Analytical Description of Optimally Time-Dependent Modes for Reduced-Order Modeling of Transient Instabilities

Abstract

The optimally time-dependent (OTD) modes form a time-evolving orthonormal basis that captures directions in phase space associated with transient and persistent instabilities. In the original formulation, the OTD modes are described by a set of coupled evolution equations that need to be solved along the trajectory of the system. For many applications where real-time estimation of the OTD modes is important, such as control or filtering, this is an expensive task. Here, we examine the low-dimensional structure of the OTD modes. In particular, we consider the case of slow-fast systems, and prove that OTD modes rapidly converge to a slow manifold, for which we derive an asymptotic expansion. The result is a parametric description of the OTD modes in terms of the system state in phase space. The analytical approximation of the OTD modes allows for their offline computation, making the whole framework suitable for real-time applications. In addition, we examine the accuracy of the slow-manifold approximation for systems in which there is no explicit time-scale separation. In this case, we show numerically that the asymptotic expansion of the OTD modes is still valid for regions of the phase space where strongly transient behavior is observed, and for which there is an implicit scale separation. We also find an analogy between the OTD modes and the Gram--Schmidt vectors (also known as orthogonal or backward Lyapunov vectors), and thereby establish new properties of the former. Several examples of low-dimensional systems are provided to illustrate the analytical formulation.