2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach by Elhadj Zeraoulia, Julien Clinton Sprott

By Elhadj Zeraoulia, Julien Clinton Sprott

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Dm−1 and Dm . , Dm−1 and Dm . 10. , Dm−1 and Dm . Then there exists a compact invariant set ˜ such that f |K is semi-conjugate to m-shift dynamics. 8 was used in Sec. 2 to estimate the topological entropy of Chua’s circuit in term of half-Poincar´e maps. 10 This σ ◦ h. 6 World Scientific Book - 9in x 6in ws-book9x6 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach The Sil’nikov criterion for the existence of chaos Homoclinic and heteroclinic orbits arise in the study of bifurcations and chaos, as well as in their applications to mechanics, biomathematics, and chemistry [Aulbach and Flockerzi (1989), Balmforth (1995), Feng (1998)].

If in addition, ξ has a homoclinic orbit through the origin, then ξ can be infinitesimally perturbed into a nearby vector field ξ ′ with a countable set of horseshoes. Using this version, [Chua et al. 13 is satisfied. In particular, for proving the existence of a homoclinic orbit, they proved that there exist parameters such that the trajectory along the unstable real eigenvector associated with the origin enters the complex stable eigenspace, and hence returns to the origin as shown in Sec. 5.

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. 8. Again the inputs of this ANN are the feedback signals from the chaotic system outputs with time delay.

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