables.