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By Elhadj Zeraoulia, Julien Clinton Sprott

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Extra info for 2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach

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F ) f (D) is equivalent to the shift map in the symbolic space. For more detail, see [Banks and Dragan (1994)]. These principal features give the idea that symbolic names can be given to all the orbits that remain in the neighborhood. , m of the orbits provide a symbolic representation of the dynamics, known as symbolic dynamics introduced in Sec. 3. 14. A dynamical system is chaotic in the sense of Smale if it has horseshoes of the Smale type. 14 with the so-called shadowing lemma [Stoffer and Palmer (1999)] was used to prove that the H´enon [H´enon (1976)] map is chaotic in the sense of Smale as shown in Sec.

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 (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. (b) The singular values of A are the positive square roots of the eigenvalues of AT A or AAT .

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