By Elhadj Zeraoulia, Julien Clinton Sprott

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3) Prove the correctness of the underlying checking algorithms and the optimization model. The method given in [Zgliczynski (1997a)] works only with human interaction because of difficulties related to some mathematical relations. The actual method [Tibor et al. (2006)] finds chaotic regions automatically without any human assistance, where the techniques used are a combinations of interval arithmetic [Alefeld and Herzberger (1983), Neumaier (1990)] and adaptive branch-and-bound subdivision of the region of interest [Dellnitz and Junge (2002)].

Then there is morphism, and let Λ ˜ such that for every δ > 0 there is an ǫ > 0 for which a neighborhood U of Λ every ǫ-orbit in U is δ-shadowed by an orbit of f . Moreover, there is a δ 0 > 0 such that, if δ < δ 0 and if the pseudo-orbit is ˜ has a local product bi-infinite, then the shadowing orbit is unique, and if Λ ˜ structure, then the shadowing orbit is in Λ. Generally, the standard use of the shadowing lemma in dynamical systems theory is to prove density of periodic points, and because our interest is the study of the shadowing lemma in real systems, namely 2-D quadratic maps and the Chua systems as an example of ODE dynamics, then Ω = Rn , n = 2, 3, and M is a Riemannian manifold.

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)].