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

By Elhadj Zeraoulia, Julien Clinton Sprott

ISBN-10: 9814307742

ISBN-13: 9789814307741

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Extra resources for 2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach

Example text

Let W s (Q) be the stable manifold and W u (Q) be the unstable manifold, and let x be the homoclinic point farthest from the fixed point Q along the unstable manifold W u (Q). Since W s (Q) and W u (Q) are invariant manifolds, f (x) ∈ W s (Q) ∩ W u (Q). Thus f (x) is either a homoclinic point or the fixed point Q. Since f takes a point on the unstable manifold W u (Q) away from the fixed point Q, then f (x) cannot be the fixed point. Thus it is a homoclinic point. For the same reason, it is not one of the n considered homoclinic points.

27. (a) The left and right singular vectors form a basis for the row-space and the column-space of A, respectively. (b) The singular values of A are the positive square roots of the eigenvalues of AT A or AAT . (c) The number of nonzero singular values gives the rank of the matrix A. The matrix configuration A given by Eq. 8. (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.

The decomposed conjugate vector pairs u1 and v1 have related physical meaning [Good (1969), Golub and VanLoan (1989), Kanjilal (1995)]. 27. (a) The left and right singular vectors form a basis for the row-space and the column-space of A, respectively. (b) The singular values of A are the positive square roots of the eigenvalues of AT A or AAT . (c) The number of nonzero singular values gives the rank of the matrix A. The matrix configuration A given by Eq. 8. (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 .

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


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