Backwards theory supports modelling via invariant manifolds for non-autonomous dynamical systems

This article establishes the foundation for a new theory of invariant/integral manifolds for non-autonomous dynamical systems. Current rigorous support for dimensional reduction modelling of slow-fast systems is limited by the rare events in stochastic systems that may cause escape, and limited in many applications by the unbounded nature of PDE operators. To circumvent such limitations, we initiate developing a backward theory of invariant/integral manifolds that complements extant forward theory. Here, for deterministic non-autonomous ODE systems, we construct a conjugacy with a normal form system to establish the existence, emergence and exact construction of center manifolds in a finite domain for systems `arbitrarily close' to that specified. A benefit is that the constructed invariant manifolds are known to be exact for systems `close' to the one specified, and hence the only error is in determining how close over the domain of interest for any specific application. Built on the base developed here, planned future research should develop a theory for stochastic and/or PDE systems that is useful in a wide range of modelling applications

A state-of-the-art review on theory and engineering applications of eigenvalue and eigenvector derivatives

Eigenvalue and eigenvector derivatives with respect to system design variables and their applications have been and continue to be one of the core issues in the design, control and identification of practical engineering systems. Many different numerical methods have been developed to compute accurately and efficiently these required derivatives from which, a wide range of successful applications have been established. This paper reviews and examines these methods of computing eigenderivatives for undamped, viscously damped, nonviscously damped, fractional and nonlinear vibration systems, as well as defective systems, for both distinct and repeated eigenvalues. The underlying mathematical relationships among these methods are discussed, together with new theoretical developments. Major important applications of eigenderivatives to finite element model updating, structural design and modification prediction, performance optimization of structures and systems, optimal control system design, damage detection and fault diagnosis, as well as turbine bladed disk vibrations are examined. Existing difficulties are identified and measures are proposed to rectify them. Various examples are given to demonstrate the key theoretical concepts and major practical applications of concern. Potential further research challenges are identified with the purpose of concentrating future research effort in the most fruitful directions.Ministry of Education (MOE)The first and third authors gratefully acknowledge the financial support from the Singapore Ministry of Education through the award of research project grant AcRF Tier 1 RG183/17