By Elhadj Zeraoulia, Julien Clinton Sprott
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Extra resources for 2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach
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D) Search for a hyperchaotic attractor. (e) Compare the behavior of Eq. 1) with the original Smale-Williams attractor. 22 x(i) here is a component of a map. , x (i + 1) versus x (i) which gives an attractor in the sense of dynamical systems. 14). 2. 2. 20) and that the image of N is contained in the set which can be continuously deformed to the horizontal strip without any intersection with the horizontal edges of N . 6(a). 7 using the geometrical formation of Smale’s horseshoe map f . 36). ˜0 and N ˜1 .
27. (a) The left and right singular vectors form a basis for the row-space and the column-space of A, respectively. (b) The singular values of A are the positive square roots of the eigenvalues of AT A or AAT . (c) The number of nonzero singular values gives the rank of the matrix A. The matrix configuration A given by Eq. 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.
3) April 27, 2010 14:29 52 World Scientific Book - 9in x 6in ws-book9x6 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach Fig. 1) [Sprott (1993b)]. then we say that f1 and g1 are affinely conjugate if there exists an affine transformation h such that g1 ◦ h (x, y) = h ◦ f1 (x, y) , for all (x, y) ∈ R2 . 4) April 27, 2010 14:29 World Scientific Book - 9in x 6in 2-D quadratic maps: The invertible case Fig. 1) [Sprott (1993b)]. 5) with the condition d = e1 l2 − e2 l1 = 0. 6) April 27, 2010 14:29 54 World Scientific Book - 9in x 6in 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach Fig.