Differentiable generalized synchronization of chaos

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 55, Issue 4, 1997, Pages 4029-4034

  Hunt, Brian R ^a   Ott, Edward ^a   Yorke, James A ^a  

^a UNIVERSITY OF MARYLAND   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ASYMPTOTIC STABILITY; BIFURCATION (MATHEMATICS); DIFFERENTIATION (CALCULUS); EIGENVALUES AND EIGENFUNCTIONS; ELECTRONIC DENSITY OF STATES; FRACTALS; LYAPUNOV METHODS; STATISTICAL MECHANICS; SYNCHRONIZATION;

DIFFERENTIABLE GENERALIZED SYNCHRONIZATION; HOLDER EXPONENT; KAPLAN-YORKE FORMULA; LYAPUNOV NUMBER;

CHAOS THEORY;

EID: 0031122644     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.55.4029     Document Type: Article

References (27)

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17. R. J. Sacker [J. Math. Mech. 18, 705 (1969)] and N. Fenichel [Indiana Univ. Math. J.21, 193 (1971)] obtain Lyapunov exponent-based conditions analogous to (ii) for the persistence of smooth invariant manifolds under perturbation;
18. these results imply (ii) in the case where the drive attractor is a compact manifold and the response system is linear in y. M. W. Hirsch and C. C. Pugh [Proc. Symp. Pure Math.14, 133 (1970)] and M. W. Hirsch, C. C. Pugh, and M. Shub [Invariant Manifolds, Lecture Notes in Mathematics Vol. 583 (Springer-Verlag, New York, 1977)] give results like the lower bound on γ(x) in (i) and (ii), but requiring uniform (Lipschitz-like) bounds on the contraction rates, again assuming in the case of (ii) that the drive attractor is a compact manifold. More recently J. Stark has considered the case of general drive attractors and has independently obtained results like (i) and (ii) concerning the regularity of φ at 'almost every' point;
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26. For some values of α>0.2, Eq. (24) does not hold, and the information dimension of the attractor is not preserved, though Figs. 2(b) and 2(c) are unaffected.
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