Reconstructing the state space of continuous time chaotic systems using power spectra

We introduce the concept of an envelope in which the high frequency region of the power spectrum of a continuous time system must lie. The position of the spectrum in relation to the envelope can help determine the nature of the solution whether it be periodic, chaotic, deterministic and noisy, or random. By fitting simple functions to the envelope, the sum of the positive Lyapunov exponents (the Kolmogorov-Sinai entropy) and a dimension estimate can be determined. Many different system types with a range of additive noise and sample sizes were studied to validate the proposal for choosing both the delay and the embedding dimension. The power spectrum therefore provides a fast, simple and reliable method for reconstructing the state space, even when the time series is noise corrupted and the number of samples is small.