Gérard Iooss

A simple global characterization for normal forms of singular vector fields

We 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, we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator. This leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form. Explicit examples, and results obtained when there are additional symmetries, are also presented.

Topics in bifurcation theory and applications

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.

Center Manifold Theory in Infinite Dimensions

Center manifold theory forms one of the cornerstones of the theory of dynamical systems. This is already true for finite-dimensional systems, but it holds a fortiori in the infinite-dimensional case. In its simplest form center manifold theory reduces the study of a system near a (non-hyperbolic) equilibrium point to that of an ordinary differential equation on a low-dimensional invariant center manifold. For finite-dimensional systems this means a (sometimes considerable) reduction of the dimension, leading to simpler calculations and a better geometric insight. When the starting point is an infinite-dimensional problem, such as a partial, a functional or an integro differential equation, then the reduction forms also a qualitative simplification. Indeed, most infinite-dimensional systems lack some of the nice properties which we use almost automatically in the case of finite-dimensional flows. For example, the initial value problem may not be well posed, or backward solutions may not exist; and one has to worry about the domains of operators or the regularity of solutions. Therefore the reduction to a finite-dimensional center manifold, when it is possible, forms a most welcome tool, since it allows us to recover the familiar and easy setting of an ordinary differential equation.

Travelling waves in the Fermi-Pasta-Ulam lattice

We consider travelling wave solutions on a one-dimensional lattice, corresponding to mass particles interacting nonlinearly with their nearest neighbour (the Fermi-Pasta-Ulam model). A constructive method is given, for obtaining all small bounded travelling waves for generic potentials, near the first critical value of the velocity. They all are given by solutions of a finite-dimensional reversible ordinary differential equation . In particular, near (above) the first critical velocity of the waves, we construct the solitary waves (localized waves with the basic state at infinity) whose global existence was proved by Friesecke and Wattis, using a variational approach. In addition, we find other travelling waves such as (a) a superposition of a periodic oscillation with a non-zero uniform stretching or compression between particles, (b) mainly localized waves which tend towards a uniformly stretched or compressed lattice at infinity, (c) heteroclinic solutions connecting a stretched pattern with a compressed one.

Water-Waves as a Spatial Dynamical System

Abstract The mathematical study of travelling waves, in the context of two-dimensional potential flows in one or several layers of perfect fluid(s) and in the presence of free surface and interfaces, can be formulated as an ill-posed evolution problem, where the horizontal space variable plays the role of “time”. In the finite depth case, the study of near equilibria waves reduces to a low-dimensional reversible ordinary differential equation. In most cases, it appears that the problem is a perturbation of an integrable system, where all types of solutions are known. We describe the method of study and review typical results. In addition, we study the infinite depth limit, which is indeed a case of physical interest. In such a case, the above reduction technique fails because the linearized operator possesses an essential spectrum filling the whole real axis, and new adapted tools are necessary. We also discuss the latest results on the existence of travelling waves in stratified fluids and on three-dimensional travelling waves, in the same spirit of reversible dynamical systems. Finally, we review the recent results on the classical two-dimensional standing wave problem.

Capillary-gravity solitary waves with damped oscillations

Abstract Capillary-gravity solitary waves with damped oscillations are studied analytically. The analysis follows the work of Iooss and Kirchgassner who proved that these waves exist for all values of the Froude number smaller than one. The water-wave problem is reduced to a system of ordinary differential equations by using the center manifold theorem. The normal form of this reduced system can be obtained and a good approximation to these waves for small amplitude is constructed. The limit as the water depth becomes infinite is considered as a special case. A comparison with existing numerical results is made for small-amplitude waves.

Primary and Secondary Bifurcations in the Couette-Taylor Problem

We study the preturbulent transitions for the Couette-Taylor flows via bifurcation theory in the presence of symmetry. The difficulty is that the linearized stability analysis leads to multiple eigenvalues for the most simple flows. Only a consideration of the symmetry-group action on the critical eigenvectors allows us to derive and to solve the bifurcation equations. We recover through this analysis the different patterns which are observed in experiments as the Reynolds number is increased: Steady Taylor vortices and bifurcation of either wavy, or twisted vortices from the Taylor vortex flow in the case of co-rotating cylinders; spiral vortices in the case of (strongly) counterrotating cylinders, and ribbon-cells, which have not yet been observed in experiments. Then we show that, under natural assumptions on the loss of stability of these oscillatory flows, the next bifurcation leads to quasi-periodic flows without frequency locking, whose different patterns are studied.

Bifurcating time-periodic solutions of Navier-Stokes equations in infinite cylinders

SummaryFor the problem of hydrodynamical stability in an infinite cylindrical domain, we investigate all time-periodic solutions, not only spatially periodic ones, when a Hopf bifurcation occurs. When reflection symmetry is present, we show the existence of spatially quasiperiodic flows. We also show the existence of heteroclinic solutions connecting two symmetrically traveling waves that stay at each end of the cylinders (“defect” solutions). The technique we use rests on (i) a center manifold argument in a space of time-periodic vector fields, (ii) symmetry and normal form arguments for the reduced ordinary differential equation in two dimensions (without reflection symmetry) or in four dimensions (with reflection symmetry), and (iii) the integrability of the associated normal form. It then remains to prove a persistence result when we add the higher-order terms of the vector field.

Localized waves in nonlinear oscillator chains

This paper reviews results about the existence of spatially localized waves in nonlinear chains of coupled oscillators, and provides new results for the Fermi-Pasta-Ulam (FPU) lattice. Localized solutions include solitary waves of permanent form and traveling breathers which appear time periodic in a system of reference moving at constant velocity. For FPU lattices we analyze the case when the breather period and the inverse velocity are commensurate. We employ a center manifold reduction method introduced by Iooss and Kirchgassner in the case of traveling waves, which reduces the problem locally to a finite dimensional reversible differential equation. The principal part of the reduced system is integrable and admits solutions homoclinic to quasi-periodic orbits if a hardening condition on the interaction potential is satisfied. These orbits correspond to approximate travelling breather solutions superposed on a quasi-periodic oscillatory tail. The problem of their persistence for the full system is still open in the general case. We solve this problem for an even potential if the breather period equals twice the inverse velocity, and prove in that case the existence of exact traveling breather solutions superposed on an exponentially small periodic tail.

Capillary Gravity Waves on the Free Surface of an Inviscid Fluid of Infinite Depth. Existence of Solitary Waves

Permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated. An application of the hodograph transform converts the free boundary-value problem into a boundary-value problem for the Cauchy-Riemann equations in the lower halfplane with nonlinear differential boundary conditions. This can be converted to an integro-differential equation with symbol −k2+4|k|−4(1+μ), where μ is a bifurcation parameter. A normal-form analysis is presented which shows that the boundary-value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for |μ| small. This normal form system has been studied thoroughly by several authors (Iooss &Kirchgässner [8],Iooss &Pérouème [10],Dias &Iooss [5]). It admits a pair of solitary-wave solutions which are reversible in the sense ofKirchgässner [11]. By applying a method introduced in [11], it is shown that this pair of reversible solitary waves persists for the boundary-value problem, and that the decay at infinity of these solitary waves is at least like 1/|x|.