# Research

My research focusses on the following, overlapping main themes:

**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

H.W. Broer and F. Takens (2011) Dynamical Systems and Chaos. Applied Mathematical Sciences 172, Springer, 2011.
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H.W. Broer, B. Hasselblatt, and F. Takens (eds.) (2010) Handbook of Dynamical Systems. North-Holland, 2010.
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H.W. Broer, M.C. Ciocci, H. Hanßmann, and A. Vanderbauwhede. Quasi-periodic stability of normally resonant tori. Physica D, 238:309–318, 2009.
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H.W. Broer, H. Hanßmann, and J. Hoo. The quasi-periodic Hamiltonian Hopf bifurcations. Nonlinearity, 20:417–460, 2007.
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H.W. Broer, J. Hoo, and V. Naudot. Normal linear stability of quasi-periodic tori. J. Diff. Eqns., 232(2):355–418, 2007.
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**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

H. W. Broer, S.J. Holtman, and G. Vegter. Recognition of resonance type in periodically forced oscillators. Physica D, 239(17):1627–1636, 2010.
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H.W. Broer, S.J. Holtman, G. Vegter, and R. Vitolo. Geometry and dynamics of mildly degenerate Hopf-Neimarck-Sacker families near resonance. Nonlinearity, 22:2161–2200, 2009.
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H.W. Broer and G. Vegter. Generic Hopf-Ne\u\imark-Sacker bifurcations in feed forward systems. Nonlinearity, 21:1547–1578, 2008.
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H.W. Broer, M. Golubitsky, and G. Vegter. The geometry of resonance tongues: A Singularity Theory approach. Nonlinearity, 16:1511–1538, 2003.
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H.W. Broer, J. Puig, and C. Simó. Resonance tongues and instability pockets in the quasi-periodic Hill-Schroedinger equation. Commun. Mathematical Physics, 241:467–503, 2003.
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**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

D.G.M. Beersma, H.W. Broer, K. Efstathiou, K.A. Gargar, and I. Hoveijn. Pacer cell response to periodic Zeitgebers. Physica D, 19:1516–1527, 2011.
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A.E. Sterk, R. Vitolo, H.W. Broer, Simó , and H.A. Dijkstra. New nonlinear mechanisms of midlatitude atmospheric low-frequency variability. Physica D, 239(10):702–718, 2010.
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H.W. Broer, C. Simó, and R. Vitolo. Chaos and quasi-periodicity in diffeomorphisms of the solid torus. Discrete Continuous Dynamical Systems - Series B, 14(3):871–905, 2010.
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H.W. Broer, K. Efstathiou, and E. Subramanian. Robustness of unstable attractors in arbitrarily sized pulse-coupled systems with delay. Nonlinearity, 21:13–49, 2008.
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H.W. Broer, C. Simó, and R. Vitolo. Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing. Nonlinearity, 15(4):1205–1267, 2003.
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