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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
62) must be solved simultaneously. However, not all critical points are maxima or minima. For example, the function h(x) = x3 has critical point at x = 0, but it is neither a maximum nor a minimum, but rather it is an inflection point. Let (xc , yc ) be a critical point, and define ∂ 2 h (x, y) ∂ 2 h (x, y) (xc , yc ) (xc , yc ) − 2 ∂x ∂y 2 2 ∂ 2 h (x, y) (xc , yc ) . 25. (1) If dh (xc , yc ) > 0 and ∂ h(x,y) (xc , yc ) < 0, then h (x, y) ∂x2 has a relative maximum at (xc , yc ). 2 (2) If dh (xc , yc ) > 0 and ∂ h(x,y) (xc , yc ) > 0, then h (x, y) has a relative ∂x2 minimum at (xc , yc ).
21 M different values of row length n, and M sets of R singular values are obtained. April 27, 2010 14:29 46 World Scientific Book - 9in x 6in ws-book9x6 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach Fig. 9 (a) The learning mechanism of the Wiener model. (b) The learning mechanism of the Hammerstein model. probability distributions are equal. , χ (x) = 1 for x > 0, and χ (x) = 0 for x ≤ 0). For large N2 (which in practical terms means a few tens), the Mann-Whitney (M-W) rank-sum statistic is given by Z= W − 21 N1 N2 1 12 N1 N2 .
Si ∈ Σm . 15. The distance between s and s is defined as ∞ 1 |si − si | d (s, s) = . 6. The space Σm is compact, totally disconnected, and perfect. 6 is a Cantor set, which frequently appears in the characterization of the complex structure of an invariant set in a chaotic dynamical system. 7. (a) σ(Σm ) = Σm , and σ is continuous. (b) The shift map σ as a dynamical system defined on Σm has the following properties: (b-1) σ has a countable infinity of periodic orbits consisting of orbits of all periods; (b-2) σ has an uncountable infinity of nonperiodic orbits; and (b-3) σ has a dense orbit.
2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach by Elhadj Zeraoulia, Julien Clinton Sprott