Chaos is continuous exponential divergence — neighboring trajectories separate at a rate measured by the Lyapunov exponent, and the separation grows smoothly in time. This is the standard picture. The Lyapunov exponent averages the divergence rate over the trajectory, and the averaging produces a smooth exponential envelope.
Salasnich and Sattin (arXiv:2602.20682) look at the geometry beneath the average. Using the Jacobi-Levi-Civita equation to track trajectory separation in two low-dimensional Hamiltonian systems, they find that the divergence is not continuous. It is closer to a multiplicative discrete process. Separation increases abruptly at turning points — the moments when trajectories bounce off energetic boundaries. Between turning points, separation grows modestly or not at all. The chaos is concentrated in the scattering events.
Whether a trajectory is chaotic depends on the details of these boundary encounters. The turning point geometry — how the trajectory meets the potential boundary, at what angle, with what energy — determines whether that particular scatter amplifies or attenuates the divergence. Parametric resonance theory and the Mathieu equation connect these discrete scattering events to the emergence of sustained exponential growth.
The Lyapunov exponent is an average over many such events. It summarizes the net effect of discrete multiplicative kicks. The smoothness is in the summary, not in the process.
The general observation: a continuous-looking quantity can be the average of a discrete process. The exponential divergence that defines chaos is not continuous divergence at each moment — it is the cumulative effect of punctuated amplifications. Smooth macroscopic behavior can be the envelope of discontinuous microscopic events.