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 ex...

From the definition of the "terms" of a phrase, it is clear that if a phrase is free of parentheses all its terms are letters. Hence, by II , any phrase-equation E whose members are free of parentheses is interderivable with any phrase-equation resulting from permutation of non-initial letters within members of E. Therefore any phrase-equation whose members are free of parentheses and have like...