Trajectory-based functionals for detecting exponential growth in dynamical systems

We study exponential growth from finite-time trajectory observations in continuous-time dynamical systems. We introduce a dimensionless endpoint velocity-increment functional 𝑉 𝐿 ⁡ ( 𝜏 ) computed from the endpoint velocities of a single trajectory. For bounded velocity, the functional decays inversely with the observation time, whereas, under the stated asymptotic assumptions for exponentially growing trajectories, its logarithmic limsup recovers the corresponding observable growth rate. The functional requires only endpoint velocity observations and is exactly invariant under a constant additive velocity offset. Alongside 𝑉 𝐿 , we consider the endpoint state-amplification measure 𝐴 𝑁 , based on the ratio of endpoint state norms, and show that the two quantities are distinct finite-time observables within the same asymptotic growth class. Accordingly, 𝑉 𝐿 is interpreted as an endpoint observable with an asymptotically consistent growth rate under the stated assumptions, rather than as a universally superior growth-rate estimator. For finite-time classification, we adopt a horizon-stability criterion, and weak instabilities may remain unresolved over finite observation intervals. Mathieu-equation experiments compare the two endpoint quantities with Floquet stability information, while a four-dimensional coupled Mathieu system illustrates that, in the presence of multiple unstable modes, a single scalar 𝑉 𝐿 reflects the largest unstable rate that is excited and visible in the observed velocity rather than the complete Floquet spectrum.