For how long time evolution of chaotic or random systems can be predicted
Leonid Bunimovich, Kirill Kovalenko
TL;DR
The paper addresses finite-time predictability in ergodic, open dynamical systems by studying first-hitting probabilities $h_w(n)$ associated with holes encoded by words in a finite Markov partition. It develops a generating-function framework $GF_w(x)=\frac{1}{1+(x-q)A_w(x)}$ with inverse autocorrelation polynomials $A_w(x)$, and analyzes the full set of roots of the recurrence polynomial $P_w(x)$ to isolate a single dominant root $\alpha_m$ near $q$. The main result shows that the time interval during which finite-time predictions are reliable grows exponentially with partition refinement level $k$, and it provides precise asymptotics for the dominant exponent, its coefficient, and the intersection time $n^*$ between different words’ first-hitting curves. The work combines root-bounding (Rouché), Faà di Bruno’s formulas for coefficients, and careful asymptotics to prove that the exponential growth is robust and that the intersection time scales as $n^* = C_{A_w/A_{w'}}\cdot q A_w(q)\,(1+O(k q^{-k/3+4}))$, with $C_{A_w/A_{w'}}$ bounded between $\frac{\ln q}{q}$ and $1$. These results quantify the extent of finite-time predictability in the most chaotic-like systems and reveal a sharp exponential-in-$k$ growth of the predictive window as the partition becomes finer.
Abstract
Traditionally, Probability theory was dealing with limit theorems where 'limit" means that time tends to infinity. Questions about finite time dynamics (evolution) were always considered as, although important for practical applications, but untreatable rigorously (mathematically). The same attitude was in the theory of strongly chaotic dynamical systems, which evolve similarly to stochastic processes. However, a natural question on dependence of the process of escape on a position of a "hole" in the state (phase) space, which was never asked in mathematical theory of open dynamical systems, opened up a new direction of research, which was dealing with finite time predictions of evolutions of such systems. It turned out, that transport of orbits in the phase space of the "most strongly chaotic" dynamical systems has three different stages. In the first stage there is a hierarchy of the first hitting probabilities, that shows which parts of the phase space the orbits of a system, which is an equilibrium state, will be more likely to visit the first. A principal (and the most important for applications) question was how the length of this interval changes with more refinement observations of the positions of the orbits in the phase space. Surprisingly, it turned out that the length of the time interval, where finite time predictions are possible, increases (rather to be shrinking), which, at the first sight, seems to be natural. However, this increase of the length of the time interval, where finite time predictions are possible, was rather slow (just linear) with respect to the growth of precision (partition of the phase space) of observations. In the present paper it is proved (by totally different technique) that this growth is actually exponential.