My research so far has been concentrated on low-dimensional chaos.
It has dealt with the analysis of low-dimensional non-linear systems in
terms of periodic orbits. This has taken two closely related paths; the
construction of symbolic dynamics and analysis in terms of zeta-functions
and Perron-Frobenius operators.
Symbolic dynamics is the description of the real dynamics in topological terms. Put in other words, to each topologically different region of phase space a symbol is assigned, thus enabling the translation of a physical trajectory to a string of symbols. The analysis and manipulation of such strings of symbols is then used to extract information about the original dynamical system.
Zeta-functions and Perron-Frobenius operators are highly efficient tools for the analysis of chaotic systems. With the use of periodic orbits, their stabilities and periods, a number of physically relevant quantities can be extracted, among which is the semi-classical energy spectrum, as well as fractal dimensions and entropies.
My most recent work has been on the analysis of Hamiltonian systems with mixed phase space structure of stable islands in chaotic regions.
Through the use of homoclinic tangencies and symmetry lines of such a system, a partition can be constructed that successfully partitions both the chaotic component of the phase space and the stable islands. The method has been implemented for the Hamiltonian standard map but is quite general and should be applicable to generic Hamiltonian systems with 2-d Poincare' surfaces of section. >/p> Furthermore, I have worked on extending the methods described above to higher dimensional chaotic systems, such as nonlinear PDEs. This is of importance if periodic orbit theory is to be used for a wider variety of physical systems. However, there are a number of questions that need to be answered to make such an extension possible:
Topology: periodic orbit analysis essentially works when one has a good handle on topology, f.ex. in the form of symbolic dynamics. Is it possible to gain such an understanding of the topology in higher dimensional systems?
Poincare' recurrence: does it make sense to take averages over periodic orbits, if the phase space is so large that close recurrences does not realistically occur?
Number of orbits: if the number of short periodic orbits are overwhelming, can one still get meaningful results? Can one find an efficient way to search for periodic orbits in higher dimensions.
Determination of correlation spectra in chaotic systems; F. Christiansen, G. Paladin and H.H. Rugh, Phys. Rev. Lett. 65, 2087-2090 (1990).
The spectrum of the period doubling operator in terms of cycles; F. Christiansen, P. Cvitanovic' and H.H. Rugh, J. Phys. A 23, L713S-L717S (1990).
Mixing rates and exterior forms in chaotic systems; F. Christiansen, S. Isola, G. Paladin and H. H. Rugh, J. Phys. A 23, L1301-L1308 (1990).
Unstable periodic orbits in the parametrically excited pendulum; W. v. d. Water, M. Hoppenbrouwers and F. Christiansen, Phys. Rev. A., 44, 6388-6398 (1991).
Periodic orbit quantization of the anisotropic Kepler problem; F. Christiansen and P. Cvitanovic', Chaos 2, 61-69 (1992).
Escape and sensitive dependence on initial conditions in a symplectic repeller; F. Christiansen and P. Grassberger, Physics Letters A 181 , 47-53 (1993).
Non-integrability of the Mixmaster Universe; F. Christiansen, H. H. Rugh and S. E. Rugh, J. Phys. A 28, 657-667 (1995). http://xxx.lanl.gov/ps/solv-int/9406002
A generating partition for the standard map; F. Christiansen and A Politi , Physical Review E, 51 R3811-R3814 (1995). http://xxx.lanl.gov/ps/chao-dyn/9611014
Symbolic encoding in symplectic maps; F. Christiansen and A. Politi, Nonlinearity 9, 1623-1640 (1996).
Spatiotemporal chaos in terms of unstable recurrent patterns; F. Christiansen, P. Cvitanovic' and V. Putkaradze, Nonlinearity 10, 55-70 (1997). http://www.mpipks-dresden.mpg.de/eprint/freddy/9702015/9702015.ps.gz
Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization; F. Christiansen and H. H. Rugh, Nonlinearity 10, 1063-1072 (1997). http://www.mpipks-dresden.mpg.de/eprint/freddy/9702017/9702017.ps
Guidelines for the construction of a generating partition in the standard map; F. Christiansen and A. Politi, Physica D 109, 32-41 (1997). http://www/mpipks-dresden.mpg.de/eprint/freddy/9702016/9702016.ps.gz
Analysis of chaotic dynamical systems in terms of cycles; Cand. Scient. - Master's thesis, 60 p., June 10, 1989, The Niels Bohr Institute, University of Copenhagen.
Periodic orbits in classical and quantum chaos; Ph.D. - thesis, 79 p., December 19, 1992, The Niels Bohr Institute, University of Copenhagen.