A new invariance property of Lyapunov characteristic directions

Lyapunov exponents and direction fields are used to characterize the time-scales and geometry of general linear time-varying (LTV) systems of differential equations. We bring to light new invariance properties of Lyapunov direction fields to show that they are analogous to the Schur vectors of an linear time invariant (LTI) system and reduce to the Schur vectors when computed for LTI systems. We also show that the Lyapunov direction field corresponding to the smallest Lyapunov exponent when computed for an LTI system (with real distinct eigenvalues) reduces to the eigenvector corresponding to the smallest eigenvalue and when computed for a periodic LTV system (with real distinct Floquet exponents), reduces to the Floquet direction field corresponding to the smallest Floquet exponent.