The first step in resolving the question of whether the observed dynamics are chaotic or not, is taken by reconstruction of the phase space or state space for the dynamics.
Specifically, under the conditions stated below, a map exists between the original phase space and the reconstructed phase space that is an embedding, i. e. the map is smooth, performs a one-to-one coordinate transformation and has a smooth inverse.
This means that the map preserves topological information about the dynamical system under the mapping, e.g. the dimension, the entropy and the Lyapunov exponents which are to be defined below. Phase space reconstruction was introduced as a tool in dynamical systems theory
by Packard et al. (1980), David Ruelle and Takens (1981) independently. The method was demonstrated numerically by Packard et al. (1980) and was formally proven by Takens (1981).
The most widely used method of phase space reconstruction is delay coordinates. According to this method, the past and the future of an observed scalar time series contain information about unobserved state variables that can be used to define a state at the present time.
Other methods of phase space reconstruction are derivative coordinates of which Packard et al. (1980) is an example, and principal value decomposition which was originally proposed by Broomhead and King (1986). Both methods use the information in delay coordinates as a starting point for a further transformation to a new coordinate system
(Casdagli et al., 1991).
Prof. Mikael Bask
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