A Note on Noisy Chaos

SUMMARY We prove that, under appropriate conditions, in a noisy environment an embedded deterministic dynamical system which admits a compact attractor can give rise to an ergodic stochastic system. This observation justifies the stochastic set-up in the study of deterministic chaos. We also clarify a folklore concerning polynomial autoregression. chaos and its relationship to statistical non-linear time series modelling. Loosely, a chaotic system may be defined as a dynamical system such that, starting from points over a certain region, all trajectories (solutions) are uniformly bounded but nearby trajectories initially diverge from each other exponentially and yet the trajec- tories have identical statistical properties. A trajectory obtained from a chaotic deterministic system thus looks like a realization from some random process. The theory of chaos casts the hope that an erratic time series could be fitted with some non-linear model so that highly accurate short-term forecasts may be available. The essential part of the theory of chaos consists of quantifying the chaoticness of a

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