The Method of Close Returns

For a discretely sampled time series x, containing N points, calculate the phase space distance d between x(i) and x(i+p), where i =1,2,3,�N-p is the starting location in the time series and p is a lag between two points being compared.

A two-dimensional phase space can be reconstructed from one-dimensional time series data by making the reasonable assumption that x�=x(i)-x(i-1).

Result is a two-dimensional (i vs. p) grid of d-values. Close returns occur when d

Periodic signals produce horizontal lines of close returns, whereas chaotic flows can visit unstable periodic orbits for short times.

(See Mindlin & Gilmore, 1992; Boyd, Mindlin, Gilmore & Solari, 1994)

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