We generalize the notion of Lyapunov exponents to higher order derivatives. For fixed points and periodic orbits we derive a relationship for the higher order Lyapunov spectrum in terms of the usual first order Lyapunov spectrum. Based on numerical experiments as well as general arguments, we conjecture that this relationship also holds for chaotic orbits. This work is relevant to a priori error estimates for time series forecasting.
Dressler, U., & Farmer, J.D. (1992). 'Generalized Lyapunov Exponents Corresponding to Higher Order Derivatives.' Physica D, 59, pp.365-377.