Intenational Summer School in Nonlinear Dynamics

Peyresq - 21-28 August 2015

School objectives

This thematic summer school aims at providing a multidisciplinary lecture program to enable the understanding, the study, and the development of research in the field of nonlinear dynamics. The focus is on describing the field from numerous viewpoints including physics, mathematics, mechanics, chemistry, biology, optics, electronics, signal processing, etc. The teaching is composed of four lectures: two fundamental courses and two specialised courses. Both fundamental courses are dedicated to basic notions of non-linear science. Additionnally experimental demonstrations allow to show, on practical examples, how the fundamental notions can be used in practice and how they are developed in a real physical system. Introductive small courses are organised in order to give a quick and clear vision on selected hot topics, as well as experimental illustrations. Beside the courses, many time slots are programmed for discussions on topics proposed by the participants, potentially useful for their own research (complements of courses, examples of calculus, interpretation of personal cases, theory, experiments), and also through seminars that participants are invited to present.


Lectures program

Fundamental courses

A brief introduction to dynamical systems and local bifurcations
Jens Rademacher
Dept. of Mathematics, Bremen, Germany

The lectures give a brief introduction to smooth dynamical systems and then focus on two broadly applicable mathematical methods of local bifurcation theory: Lyapunov-Schmidt and center manifold reduction. These allow to reduce bifurcation problems to usually much lower dimensions for further analysis, in particular by normal form transformations. Starting with the simplest examples in ordinary differential equations, some more complex applications to partial differential equation models from fluids and active media will be considered to highlight the power of these methods. While various mathematical notions and theorems will be explained and applied, only some ideas for proofs will be given during the lectures.

Spatio-temporal instabilities in hydrodynamics
Francois Charru
IMFT, Toulouse, France

Classical results on instabilities of open flows will be first presented, with emphasis on dimensional analysis and physical mechanisms: convective and absolute instabilities of open flows, temporal and spatial instabilities, inviscid instability of shear layers (Kelvin-Helmholtz), viscous instability of wall-bounded flows (Poiseuille flow, boundary layers). The successes and failures of basic approaches to account for the experimental observations will be discussed. More elaborated concepts will then be presented: weakly nonlinear developments (Ginzburg- Landau) and secondary instabilities, transient growth and streaks, weakly unstable nonlinear structures, intermittency.

Specialized courses

Spatiotemporal chaos
Paul Manneville
LadHyX, Ecole Polytechnique, France

We shall discuss the emergence of soft turbulence in systems with dimensions large when compared to intrinsic scales generated by instabilities. The key characteristic is the slow dynamics in time and space that results from the proximity of a bifurcation and the continuous symmetries, especially translational. An equally important factor is the continuous or discontinuous nature of the underlying bifurcation. The globally supercritical scenario occurring in the first case is amenable to analysis via multiple-scale expansions, introducing envelopes and phases, as natural frameworks for pattern formation and phase turbulence. The subcritical case implies the coexistence of separate stable states in both local phase space and physical space. In extended systems, it lends itself to spin-like reduction to be studied within the framework of statistical physics, hence an analysis in terms of phase transitions and critical phenomena of, e.g., spatiotemporal intermittency.

Machine Learning Control
Bernd Noack
Institut PPRIME, Poitiers, France & TU Braunschweig, Germany
Closed-loop turbulence control is an academically fascinating topic with potential engineering applications of epic proportion. The current paradigm of modelling and model-based control has been successful for linear dynamics of laminar flows, but is strongly challenged for nonlinear turbulent flows. We shall discuss a novel machine learning strategy for modelling and controlling turbulent flows in an automatic unsupervised manner. We show how optimal nonlinear control laws can be obtained from experimental measurements using genetic programming without any model of the system. The nonlinear mechanisms and control opportunities are identified by a clustering of snapshots and a Markov model for the transition probabilities between corresponding clusters. All presented approaches are data-driven and can be applied to model and to control any dynamical system and any experiments even with unknown equations of motion.

Discussions and short seminars

You are invited to bring any personal material (results from computations or experiments) which could be used for discussions or short seminars. A mid-day break is proposed for relaxation or treks from the village.

Introductory short courses

Yohann Duguet (LIMSI-CNRS, Orsay, France) Nonlinear models of shear flow transition

Björn Hof (Nonlinear Dynamics and Turbulence, IST Austria) How fully turbulent flow arises and how it can be controlled

Yuri Maistrenko (Laboratory of Mathematical Modeling of Nonlinear Processes, Kiev, Ukraine) Chimera states: a new paradigm in nonlinear science

Patrice Le Gal (IRPHE, Marseille, France) Rotating stratified flows: from lab to Jupiter red spot

Dates & pre-registration

Arrival in Peyresq Friday 21st August 2015 at ca. 8 pm
Beginning of the lecture Saturday 22nd August 2015 at 9 am
Departure from Peyresq Friday 28th August 2015 at about noon

Information & registration Please contact Stéphane METENS (stephane.metens (at)
for pre-registration Deadline: 15th June 2015

Registrations fees Inscription fees are 300 € for the stay, food and lectures. The fees must be paid at the registration step.

Traveling fees Doctoral schools may provide financial support to students.

Organisation committee

Organizing commitee Axelle AMON (IPR, Univ. Rennes 1)
Yanne CHEMBO (FEMTO-ST, UFC, Besançon)
Mariana HARAGUS (LMB, UFC, Besançon)
Laurent LARGER (FEMTO-ST, UFC, Besançon)
Stéphane METENS (MSC, Univ. Paris 7)
Luc PASTUR (LIMSI, Univ. Paris Sud 11)

Scientific committee/b> Yacine CHITOUR (L2S, Univ. Paris Sud 11)
Luca GRECO (L2S, Univ. Paris Sud 11)
Lionel MATHELIN (LIMSI, Univ. Paris Sud 11)
Jens RADEMACHER (Dept of Mathematics, Bremen, Germany)
Sami TLIBA (L2S, Univ. Paris Sud 11)

Useful numbers in Peyresq tel : 04 92 83 37 32
fax : 04 92 83 37 67

The place

A history

Peyresq Centre is held by the Nicolas-Claude Fabri de Peiresc Fondation .
Peyresq (not anymore Peiresc!) is a little village of Alpes de Haute Provence located 100km north of Nice and 75 km of Digne, in France.
The village was completely restaured, on a rock at 1528 m high the in pre-Alpes mountains. The village overlooks both Verdon and Vaïre valleys, in a preserved nature, favorable to hiking. Stone ('pierre' in French), the dominant element of the site, gave its name to the village. The village has been founded at the begining of the XIIIth century and it has preserved its old roman church. The place became a famous center for scientific meetings. It offers high quality accomodation, services and exceptional landscapes, promoting intellectual work, relaxation and reflexion.


Landscape contemplation, walks in the mountain... Ideal to gain in serenity and plenitude! To fully benefit from it, it is better to plan to come with mountain equipement, good shoes, etc. In altitude, at 1528 m, nights are cool, as well as days sometimes!


Shared rooms, in general with two beds, and in some cases 4 to 6 beds. Bedsheet, linen, towels and soap are provided by the Center.

How to get to Peyresq?

From Nice to Peyresq (and reciprocally)

A free shuttle is at disposal. It stops by the railway station (Nice-Ville) and by the airport (Terminal 1 and 2).
Details of the schedule will be communicated later to participants. A safe schedule is to plan your arrival in Nice the latest at 4pm on the first day of the school. The trip back to Nice from Peyresq on the last day, is around 2 to 3pm.

By the road