Saddle Point Analysis for an Ordinary Differential Equation in a Banach Space, and an Application to Dynamic Buckling of a Beam

Publisher Summary This chapter describes saddle point analysis for an ordinary differential equation in a Banach space and an application to dynamic buckling of a beam. Saddle point analysis originated in the context of an ordinary differential equation (ODE) in Rn, but has been extended to neutral functional differential equations. The chapter also presents an extension to a class of ODE in a Banach space As hypotheses, it is assumed in the chapter that both the equation may be written in variation of constants form and that an exponential decomposition holds for the linearized equation. With these hypotheses, certain proofs for ODE in Rn carry over to ODE in a Banach space almost word for word. Whether a solution to an ODE in a Banach space satisfies the corresponding variation of constants formula, and vice-versa, seem to be delicate questions. The combined use of an invariance principle and saddle point analysis may find other applications to problems in continuum mechanics, whose characteristic feature is a multiplicity of equilibrium states, steady states, or periodic solutions.