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Gérard Iooss

Bio: Gérard Iooss is an academic researcher from Centre national de la recherche scientifique. The author has contributed to research in topics: Bifurcation & Hopf bifurcation. The author has an hindex of 36, co-authored 119 publications receiving 6342 citations. Previous affiliations of Gérard Iooss include Institut Universitaire de France & University of Nice Sophia Antipolis.


Papers
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Book
01 Jan 1980
TL;DR: Asymptotic solutions of evolution problems bifurcation and stability of steady solution of evolution equations in one dimension imperfection theory and isolated solutions which perturb bifurbcation stability of stable solutions in two dimensions and n dimensions appendices as discussed by the authors.
Abstract: Asymptotic solutions of evolution problems bifurcation and stability of steady solutions of evolution equations in one dimension imperfection theory and isolated solutions which perturb bifurcation stability of steady solutions of evolution equations in two dimensions and n dimensions appendices - bifurcation of steady solution in two dimensions and the stability of the bifurcating solutions appendix - methods of projection for general problems of bifurcation into steady solutions bifurcation of periodic solutions from steady ones (Hopf Bifurcation) in two dimensions bifurcation of periodic solutions in the general case subharmonic bifurcation of forced T-periodic solutions subharmonic bifurcation of forced T-periodic solutions into asymptotically quasi-periodic solutions appendix - secondary subharmonic and symptotically quasi-periodic bifurcation of periodic solutions (of Hopf's type) in the autonomous case stability and bifurcation in conservative systems.

1,180 citations

Journal ArticleDOI
TL;DR: In this article, the authors derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems, using an appropriate scalar product in the space of homogeneous vector polynomials, and show that the resonant terms commute with the group generated by the original critical linear operator.

433 citations

Book ChapterDOI
01 May 1992

354 citations

Book
01 Jan 1994
TL;DR: In this article, a unified approach to the non-linear stability problem and transitions in the Couette-Taylor problem is presented by means of analytic and constructive methods, which can be applied to other hydrodynamical instabilities, or more generally to physical problems modelled by partial differential equations.
Abstract: This monograph presents a systematic and unified approach to the non-linear stability problem and transitions in the Couette-Taylor problem, by the means of analytic and constructive methods. The most "elementary" one-parameter theory is first presented. More complex situations are then analyzed (mode interactions, imperfections, non-spatially periodic patterns). The whole analysis is based on the mathematically rigorous theory of centre manifold and normal forms, and symmetries are fully taken into account. These methods are very general and can be applied to other hydrodynamical instabilities, or more generally to physical problems modelled by partial differential equations.

353 citations

Book
25 Jan 1999
TL;DR: In this article, Couette flow bifurcations from Taylor vortex flow center manifolds, normal forms, and bifurlcations of vector fields near closed orbits are adapted to the unperturbed case perturbed case.
Abstract: Centre manifolds normal forms, and bifurcations of vector fields near critical points - unperturbed vector fields perturbed vector fields Couette-Taylor problem - formulation of the problem Couette flow bifurcations from Couette flow bifurcations from Taylor vortex flow centre manifolds, normal forms, and bifurcations of vector fields near closed orbits - preliminaries adaptation of Floquet thoery unperturbed case perturbed case.

309 citations


Cited by
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Journal ArticleDOI
TL;DR: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented in this article, with emphasis on comparisons between theory and quantitative experiments, and a classification of patterns in terms of the characteristic wave vector q 0 and frequency ω 0 of the instability.
Abstract: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

6,145 citations

Book
01 Jan 1996
TL;DR: In this article, the authors present a review of rigor properties of low-dimensional models and their applications in the field of fluid mechanics. But they do not consider the effects of random perturbation on models.
Abstract: Preface Part I. Turbulence: 1. Introduction 2. Coherent structures 3. Proper orthogonal decomposition 4. Galerkin projection Part II. Dynamical Systems: 5. Qualitative theory 6. Symmetry 7. One-dimensional 'turbulence' 8. Randomly perturbed systems Part III. 9. Low-dimensional Models: 10. Behaviour of the models Part IV. Other Applications and Related Work: 11. Some other fluid problems 12. Review: prospects for rigor Bibliography.

2,920 citations

Journal ArticleDOI
TL;DR: In this article, simple dissipative dynamical systems exhibiting a transition from a stable periodic behavior to a chaotic one were studied, where the inverse coherence time grows continuously from zero to zero due to the random occurrence of widely separated bursts in the time record.
Abstract: We study some simple dissipative dynamical systems exhibiting a transition from a stable periodic behavior to a chaotic one. At that transition, the inverse coherence time grows continuously from zero due to the random occurrence of widely separated bursts in the time record.

1,753 citations

Journal ArticleDOI
TL;DR: In this article, a new conceptual model for ENSO has been constructed based upon the positive feedback of tropical ocean atmosphere interaction proposed by Bjerknes as the growth mechanism and the recharge discharge of the equatorial heat content as the phase transition mechanism suggested by Cane and Zebiak and by Wyrtki.
Abstract: A new conceptual model for ENSO has been constructed based upon the positive feedback of tropical ocean‐ atmosphere interaction proposed by Bjerknes as the growth mechanism and the recharge‐discharge of the equatorial heat content as the phase-transition mechanism suggested by Cane and Zebiak and by Wyrtki. This model combines SST dynamics and ocean adjustment dynamics into a coupled basinwide recharge oscillator that relies on the nonequilibrium between the zonal mean equatorial thermocline depth and wind stress. Over a wide range of the relative coupling coefficient, this recharge oscillator can be either self-excited or stochastically sustained. Its period is robust in the range of 3‐5 years. This recharge oscillator model clearly depicts the slow physics of ENSO and also embodies the delayed oscillator (Schopf and Suarez; Battisti and Hirst) without requiring an explicit wave delay. It can also be viewed as a mixed SST‐ocean dynamics oscillator due to the fact that it arises from the merging of two uncoupled modes, a decaying SST mode and a basinwide ocean adjustment mode, through the tropical ocean‐atmosphere coupling. The basic characteristics of this recharge oscillator, including the relationship between the equatorial western Pacific thermocline depth and the eastern Pacific SST anomalies, are in agreement with those of ENSO variability in the observations and simulations with the Zebiak‐Cane model.

1,322 citations

Journal ArticleDOI
TL;DR: In this paper, a flow visualization and spectral studies of flow between concentric independently rotating cylinders have revealed a surprisingly large variety of different flow states, including Taylor vortices, wavy vortice, modulated wavy vectors, outflow boundaries and internal waves.
Abstract: Our flow-visualization and spectral studies of flow between concentric independently rotating cylinders have revealed a surprisingly large variety of different flow states. (The system studied has radius ratio 0.883, aspect ratios ranging from 20 to 48, and the end boundaries were attached to the outer cylinder.) Different states were distinguished by their symmetry under rotation and reflection, by their azimuthal and axial wavenumbers, and by the rotation frequencies of the azimuthal travelling waves. Transitions between states were determined as functions of the inner- and outer-cylinder Reynolds numbers, Ri and Ro, respectively. The transitions were located by fixing Ro and slowly increasing Ri. Observed states include Taylor vortices, wavy vortices, modulated wavy vortices, vortices with wavy outflow boundaries, vortices with wavy inflow boundaries, vortices with flat boundaries and internal waves (twists), laminar spirals, interpenetrating spirals, waves on interpenetrating spirals, spiral turbulence, a flow with intermittent turbulent spots, turbulent Taylor vortices, a turbulent flow with no large-scale features, and various combinations of these flows. Some of these flow states have not been previously described, and even for those states that were previously described the present work provides the first coherent characterization of the states and the transitions between them. These flow states are all stable to small perturbations, and the transition boundaries between the states are reproducible. These observations can serve as a challenge and test for future analytic and numerical studies, and the map of the transitions provides several possible codimension-2 bifurcations that warrant further study.

1,076 citations