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
36%. 1), and it was found that the average correlation dimension scales approximately as the square root of ws-book9x6 April 27, 2010 14:29 50 World Scientific Book - 9in x 6in ws-book9x6 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach the dimension of the system with a small variation. 1) with randomly chosen coefficients was performed using a simple computer program that produces an enormous collection of strange attractors. 1) were studied in detail in [H´enon (1969-1976), Aronson et al.
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. (c) A homoclinic point is a point x, different from a fixed point, that lies on both a stable manifold and an unstable manifold of the same fixed point Q. 19. (Poincar´e, 1890).
5 Part of a homoclinic tangency: Q is the hyperbolic fixed point, W s (Q) , W u (Q) are the stable and unstable manifolds, respectively, x is a homoclinic point, and thus f (x) is also a homoclinic point. Therefore, the number of homoclinic points on the corresponding invariant manifolds is infinite. 20. If f is a diffeomorphism and has a transversal homoclinic point, then there exists a Cantor set Λ ⊂ Rn in which f m is topologically equivalent to the shift automorphism for some m. The existence of the shift automorphism implies the existence of a dense set of periodic orbits and an uncountably infinite collection of asymptotically aperiodic points of the map f within the set Λ.
2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach by Elhadj Zeraoulia, Julien Clinton Sprott