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Extra info for 2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach
An example of how to apply an algorithm based on the Wiener model to the H´enon map is discussed in Sec. 4. April 27, 2010 14:29 42 World Scientific Book - 9in x 6in 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach Fig. 2 ws-book9x6 The three-layer feed-forward ANN. Algorithm based on the Hammerstein model The architecture of the Hammerstein model is shown in Fig. 9(b). Similar to the Wiener model, a simple linear plant is chosen to be completely controllable and to have the same dimension as the given chaotic system, and a simple three-layer feed-forward ANN is used to approximate the static nonlinearity, which is the same as that shown in Fig.
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. 7 is that the dynamics generated by the shift map σ are sensitive to initial conditions, and therefore it is chaotic in the commonly accepted sense. Now, let X be a separable metric space and f be a continuous map ˜ → X, where Q ˜ ⊂ X is locally connected and compact.
15 was used in [Marotto (1979b)] to prove that the H´enon map is chaotic as shown in Sec. 3. 8 World Scientific Book - 9in x 6in 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach The verified optimization technique The verified optimization technique was introduced in [Tibor et al. (2006)], where a new version of this method was employed with some sufficient conditions to find chaotic regions for the H´enon map [H´enon (1976)]. The method is as follows: (1) Check the set theoretical conditions of a respective theorem in a reliable way by computer programs.