TY - JOUR A1 - Letellier, Christophe A1 - Abraham, Ralph A1 - Shepelyansky, Dima L. A1 - Rossler, Otto E. A1 - Holmes, Philip A1 - Lozi, Rene A1 - Glass, Leon A1 - Pikovsky, Arkady A1 - Olsen, Lars F. A1 - Tsuda, Ichiro A1 - Grebogi, Celso A1 - Parlitz, Ulrich A1 - Gilmore, Robert A1 - Pecora, Louis M. A1 - Carroll, Thomas L. T1 - Some elements for a history of the dynamical systems theory JF - Chaos : an interdisciplinary journal of nonlinear science N2 - Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to "reconstruct" some supposed influences. In the 1970s, a new way of performing science under the name "chaos" emerged, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements-which were never published-illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed. Y1 - 2021 U6 - https://doi.org/10.1063/5.0047851 SN - 1054-1500 SN - 1089-7682 VL - 31 IS - 5 PB - AIP Publishing CY - Melville ER - TY - JOUR A1 - Baptista, Murilo da Silva A1 - Grebogi, Celso A1 - Koberle, Roland T1 - Dynamically multilayered visual system of the multifractal fly JF - Physical review letters Y1 - 2006 U6 - https://doi.org/10.1103/PhysRevLett.97.178102 SN - 0031-9007 VL - 97 IS - 17 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Hunt, Brain R. A1 - Grebogi, Celso A1 - Barreto, Ernest A1 - Yorke, James A. T1 - From high dimensional chaos to stable periodic orbits : the structure of parameter space Y1 - 1997 ER - TY - JOUR A1 - Sauer, T. A1 - Grebogi, Celso A1 - Yorke, J. A. T1 - How long do numerical chaotic solutions remain valid? Y1 - 1997 ER - TY - JOUR A1 - Rosa, Epaminondas A1 - Hayes, S. A1 - Grebogi, Celso T1 - Noise filtering in communication with chaos Y1 - 1997 ER - TY - JOUR A1 - Lai, Ying Cheng A1 - Nagai, Y. A1 - Grebogi, Celso T1 - Characterization of the natural measure by unstable periodic orbits in chaotic attractors Y1 - 1997 ER - TY - JOUR A1 - Grebogi, Celso A1 - Lai, Ying Cheng A1 - Hayes, S. T1 - Control and applications of chaos Y1 - 1997 SN - 0016-0032 ER - TY - JOUR A1 - Grebogi, Celso A1 - Lai, Ying Cheng T1 - Controlling chaotic dynamical systems Y1 - 1997 ER - TY - JOUR A1 - Grebogi, Celso A1 - Lai, Ying Cheng T1 - Controlling chaos in high dimensions Y1 - 1997 ER - TY - JOUR A1 - Feudel, Ulrike A1 - Grebogi, Celso A1 - Ott, E. T1 - Phase-locking in quasiperiodically forced systems Y1 - 1997 ER - TY - JOUR A1 - Feudel, Ulrike A1 - Grebogi, Celso T1 - Multistability and the control of complexity Y1 - 1997 SN - 1054-1500 ER - TY - JOUR A1 - Bolt, Eric A1 - Lai, Ying Cheng A1 - Grebogi, Celso T1 - Coding, channel capacity and noise resistance in communication with chaos Y1 - 1997 ER - TY - JOUR A1 - Lai, Ying Cheng A1 - Grebogi, Celso A1 - Feudel, Ulrike A1 - Witt, Annette T1 - Basin bifurcation in quasiperiodically forced systems Y1 - 1998 ER - TY - JOUR A1 - Poon, L. A1 - Grebogi, Celso A1 - Feudel, Ulrike A1 - Yorke, J. A. T1 - Dynamical properties of a simple mechanical system with a large number of coexisting periodic attractors Y1 - 1998 ER - TY - JOUR A1 - Karolyi, György A1 - Pentek, Aron A1 - Toroczkai, Zoltán A1 - Tél, Tómas A1 - Grebogi, Celso T1 - Advection of active particles in open chaotic flows Y1 - 1998 SN - 0031-9007 ER - TY - JOUR A1 - Schwarz, Udo A1 - Spahn, Frank A1 - Grebogi, Celso A1 - Kurths, Jürgen A1 - Petzschmann, Olaf T1 - Length scales of clustering in granular gases Y1 - 1999 ER - TY - JOUR A1 - Lai, Ying Cheng A1 - Grebogi, Celso A1 - Kurths, Jürgen T1 - Modeling of deterministic chaotic systems Y1 - 1999 ER - TY - JOUR A1 - Kraut, Suso A1 - Feudel, Ulrike A1 - Grebogi, Celso T1 - Preference of attractors in noisy multistable systems Y1 - 1999 ER - TY - JOUR A1 - Viana, R. L. A1 - Grebogi, Celso A1 - Pinto, Seds A1 - Lopes, S. R. A1 - Batista, A. M. A1 - Kurths, Jürgen T1 - Validity of numerical trajectories in the synchronization transition of complex systems N2 - We investigate the relationship between the loss of synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems. In the neighborhood of the critical transition to strongly nonhyperbolic behavior, the system undergoes on-off intermittency with respect to the synchronization state. There are potentially severe consequences of these facts on the validity of the computer-generated trajectories obtained from dynamical systems whose synchronization manifolds share the same nonhyperbolic properties Y1 - 2003 SN - 1063-651X ER - TY - JOUR A1 - Viana, Ricardo L. A1 - Barbosa, José R. R. A1 - Grebogi, Celso T1 - Unstable dimension variability and codimension-one bifurcations of two-dimensional maps N2 - Unstable dimension variability is a mechanism whereby an invariant set of a dynamical system, like a chaotic attractor or a strange saddle, loses hyperbolicity in a severe way, with serious consequences on the shadowability properties of numerically generated trajectories. In dynamical systems possessing a variable parameter, this phenomenon can be triggered by the bifurcation of an unstable periodic orbit. This Letter aims at discussing the possible types of codimension-one bifurcations leading to unstable dimension variability in a two-dimensional map, presenting illustrative examples and displaying numerical evidences of this fact by computing finite-time Lyapunov exponents. (C) 2004 Elsevier B.V. All rights reserved Y1 - 2004 SN - 0375-9601 ER - TY - JOUR A1 - Karolyi, G. A1 - Tel, Tomas A1 - de Moura, A. P. S. A1 - Grebogi, Celso T1 - Reactive particles in random flows N2 - We study the dynamics of chemically or biologically active particles advected by open flows of chaotic time dependence, which can be modeled by a random time dependence of the parameters on a stroboscopic map. We develop a general theory for reactions in such random flows, and derive the reaction equation for this case. We show that there is a singular enhancement of the reaction in random flows, and this enhancement is increased as compared to the nonrandom case. We verify our theory in a model flow generated by four point vortices moving chaotically Y1 - 2004 SN - 0031-9007 ER - TY - JOUR A1 - De Freitas, M. S. T. A1 - Viana, R. L. A1 - Grebogi, Celso T1 - Basins of attraction of periodic oscillations in suspension bridges N2 - We consider the dynamics of the lowest order transversal vibration mode of a suspension bridge, for which the hangers are treated as one-sided springs, according to the model of Lazer and McKeena [SIAM Review 58, 1990, 537]. We analyze in particular the multi-stability of periodic attractors and the basin of attraction structure in phase space and its dependence with the model parameters. The parameter values used in numerical simulations have been estimated from a number of bridges built in the United States and in the United Kingdom, thus taking into account realistic, yet sometimes simplified, structural, aerodynamical, and physical considerations Y1 - 2004 SN - 0924-090X ER - 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 -