Analysis of non-autonomous linear ODE systems in bifurcation problems via lie group geometric numerical integrators

A bifurcation analysis developed within the context of the nonlinear theory of elasticity usually leads to the study of systems of second-order ODE’s characterized by matrices with varying coefficients. A common practice is that of reducing the governing set of differential equations to a simpler non-autonomous first order linear ODE system. Here, we investigate the potentiality of alternative geometric numerical integrators, based on Lie Group methods, which furnish approximate exponential representations of the matricant of first order linear ODE systems. Within such numerical schemes, the Magnus expansion seems to be very efficient since it features the determination of approximate solutions that preserve at any order of approximation the same qualitative properties of the exact (but unknown) solution and it also exhibits an improved accuracy with respect to other frequently used numerical schemes. As applications of the Magnus method, we study certain paradigmatic bifurcation problems with the major aim of investigating whether solid bodies may support certain instabilities that have common features to some classical hydrodynamic instabilities observed in viscous fluids.