@article{ChigarevKazakovPikovskij2021,
  author    = {Chigarev, Vladimir and Kazakov, Alexey and Pikovskij, Arkadij},
  title     = {Mutual singularities of overlapping attractor and repeller},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {31},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {8},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/5.0056891},
  pages     = {10},
  year      = {2021},
  abstract  = {We apply the concepts of relative dimensions and mutual singularities to characterize the fractal properties of overlapping attractor and repeller in chaotic dynamical systems. We consider one analytically solvable example (a generalized baker's map); two other examples, the Anosov-Mobius and the Chirikov-Mobius maps, which possess fractal attractor and repeller on a two-dimensional torus, are explored numerically. We demonstrate that although for these maps the stable and unstable directions are not orthogonal to each other, the relative Renyi and Kullback-Leibler dimensions as well as the mutual singularity spectra for the attractor and repeller can be well approximated under orthogonality assumption of two fractals.},
  language  = {en}
}
@article{ChigarevKazakovPikovsky2020,
  author    = {Chigarev, Vladimir and Kazakov, Alexey and Pikovsky, Arkady},
  title     = {Kantorovich-Rubinstein-Wasserstein distance between overlapping attractor and repeller},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {30},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {7},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/5.0007230},
  pages     = {10},
  year      = {2020},
  abstract  = {We consider several examples of dynamical systems demonstrating overlapping attractor and repeller. These systems are constructed via introducing controllable dissipation to prototypic models with chaotic dynamics (Anosov cat map, Chirikov standard map, and incompressible three-dimensional flow of the ABC-type on a three-torus) and ergodic non-chaotic behavior (skew-shift map). We employ the Kantorovich-Rubinstein-Wasserstein distance to characterize the difference between the attractor and the repeller, in dependence on the dissipation level.},
  language  = {en}
}