Graduate School in Nonlinear Science

Sponsored by the Danish Research Academy

MIDIT                               OFD                           CATS
Modelling, Nonlinear Dynamics       Optics and Fluid Dynamics     Chaos and Turbulence Studies
and Irreversible Thermodynamics     Risø National Laboratory      Niels Bohr Institute and 
Technical University of Denmark     Building 128                  Department of Chemistry
Building 321                        P.O. Box 49                   University of Copenhagen 
DK-2800 Lyngby                      DK-4000 Roskilde              DK-2100 Copenhagen Ø
Denmark                             Denmark                       Denmark


The Center for Modelling, Nonlinear Dynamics and Irreversible Thermodynamics (MIDIT) at the Technical University of Denmark (DTU), the Department of Optics and Fluid Dynamics (OFD) at Risø National Laboratory (Risø), and the Center of Chaos and Turbulence Studies (CATS) at Copenhagen University (KU) conduct a Graduate School in Nonlinear Science. In collaboration with industry, this school provides a Danish Ph.D. program at the highest international level with the aim of preparing students to apply recent advances in Nonlinear Science to outstanding problems of science and technology and to strengthen international exchange of Ph.D. students.
Over the past three decades, science has experienced a revolutionary shift in its fundamental paradigms. Primarily based on linear models through the 1960s, scientific research is now commonly motivated by nonlinear concepts, in which the whole is more than the sum of its parts, and the emergence of qualitatively new phenomena is anticipated and made welcome. The basic theme of modern Nonlinear Science is the interplay between chaos and coherent structures. Formerly deemed unworthy of the attention of a serious scientist, low order systems of nonlinear ordinary differential equations are now known to exhibit explosive behavior, leading to the emergence of strange attractors upon which phase space trajectories wander aimlessly until the end of time. Largely ignored as being far too difficult to solve analytically, nonlinear partial differential equations have been found to generate the emergence of solitary waves, which interact as new dynamic entities at higher levels of description. These new paradigms lead to significant advances in our understanding of a number of observed phenomena in physics, chemistry, and biology. From the theory, methods for constructive applications in the engineering sciences have been developed, using computer technology.
In the 1980's a number of centers for Nonlinear Science were established at universities and research institutions in U.S.A., the European countries, and in Japan. In Denmark, MIDIT was formed at DTU in 1985 and CATS at KU in 1993. In close collaboration with other Danish universities and research institutions - e.g. OFD at RISØ - research as well as Ph.D. and Master's Degree education were initiated at a very early stage. Jointly, MIDIT, OFD, and CATS form the major part of the national activity within Nonlinear Science. The early and persistent effort has given Denmark a central placing and recognition within the international Nonlinear Science community.
It is in the spirit of this growing activity that the Graduate School will operate. Although seemingly diverse at first reading, the facets of this activity are, in fact, closely interrelated components of modern research in nonlinear dynamics including scientific computing and experimental research.


The Graduate School in Nonlinear Science was established in 1997 by The Danish Research Academy as a collaboration between MIDIT, OFD and CATS.
MIDIT was established at DTU in 1985. Today scientists from the DTU departments of Mathematics, Mathematical Modelling, Physics, Chemistry, Chemical and Energy Engineering collaborate within the Centre.
The scientific background for MIDIT is the decisive break-throughs that have occurred in the last three decades in nonlinear dynamics and far-from-equilibrium statistical physics, combined with the possibilities offered by modern computer technology for detailed modelling and simulation of complex phenomena.
The applications of the new tools from Nonlinear Science within the boundary areas between mathematics and physics on one hand and chemistry, technological sciences, biology and economy on the other are manifold. MIDIT stimulates research and engineering education within Nonlinear Science and its applications.
Chairman: Erik Mosekilde. Scientific Staff: 27
OFD at Risø performs research and development within systems, structures and materials for optical measurements and information processing based on diffractive optics and nonlinear dynamics. OFD hosts close interaction between theory and experiments.
Core competences within physical optics (electromagnetic interaction with matter, materials with special optical properties), nonlinear continuum dynamics (self-organization, pattern formation, parametric oscillations and amplification, soliton dynamics, collapse and turbulence in fluids and plasmas), scientific computing and information processing.
Head of Department: Lars Lading. Scientific staff: 25
CATS was established in 1993 by the Danish Natural Science Research Council (SNF). The Center consists of groups from the Niels Bohr Institute/Nordita (NBI), Department of Chemistry, H. C. Ørsted Institute (HCØ), KU, and Department of Physics, DTU.
The scientific background for CATS is the fundamental insight that the simplest laws of nature can lead to bewildering complex dynamics, and that at the same time such dynamics exhibit universal features largely independent of the details of the particular underlying dynamical system.
The Center pursues theoretical and experimental research in chaos, turbulence and related problems in physics, chemistry and biology. The Center brings together the groups which are actively conducting research and education in Nonlinear Science.
Head of Center: Predrag Cvitanovic'. Scientific Staff: 20
Head of the Graduate School in Nonlinear Science is professor, dr. techn. Peter L. Christiansen, Department of Mathematical Modelling, DTU.
Christiansen holds Master of Science, Ph. D. and dr. techn. degrees from DTU. Since 1966 he has worked at DTU and served as Head of the Laboratory of Applied Mathematical Physics 1977-79 and 81-89. Since its beginning in 1985 he has been coordinating secretary for the DTU Center MIDIT. He has been a research associate and a visiting professor at the Universities of Michigan, New York (Courant Institute of Mathematical Sciences), Minnesota, Arizona, and Naples and at Los Alamos National Laboratory. He is author and co-author of more than 130 papers in scientific journals and is presently on the editorial board of Physica Scripta. He has taught many courses in Applied Mathematics and Nonlinear Science and supervised more than 30 Ph. D. students.
The Graduate School in Nonlinear Science hosts professional activity at DTU and KU, under the auspices of the directors of Ph.D. studies, the Ph.D. study councils, and program committees. in accordance with the University Law. The head is in charge of the daily running of the school, assisted by the executive committee of the steering committee of the Graduate School. The steering committee assists the head in the direction of the school, the planning and implementation of the activities and the contacts with foreign scientists, institutions and Ph.D. students. Members of the steering committee are: (Executive committee members are indicated by *))

  • Morten Brøns, MIDIT
  • Morten Colding-Jørgensen,
  • Sten Bay Jørgensen, MIDIT
  • Alwyn C. Scott, MIDIT
  • Tomas Bohr, MIDIT
  • Erik Mosekilde*), MIDIT
  • Jens Juul Rasmussen*), OFD
  • Peter L. Christiansen*),
  • Mogens Høgh Jensen, NBI
  • Ole Mouritsen, MIDIT
  • Preben Graae Sørensen*), HCØ
  • Per Michael Johansen, OFD
  • Jesper Mygind, MIDIT
  • Jens Nørkær Sørensen, MIDIT

    36 Danish professors participate in the Graduate School in Nonlinear Science.

  • Jørn Bindslev Hansen
  • Poul Scheel Larsen
  • Mogens Samuelsen
  • Tomas Bohr
  • Vagn Lundsgaard Hansen
  • Erik Mosekilde
  • Alwyn C. Scott
  • Bodil Branner
  • Poul Hjorth
  • Ole Mouritsen
  • Jens Nørkær Sørensen
  • Morten Brøns
  • Claus Schelde Jacobsen
  • Jesper Mygind
  • Mads Peter Sørensen
  • Peter L. Christiansen
  • Sten Bay Jørgensen
  • Niels Falsig Pedersen
  • Hans True
  • Poul Michael Petersen
  • CATS, NBI and HCØ
  • Finn Hynne
  • Per Michael Johansen
  • P. S. Ramanujam
  • Preben Alstrøm
  • Mogens Høgh Jensen
  • Jens-Peter Lynov
  • Jens Juul Rasmussen
  • Predrag Cvitanovic
  • Mogens T. Levinsen
  • Poul Michelsen
  • Mark Saffman
  • Clive Ellegaard
  • Preben Graae Sørensen
  • Anders H. Nielsen
  • Jørgen Schou
  • Axel Hunding
  • CATS, MIDIT and OFD collaborate within Framework Programs sponsored by SNF and participate in EU-, NATO-, and ESF-sponsored international programs and networks.
    The Graduate School in Nonlinear Science has collaboration agreements with important research institutions and universities which send their leading scientists and Ph.D. students to attend the school.
    MIDIT, CATS, and OFD collaborate with industrial companies in co-sponsored Ph.D.projects.


    PhD Courses and Workshops

    Core Courses for the Graduate School in Nonlinear Science

    Guest Professors


    To enter the Graduate School in Nonlinear Science the candidate must have passed courses in nonlinear dynamics at Master's degree level. At DTU and KU such courses are given under the Mathematics, Informatics, Physics, and Chemistry programs.
    Applicants for a full Ph.D. study at the school will be evaluated by program committees. At DTU the admission of Ph.D. students is essentially based on the average of grades obtained in the Master's degree courses. At Risø the applicants are interviewed in addition. At KU, the Master's thesis is particularly emphasized.
    Co-sponsored grants will be advertised internationally. In connection with the selection of Ph.D. students for these grants we plan to interview as many as possible of the applicants.


    Danish Ph.D. students

  • MIDIT, DTU: Total 17 Danish Ph.D.s
    5 Ph.D.'s co-sponsored with: Danish Research Academy, Research Centre Bygholm, Danish Fundamental Metrology Institute, Force Institute, Topsil, Danish Technical Research Council, Academy of Technical Sciences, and European Union
    (1993-97 MIDIT, DTU: Total: 22 Danish Ph.D.s)
  • OFD, RISØ: Total 10 Danish Ph.D.s
    9 Ph.D.'s co-sponsored with: Danish Research Academy, Danish Natural Science Research Council, and UNIC-C,
    (1993-97 OFD, RISØ: Total 8 Danish Ph.D.s)
  • CATS, KU: Total 9 Danish Ph.D.s
    4 Ph.D.'s co-sponsored with: Danish Research Academy, Danish Natural Science Research Council, UNIC-C, DTU, Risø, European Union, and FOFT
    (1993-97 CATS, KU: Total 16 Ph.D.s)

  • Total 35 Danish Ph.D.s

    About 10 Danish Ph.D. students are expected to enter the school each year. Thus the total enrolment over a five year period amounts to about 50 Danish Ph.D. students.

    Foreign Ph.D. students

  • MIDIT, OFD, CATS: Total 29 Foreign Ph.D.'s

    Many of the foreign Ph.D. students come from the home institutions of our Guest Professors (who recommend the students) or from other institutions in contact with our school. The students typically spend a couple of months here, participating in the activities of the school and working on their Ph.D. thesis. Their stay constitutes an important part of their Ph.D. training. We issue a diploma recognizing their achievements here.

    Post Docs

    15 Post Docs, sponsored by various sources, are working in connection with the Graduate School (MIDIT: 3, OFD: 3, and CATS: 9).