A New Invariance Property of Lyapunov Characteristic Directions

Lyapunov exponents and direction elds are used to characterize the time-scales and geometry of general linear time-varying (LTV) systems of di erential equations. Lyapunov exponents are already known to correctly characterize the time-scales present in a general LTV system; they reduce to real parts of eigenvalues when computed for linear time-invariant(LTI) systems and real parts of Floquet exponents when computed for periodic LTV systems. Here, we bring to light new invariance properties of Lyapunov direction elds to show that they are analogous to the Schur vectors of an LTI system and reduce to the Schur vectors when computed for LTI systems. We also show that the Lyapunov direction eld 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 eld corresponding to the smallest Floquet exponent.