After a brief training period devoted to solid state physics, more specifically magnetic phase transitions (experiments and mean field theory, [1]), I turned my attention to essentially theoretical work, first on arrays of 1D superconducting filaments coupled by Josephson effect [3], next on shear flow instabilities in nematic liquid crystals [4, 5, 6, 7, 8, 9].
Since 1978, I have been mostly interested in the transition to turbulence from a nonlinear dynamics perspective, the approach being different according to whether the system under consideration is ``confined'' or ``extended.'' When confined, physical systems can be reduced to dissipative dynamical systems with a small number of degrees of freedom and the concept of temporal chaos makes sense. In this framework, at the end of the 70's I worked in collaboration with Yves Pomeau on the routes to chaos via intermittency [10, 12, 13, 14, 15, 16]. In parallel, I began to study the phase instabilities occurring in extended systems as a first step towards spatio-temporal chaos [11, 17, 19]. This work culminated in the calculation of phase diffusion coefficients for Rayleigh-Bénard convection with no-slip boundary conditions [22, 23, 24].
Spatio-temporal chaos raises the problem of reduction and modelling of the dynamics of extended systems. When the transition to turbulence is essentially continuous (``supercritical'' in a wide sense), the reduction procedure generically leads to Ginzburg-Landau-like and further to Kuramoto-Sivashinsky-like equations. In this respect, I became deeply involved in the first determination of the Lyapunov spectrum characterizing the growth of spatio-temporal chaos in the Kuramoto-Sivashinsky equation [98], the determination of the bifurcation diagram in the Ginzburg-Landau equation [51], as well as the properties of its topological defects [42] and of its phase-turbulence regime [52].
Another facet of my work on extended systems relates to modelling issues. In particular, I derived the extension of the classical Swift-Hohenberg model of Rayleigh-Bénard convection by taking into account large scale flows that are crucial in explaining its non-relaxational behavior at low Prandtl numbers [29, 30]. In practice a specific transition mechanism called spatio-temporal intermittency, i.e. chaotic behavior localised in space and time, has been suggested by Pomeau, with features akin to critical phenomena in phase transitions (scaling exponents and universality). In order to study the statistical properties of very extended systems, at the so-called thermodynamic (infinite size) limit, one needs more drastically simplified models in terms of coupled map lattices and cellular automata. A large body of work, performed in collaboration with Hugues Chaté, has thus been devoted to the scaling behavior of spatio-temporal intermittency [35, 36, 37, 38, 39, 40] and to the resulting limits of the universality argument. Within the same modelling framework, the emergence of ``collective non trivial behaviour'' was identified [41, 43, 44, 45, 46]. This regime is characterized by a periodic or quasiperiodic time dependence at the thermodynamic limit, in contrast with the time-independent behavior expected from arguments imported from thermodynamics. Furthermore bifurcations between such regimes display critical properties different from those of the corresponding equilibrium phase transitions with the same symmetries [50, 55].
Most of the work described above has been done at the Service de Physique de l'État Condensé at the Saclay centre of the French Atomic Energy Commission (CEA). In 1994, I joined the newly created Laboratoire d'Hydrodynamique (LadHyX) of the École Polytechnique in Palaiseau, focussing my interests on problems of a more explicit hydrodynamical flavour, such as film flows down inclined planes or the transition to turbulence in subcritical shear flows, notably plane Couette flow. In the first case, the instability develops as a long wavelength supercritical bifurcation for which optimal models can be obtained by a combination of a gradient expansion and a Galerkin approximation, thereby yielding accurate results in a wide neighbourhood of the threshold [61, 62]. Future work on this subject will focus on secondary instabilities and the growth of spatio-temporal disorder involving solitary waves. The second topic, plane Couette flow, is interesting for complementary reasons: the flow is subcritically unstable against short-wavelength perturbations of finite amplitude. Apart from my participation in the interpretation of experiments conducted at Saclay [59, 60], my recent work on the subject has been devoted to the role of transient energy growth [57] and the derivation of a Swift-Hohenberg-like model able to capture the formation of streamwise streaks thought to be important in the transition to turbulence in wall-bounded flow [63]. Encouraging results have been obtained with this model for systems of intermediate size [120], which suggests using it for very wide systems presently out of reach of direct numerical simulations of the Navier-Stokes equations.
To summarize, I have been mostly involved in the study of nonlinear dynamics from a general point of view and in its concrete applications at the crossing of fluid mechanics and statistical physics.