@phdthesis{Zaks2001,
  author    = {Zaks, Michael A.},
  title     = {Fractal Fourier spectra in dynamical systems},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-0000500},
  school      = {Universit{\"a}t Potsdam},
  year      = {2001},
  abstract  = {Eine klassische Art, die Dynamik nichtlinearer Systeme zu beschreiben, besteht in der Analyse ihrer Fourierspektren. F{\"u}r periodische und quasiperiodische Prozesse besteht das Fourierspektrum nur aus diskreten Deltafunktionen. Das Spektrum einer chaotischen Bewegung ist hingegen durch das Vorhandensein einer stetigen Komponente gekennzeichnet. In der Arbeit geht es um einen eigenartigen, weder regul{\"a}ren noch vollst{\"a}ndig chaotischen Zustand mit sogenanntem singul{\"a}rstetigen Leistungsspektrum. Unsere Analyse ergab verschiedene F{\"a}lle aus weit auseinanderliegenden Gebieten, in denen singul{\"a}r stetige (fraktale) Spektren auftreten. Die Beispiele betreffen sowohl physikalische Prozesse, die auf iterierte diskrete Abbildungen oder gar symbolische Sequenzen reduzierbar sind, wie auch Prozesse, deren Beschreibung auf den gew{\"o}hnlichen oder partiellen Differentialgleichungen basiert.},
  subject      = {Nichtlineares dynamisches System / Harmonische Analyse / Fraktal},
  language  = {en}
}
@article{SposiniGrebenkovMetzleretal.2020,
  author    = {Sposini, Vittoria and Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb and Seno, Flavio},
  title     = {Universal spectral features of different classes of random-diffusivity processes},
  series = {New Journal of Physics},
  volume    = {22},
  journal   = {New Journal of Physics},
  number    = {6},
  publisher = {Dt. Physikalische Ges.},
  address   = {Bad Honnef},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/ab9200},
  pages     = {26},
  year      = {2020},
  abstract  = {Stochastic models based on random diffusivities, such as the diffusing-diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media. Studies of these models typically focus on the moments and the displacement probability density function. Here we develop the complementary power spectral description for a broad class of random-diffusivity processes. In our approach we cater for typical single particle tracking data in which a small number of trajectories with finite duration are garnered. Apart from the diffusing-diffusivity model we study a range of previously unconsidered random-diffusivity processes, for which we obtain exact forms of the probability density function. These new processes are different versions of jump processes as well as functionals of Brownian motion. The resulting behaviour subtly depends on the specific model details. Thus, the central part of the probability density function may be Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential, log-normal, or even power-law forms. For all these models we derive analytically the moment-generating function for the single-trajectory power spectral density. We establish the generic 1/f²-scaling of the power spectral density as function of frequency in all cases. Moreover, we establish the probability density for the amplitudes of the random power spectral density of individual trajectories. The latter functions reflect the very specific properties of the different random-diffusivity models considered here. Our exact results are in excellent agreement with extensive numerical simulations.},
  language  = {en}
}
@misc{SposiniGrebenkovMetzleretal.2020,
  author    = {Sposini, Vittoria and Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb and Seno, Flavio},
  title     = {Universal spectral features of different classes of random-diffusivity processes},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {999},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-47696},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-476960},
  pages     = {27},
  year      = {2020},
  abstract  = {Stochastic models based on random diffusivities, such as the diffusing-diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media. Studies of these models typically focus on the moments and the displacement probability density function. Here we develop the complementary power spectral description for a broad class of random-diffusivity processes. In our approach we cater for typical single particle tracking data in which a small number of trajectories with finite duration are garnered. Apart from the diffusing-diffusivity model we study a range of previously unconsidered random-diffusivity processes, for which we obtain exact forms of the probability density function. These new processes are different versions of jump processes as well as functionals of Brownian motion. The resulting behaviour subtly depends on the specific model details. Thus, the central part of the probability density function may be Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential, log-normal, or even power-law forms. For all these models we derive analytically the moment-generating function for the single-trajectory power spectral density. We establish the generic 1/f²-scaling of the power spectral density as function of frequency in all cases. Moreover, we establish the probability density for the amplitudes of the random power spectral density of individual trajectories. The latter functions reflect the very specific properties of the different random-diffusivity models considered here. Our exact results are in excellent agreement with extensive numerical simulations.},
  language  = {en}
}