Characterization of perturbative nonlinear systems using symmetry methods

This work studies the application of analytic symmetry methods to peturbative nonlinear oscillators featured prominently in dynamical systems. Examining the differential equations governing the behavior of such systems, we use the method of approximate Lie groups to develop perturbative solutions as 'deformations' of the symmetric, unperturbed system. Additionally, we connect the concept of the Renormalization Group method, which is used to improve the global nature of approximate solutions, to fundamental Lie symmetries of the equation. Namely, invariance of the governing equation with respect to translation of the independent variable is seen to allow for the construction of 'renormalized' solutions. In this way, we are able to systematically build uniformly valid, approximate solutions to perturbative nonlinear equations by exploiting their approximate and Renormalization Group symmetries. These solutions are then compared to those obtained by standard perturbation methods using a combination of analytic and numerical techniques. Ultimately, we seek to identify how the context and symmetries of the underlying system inform the relative merits of the corresponding solution methods.