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By Elhadj Zeraoulia, Julien Clinton Sprott

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April 27, 2010 14:29 44 World Scientific Book - 9in x 6in ws-book9x6 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach wide range to capture the embedded dynamics in {x (k)}. The matrix A with any dimension m × n is given by:   x (1) x (2) x (3) ... x (n)  x (n + 1) x (n + 2) x (n + 3) ... 74) A=  ... ... ...  x (1 + n (m − 1)) x (2 + n (m − 1)) ... 26. 75) where p = min (m; n), and σ 1 ≥ σ 2 ≥ .... ≥ σ p ≥ 0 are the singular values of A which are non-negative. U and V are the left and the right singular vector matrices, respectively.

X (n)  x (n + 1) x (n + 2) x (n + 3) ... 74) A=  ... ... ...  x (1 + n (m − 1)) x (2 + n (m − 1)) ... 26. 75) where p = min (m; n), and σ 1 ≥ σ 2 ≥ .... ≥ σ p ≥ 0 are the singular values of A which are non-negative. U and V are the left and the right singular vector matrices, respectively. The decomposed conjugate vector pairs u1 and v1 have related physical meaning [Good (1969), Golub and VanLoan (1989), Kanjilal (1995)]. 27. (a) The left and right singular vectors form a basis for the row-space and the column-space of A, respectively.

8. (a) If the series {x (k)} is nearly periodic with a repeating periodic pattern having periodicity n, then the m × n matrix A has nearly rank one because σ 1 ≫ σ 2 . (b) If the series {x (k)} is truly random, then A has the full rank with the singular values having comparable magnitudes. In the case where {x (k)} is nearly periodic, the vector v1 gives the pattern over periodic segments, and the elements of u1 give the scaling factors for the successive periodic segments [Kanjilal (1995)]. April 27, 2010 14:29 World Scientific Book - 9in x 6in Tools for the rigorous proof of chaos and bifurcations ws-book9x6 45 Hence the distinguishing features between deterministic and nondeterministic processes are expected to be reflected in the distribution of the nonprime singular values.

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