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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17. Assume that the underlying mapping Υ is given by an inclusion function T that has the zero convergence property, ǫ = 0, and that T (Q′ ) ⊂ O′ holds. Then the checking routine algorithm concludes after a finite number of iteration steps that the condition of chaotic behavior is fulfilled. 18. Assume that the underlying mapping Υ is given by an inclusion function T , ǫ = 0, and there exist a point x ∈ Q′ such that April 27, 2010 14:29 32 World Scientific Book - 9in x 6in 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach T (x) ∈ / O′ .

4) April 27, 2010 14:29 World Scientific Book - 9in x 6in 2-D quadratic maps: The invertible case Fig. 1) [Sprott (1993b)]. 5) with the condition d = e1 l2 − e2 l1 = 0. 6) April 27, 2010 14:29 54 World Scientific Book - 9in x 6in 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach Fig. 1) [Sprott (1993b)]. Note that if such a transformation h exists, then there is an equivalence relation and the set of all maps is divided into classes of topologically conjugate maps. This implies that f1 and g1 have identical topological properties, in particular, they have the same number of fixed and periodic points of the same stability types.

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. 14. 7 The Marotto theorem The Marotto theorem [Marotto (1978)], is best suited for predicting and analyzing discrete chaos in higher-dimensional difference equations because in practice, the homoclinic orbit or other techniques used for predicting chaos in dynamical systems are extremely difficult to compute, whereas the snap-back repellors are relatively easy, often needing only a small number of iterations.

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