Representation theory for strange attractors

Use of False Nearest Neighbours for Selecting Variables and Embedding Parameters for State Space Reconstruction

If data are generated by a system with a d-dimensional attractor, then Takens’ theorem guarantees that reconstruction that is diffeomorphic to the original attractor can be built from the single time series in 2d+1-dimensional phase space. However, under certain conditions, reconstruction is possible even in a space of smaller dimension. This topic is very important because the size of the reconstruction space relates to the effectiveness of the whole subsequent analysis. In this paper, the false nearest neighbour (FNN) methods are revisited to estimate the optimum embedding parameters and the most appropriate observables for state space reconstruction. A modification of the false nearest neighbour method is introduced. The findings contribute to evidence that the length of the embedding time window (TW) is more important than the reconstruction delay time and the embedding dimension (ED) separately. Moreover, if several time series of the same system are observed, the choice of the one that is used for the reconstruction could also be critical. The results are demonstrated on two chaotic benchmark systems.

Dressed return maps distinguish chaotic mechanisms

Chaotic data generated by a three-dimensional dynamical system can be embedded into R(3) in a number of inequivalent ways. However, when lifted into R(5) they all become equivalent, indicating that they all belong to a single universality class sharing a common chaos-generating mechanism. We present a complete invariant determining this universality class and distinguishing attractors generated by distinct mechanisms. This invariant is easily computable from an appropriately "dressed" return map of any particular three-dimensional embedding.

Optimal state-space reconstruction using derivatives on projected manifold

A paradigm for optimal state-space reconstruction with nonuniform time delays is proposed. A comparison based on a diffeomorphic measure and a smoothness cost function shows that the proposed methodology achieves a better reconstruction compared to a reconstruction based on time delays that are multiples of the first minimum of mutual information. It is also shown how the proposed methodology is a more reliable approach to determining the embedding dimension.

Complete set of representations for dissipative chaotic three-dimensional dynamical systems

Embeddings are diffeomorphisms between some dynamical phase space and a reconstructed image. Different embeddings may or may not be equivalent under isotopy. We regard embeddings as representations of the dynamical phase space. We determine the topological labels required to distinguish inequivalent representations of three-dimensional dissipative dynamical systems when the embeddings are into R(k), k=3,4,5,…. Three representation labels are required for embeddings into R³, and only one is required in R⁴. In R⁵ there is a single "universal" representation.

Differential embedding of the Lorenz attractor

Ideally an embedding of an N -dimensional dynamical system is N -dimensional. Ideally, an embedding of a dynamical system with symmetry is symmetric. Ideally, the symmetry of the embedding is the same as the symmetry of the original system. This ideal often cannot be achieved. Differential embeddings of the Lorenz system, which possesses a twofold rotation symmetry, are not ideal. While the differential embedding technique happens to yield an embedding of the Lorenz attractor in three dimensions, it does not yield an embedding of the entire flow. An embedding of the flow requires at least four dimensions. The four dimensional embedding produces a flow restricted to a twisted three dimensional manifold in R4. This inversion symmetric three-manifold cannot be projected into any three dimensional Euclidean subspace without singularities.