Higher order normal modes

TL;DR

This paper extends the concept of normal modes to systems where the potential's Taylor expansion at critical points begins with higher order terms, exploring their properties and symmetric examples.

Contribution

It introduces a generalized notion of normal modes for higher order potentials, expanding the theoretical framework beyond quadratic approximations.

Findings

Higher order normal modes share properties with standard modes

Symmetric examples illustrate the extended concept

Potential applications in complex dynamical systems

Abstract

Normal modes are intimately related to the quadratic approximation of a potential at its hyperbolic equilibria. Here we extend the notion to the case where the Taylor expansion for the potential at a critical point starts with higher order terms, and show that such an extension shares some of the properties of standard normal modes. Some symmetric examples are considered in detail.