Non-closed Curves in R with Finite Total First Curvature Arising from the Solutions of an Ode

The solution space of a constant coefficient ODE gives rise to a natural real analytic curve in Euclidean space. We give necessary and sufficient conditions on the ODE to ensure that this curve is a proper embedding of infinite length or has finite total first curvature. If all the roots of the associated characteristic polynomial are simple, we give a uniform upper bound for the total first curvature and show the optimal uniform upper bound must grow at least linearly with the order n of the ODE. We then examine the case where multiple roots are permitted. We present several examples illustrating that a curve can have finite total first curvature for positive/negative time and infinite total first curvature for negative/positive time as well as illustrating that other possibilities may occur.