Noise reduction in chaotic systems using unstable periodic orbits

In this thesis, a method is presented to approximate a noise-free low-dimensional continuous chaotic system (flow), using a single sample or multi-sampled scalar time series containing high levels of measurement noise or low levels of dynamical noise. The shadow-UPO noise reduction method (SUNR method) does not require the prior embedding of data and operates directly on the sampled time series, thus avoiding the limitations of Takens theorem and the estimation of embedding parameters when significant levels of noise are present. The method aims to overcome the well-documented severe limitations of directly filtering noise-infected chaotic time series, by focusing on nearly periodic orbit segments ('shadow-UPOs') shadowing the dense set of unstable periodic cycles (UPOs) that form the skeleton of a chaotic system, each of which is locally amenable to linear filtering techniques. The innovation is two-fold and comes from firstly deconstructing the chaotic system into approximate cycles, where we are free to directly apply signal processing techniques, based on the specific type of noise. Secondly, shadow-UPOs are detectable in the presence of high noise using the observation that histograms constructed from recurrence matrices are highly robust to noise. Shadow-UPOs are located, allocated to categorical bins, and filtered. We firstly utilise these to estimate the basis set of noise-free lower order UPOs, and estimate individual maximal Lyapunov exponents for each UPO. Secondly, we approximate the noise-free time series by replacing noise-infected near-cycles in the time series with their noise-filtered counterparts. The resultant time series are sufficiently noise-reduced that conventional algorithms can be used to subsequently estimate the Lyapunov exponents that would otherwise not be computable. The method is illustrated in detail as a case study of the Rossler system, tested for various types of noise (uniform white, Gaussian white, high-frequency, coloured and dynamical) and also on several chaotic systems with a range of differing topologies (Chua, Rabinovich-Fabrikant, Lu-Chen, Lorenz). Goodness of fit metrics are defined, measured for each system and presented. We identified limitations of the recurrence method of detecting cycles when dealing with higher instability systems, and successfully modified the SUNR method for these.