Noise reduction: multiple solutions

It is an interesting property of chaotic systems that, given a knowledge of the underlying dynamics, even a series of quite crude or noisy observations is sufficient to allow the state-space trajectory of the system to be reconstructed to a very high level of accuracy-far higher accuracy than say a simple moving time-average. The success of chaotic noise reduction is due to the stretching properties of chaotic dynamics. Each observation may define rather a large cloud of compatible points in the state-space. However as time progresses this cloud evolves with the dynamics. For chaotic dynamics this implies an exponential stretching. Unstable directions in the cloud are exponentially extended; stable directions are exponentially contracted. Combining the information from past and future observations can thus substantially reduce uncertainty in the present. The noise reduction is especially powerful for deterministic dynamics. Dynamically stable directions then continue to contract indefinitely, until they have negligible width. The whole cloud from a past observation thus eventually evolves into an arbitrarily thin surface, known as the unstable manifold corresponding to just the expanding directions of its past dynamics. Noise reduction is compared to Kalman filtering