TY  - JOUR
A1  - Viana, R. L.
A1  - Grebogi, Celso
A1  - Pinto, S. E. D.
A1  - Lopes, S. R.
A1  - Batista, A. M.
A1  - Kurths, Jürgen
T1  - Bubbling bifurcation : loss of synchronization and shadowing breakdown in complex systems
N2  - Complex dynamical systems with many degrees of freedom may exhibit a wealth of collective phenomena related to high-dimensional chaos. This paper focuses on a lattice of coupled logistic maps to investigate the relationship between the loss of chaos synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems. In the neighborhood of the critical transition to strongly non-hyperbolic behavior, the system undergoes on-off intermittency with respect to the synchronization manifold. This has been confirmed by numerical diagnostics of synchronization and non-hyperbolic behavior, the latter using the statistical properties of finite-time Lyapunov exponents. (c) 2005 Elsevier B.V. All rights reserved
Y1  - 2005
ER  - 
TY  - JOUR
A1  - Feudel, Fred
A1  - Witt, Annette
A1  - Gellert, Marcus
A1  - Kurths, Jürgen
A1  - Grebogi, Celso
A1  - Sanjuan, Miguel Angel Fernandez
T1  - Intersections of stable and unstable manifolds : the skeleton of Lagrangian chaos
N2  - We study Hamiltonian chaos generated by the dynamics of passive tracers moving in a two-dimensional fluid flow and describe the complex structure formed in a chaotic layer that separates a vortex region from the shear flow. The stable and unstable manifolds of unstable periodic orbits are computed. It is shown that their intersections in the Poincare map as an invariant set of homoclinic points constitute the backbone of the chaotic layer. Special attention is paid to the finite time properties of the chaotic layer. In particular, finite time Lyapunov exponents are computed and a scaling law of the variance of their distribution is derived. Additionally, the box counting dimension as an effective dimension to characterize the fractal properties of the layer is estimated for different duration times of simulation. Its behavior in the asymptotic time limit is discussed. By computing the Lyapunov exponents and by applying methods of symbolic dynamics, the formation of the layer as a function of the external forcing strength, which in turn represents the perturbation of the originally integrable system, is characterized. In particular, it is shown that the capture of KAM tori by the layer has a remarkable influence on the averaged Lyapunov exponents. (C) 2004 Elsevier Ltd. All rights reserved
Y1  - 2005
ER  - 
TY  - JOUR
A1  - Baptista, Murilo da Silva
A1  - Kraut, Suso
A1  - Grebogi, Celso
T1  - Poincare recurrence and measure of hyperbolic and nonhyperbolic chaotic attractors
N2  - We study Poincare recurrence of chaotic attractors for regions of finite size. Contrary to the standard case, where the size of the recurrent regions tends to zero, the measure is no longer supported solely by unstable periodic orbits of finite length inside it, but also by other special recurrent trajectories, located outside that region. The presence of the latter leads to a deviation of the distribution of the Poincare first return times from a Poissonian. Consequently, by taking into account the contribution of these special recurrent trajectories, a corrected estimate of the measure is obtained. This has wide experimental implications, as in the laboratory all returns can exclusively be observed for regions of finite size, and only unstable periodic orbits of finite length can be detected
Y1  - 2005
SN  - 0031-9007
ER  -