Henk Broer > DSMP > MATHS > JBI > FWN > RUG

Research

My research focusses on the following, overlapping main themes:


Cantorized fold

Parametrized KAM Theory. Kolmogorov Arnold Moser Theory deals with the persistent occurrence of multi-periodicity in dynamical systems. Examples can be found in the solar system, or in coupled oscillators as these occur in many contexts. Multi- or quasi-periodic dynamics usually occurs as an orderly `state' in a scenario in between orderly (say, periodic) and chaotic. Our interest is to describe these scenarios with parametrized dynamical systems.
In the mathematics of KAM Theory one meets `small divisors': certain perturbation series diverge on a dense set of resonances. In the global geometry also singularity theory shows up, where the classical semi-algebraic geometry is `Cantorized' by the resonances. Quasi-periodic dynamics takes place on invariant tori. Of special interest are the quasi-periodic bifurcations, which often involves tori of different dimension.

Literature

  1. Broer, H.W. and Takens, F. (2011) Dynamical Systems and Chaos. Applied Mathematical Sciences 172, Springer. (URL) (BibTeX)
  2. (2010) Handbook of Dynamical Systems. (Broer, H.W., Hasselblatt, B. and Takens, F., Eds.) North-Holland. (BibTeX)
  3. Broer, H.W., Ciocci, M.C., Han\ssmann, H. and Vanderbauwhede, A. (2009) Quasi-periodic stability of normally resonant tori.. . (URL) (BibTeX)
  4. Broer, H.W., Hanßmann, H. and Hoo, J. (2007) The quasi-periodic Hamiltonian Hopf bifurcations.. . (BibTeX)
  5. Broer, H.W., Hoo, J. and Naudot, V. (2007) Normal linear stability of quasi-periodic tori.. . (BibTeX)

Botafumeiro in 1 : 2 resonance (Santiago de Compostela)

Resonance. Resonance is a periodic interaction of oscillatory subsystems with rational frequency ratios. Such phenomena are most visible when these ratios have the format 1:1, 1:2, 2:3, etc., so with small integers. They occur as tidal resonances in the solar system (think of the Moon locked to the Earth always showing the same face), or in synchronized clocks according to Christiaan Huygens, as well as in many biological circumstances. Think of synchronizing fire flies on a bush or of the entrainment of biological clocks by the twenty-four hours rhythms.
Resonances are best understood in terms of parameter dependent dynamical systems. In the parameter space each resonance usually corresponds to an open domain (or `gap'), as part of a bifurcation set determined by singularity theory. The overall geometry then often is fractal, due to the Cantorization described above.

Literature

  1. Broer, H. W., Holtman, S.J. and Vegter, G. (2010) Recognition of resonance type in periodically forced oscillators.. . (BibTeX)
  2. Broer, H.W., Holtman, S.J., Vegter, G. and Vitolo, R. (2009) Geometry and dynamics of mildly degenerate Hopf-Neimarck-Sacker families near resonance.. . (URL) (BibTeX)
  3. Broer, H.W. and Vegter, G. (2008) Generic Hopf-Ne\u\imark-Sacker bifurcations in feed forward systems.. . (BibTeX)
  4. Broer, H.W., Golubitsky, M. and Vegter, G. (2003) The geometry of resonance tongues: A Singularity Theory approach.. . (URL) (BibTeX)
  5. Broer, H.W., Puig, J. and Simó, C. (2003) Resonance tongues and instability pockets in the quasi-periodic Hill-Schroedinger equation.. . (URL) (BibTeX)

Quasi-periodic Henon like strange attractor

Modelling. Dynamical systems theory is extremely useful for many modelling purposes. Apart from classical areas as mechanics, optics, etc., also meteorology and biology are suitable from mathematical modelling from dynamical systems. In meteorology we are working on several subjects regarding climate variability. A current subject regards the statistics of extreme events in cooperation with groups from the University of Exeter and the Royal Netherlands Meteorological Institute (KNMI).
Advanced dynamical systems theory is being used to describe certain phenomena, e.g., concerning the Atlantic Multidecadal Oscillation. In this description chaos and intermittency occur in several forms.
In biology the current interest is formed by applications in chronobiology, see under the item Resonance. Here we cooperate with the Groningen group Chronobiology.

Literature

  1. Beersma, D.G.M., Broer, H.W., Efstathiou, K., Gargar, K.A. and Hoveijn, I. (2011) Pacer cell response to periodic Zeitgebers.. . (URL) (BibTeX)
  2. Sterk, A.E., Vitolo, R., Broer, H.W., , Simó, C. and Dijkstra, H.A. (2010) New nonlinear mechanisms of midlatitude atmospheric low-frequency variability.. . (URL) (BibTeX)
  3. Broer, H.W., Simó, C. and Vitolo, R. (2010) Chaos and quasi-periodicity in diffeomorphisms of the solid torus.. . (URL) (BibTeX)
  4. Broer, H.W., Efstathiou, K. and Subramanian, E. (2008) Robustness of unstable attractors in arbitrarily sized pulse-coupled systems with delay.. . (BibTeX)
  5. Broer, H.W., Simó, C. and Vitolo, R. (2003) Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing.. . (URL) (BibTeX)

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