Semianalytical Measures of Nonlinearity Based on Tensor Eigenpairs

Abstract

This paper proposes a semianalytical measure of nonlinearity (MoN) based on tensor eigenpairs. Approximating nonlinear dynamics via local linearization is a common practice that is used to enable analytical methods in dynamics, control, navigation, and uncertainty propagation. However, these linear approximations are only applicable near the linearization point, and the size and characteristics of the linear region are dependent on the system’s underlying degree of nonlinearity. This paper presents a novel MoN that is based on the eigenpairs of tensors that are derived from the higher-order terms in a Taylor series expansion, e.g., the local dynamics tensors and state transition tensors. Unlike existing MoN, the tensor eigenpair measure of nonlinearity (TEMoN) is semianalytical and its computation does not require empirical sampling or numerical optimization. TEMoN can be used to quantify nonlinearity, identify directions of strong nonlinearity, predict the size of a linear region, and justify the truncation order of a Taylor series expansion by distinguishing higher-order contributions to nonlinearity. The TEMoN method is demonstrated for equilibrium points in the circular restricted three-body problem, near-rectilinear halo orbits, and nonlinear parameter transformations.