By Elhadj Zeraoulia, Julien Clinton Sprott
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Additional info for 2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach
3. The most crucial role of Smale’s horseshoe is its relationship to the homoclinic tangency11 discovered by Poincar´e while investigating the threebody problem of celestial mechanics [Tufillaro et al. (1987)]. 22. (a) The invariant manifold of a map f is a set of points X such that f (X) ⊂ X 12 . (b) Given a fixed point Y0 in the invariant manifold, the manifold Y is called stable if ∀y ∈ Y , limn→∞ f n (y) → Y0 . Similarly, a manifold is called unstable if ∀y ∈ Y , limn→∞ f −n (y) → Y0 . (c) A fixed point is hyperbolic if it is the intersection of one or more stable manifolds and one or more unstable manifolds.
1) Choose a simple and completely controllable linear plant15 . (2) Choose a static nonlinear subsystem16 . 67) (4) Train the ANN by a model-reference adaptive control method using the framework shown in Fig. 9(a). 68) 15 This plant can be in the first canonical form, with the same dimension as the unknown chaotic system to be identified. If the dimension of the chaotic system is unknown, then it is possible to increase the dimension during the training process until satisfactory results are obtained.
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 . 28. The Mann–Whitney rank-sum statistic Z is normally distributed with zero mean and unit variance under the null hypothesis that the two observed samples came from the same distribution. 96 [Zar (1984)], we can reject the null hypothesis at the 95% confidence level. An example to how to apply the algorithm based on time series analysis to the H´enon map is given in Sec.