 On might try to get around these problems by invoking the standard physicist’s assumption that an infinite-time limit can be used to effectively represent some large but finite elapsed time. However, one reason to doubt this assumption in the context of chaos is that the calculation of finite-time Lyapunov exponents do not usually lead to on-average exponential growth as characterized by global Lyapunov exponents (e.g., Smith, Ziehmann and Fraedrich 1999). In general, for finite times the propagator varies from point to point in state space (i.e., it is a function of the position \(\bx(t)\) in state space and only approaches a constant in the infinite time limit), implying that the local finite-time Lyapunov exponents vary from point to point. Therefore, trajectories diverge and converge from each other at various rates as they evolve in time—the uncertainty does not vary uniformly in the chaotic region of state space (Smith, Ziehmann and Fraedrich 1999; Smith 2000). This is in contrast to global Lyapunov exponents which are on-average global measures of trajectory divergence and which imply that uncertainty grows uniformly (for \(\lambda \gt 0)\), but such uniform growth rarely occurs outside a few simple mathematical models. For instance, the Lorenz, Moore-Spiegel, Rössler, Henon and Ikeda attractors all possess regions dominated by decreasing uncertainties in time, where uncertainties associated with different trajectories issuing forth from some small neighborhood shrink for the amount of time trajectories remain within such regions (e.g., Smith, Ziehmann and Fraedrich 1999, pp. 2870–9; Ziehmann, Smith and Kurths 2000, pp. 273–83). Hence, on-average exponential growth in trajectory divergence is not guaranteed for chaotic dynamics. Linear stability analysis can indicate when nonlinearities can be expected to dominate the dynamics, and local finite-time Lyapunov exponents can indicate regions on an attractor where these nonlinearities will cause all uncertainties to decrease—cause trajectories to converge rather than diverge—so long as trajectories remain in those regions. 