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By Elhadj Zeraoulia, Julien Clinton Sprott

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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).

A) If the series {x (k)} is nearly periodic with a repeating periodic pattern having periodicity n, then the m × n matrix A has nearly rank one because σ 1 ≫ σ 2 . (b) If the series {x (k)} is truly random, then A has the full rank with the singular values having comparable magnitudes. In the case where {x (k)} is nearly periodic, the vector v1 gives the pattern over periodic segments, and the elements of u1 give the scaling factors for the successive periodic segments [Kanjilal (1995)]. April 27, 2010 14:29 World Scientific Book - 9in x 6in Tools for the rigorous proof of chaos and bifurcations ws-book9x6 45 Hence the distinguishing features between deterministic and nondeterministic processes are expected to be reflected in the distribution of the nonprime singular values.

11). 11). Note that any quadratic planar map with a constant determinant of the Jacobian matrix can be put into the form of the H´enon map [H´enon (1969-1976)] by an affine coordinate transformation.

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