@misc{StichBeta2019,
  author    = {Stich, Michael and Beta, Carsten},
  title     = {Time-Delay Feedback Control of an Oscillatory Medium},
  series = {Biological Systems: Nonlinear Dynamics Approach},
  volume    = {20},
  journal   = {Biological Systems: Nonlinear Dynamics Approach},
  publisher = {Springer},
  address   = {Cham},
  isbn      = {978-3-030-16585-7},
  issn      = {2199-3041},
  doi       = {10.1007/978-3-030-16585-7_1},
  pages     = {1 -- 17},
  year      = {2019},
  abstract  = {The supercritical Hopf bifurcation is one of the simplest ways in which a stationary state of a nonlinear system can undergo a transition to stable self-sustained oscillations. At the bifurcation point, a small-amplitude limit cycle is born, which already at onset displays a finite frequency. If we consider a reaction-diffusion system that undergoes a supercritical Hopf bifurcation, its dynamics is described by the complex Ginzburg-Landau equation (CGLE). Here, we study such a system in the parameter regime where the CGLE shows spatio-temporal chaos. We review a type of time-delay feedback methods which is suitable to suppress chaos and replace it by other spatio-temporal solutions such as uniform oscillations, plane waves, standing waves, and the stationary state.},
  language  = {en}
}
@article{StichCasalBeta2013,
  author    = {Stich, Michael and Casal, Alfonso and Beta, Carsten},
  title     = {Stabilization of standing waves through time-delay feedback},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {88},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {4},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {1539-3755},
  doi       = {10.1103/PhysRevE.88.042910},
  pages     = {7},
  year      = {2013},
  abstract  = {Standing waves are studied as solutions of a complex Ginzburg-Landau equation subjected to local and global time-delay feedback terms. The onset is described as an instability of the uniform oscillations with respect to spatially periodic perturbations. The solution of the standing wave pattern is given analytically and studied through simulations.},
  language  = {en}
}
@article{StichBeta2010,
  author    = {Stich, Michael and Beta, Carsten},
  title     = {Control of pattern formation by time-delay feedback with global and local contributions},
  issn      = {0167-2789},
  doi       = {10.1016/j.physd.2010.05.001},
  year      = {2010},
  abstract  = {We consider the suppression of spatiotemporal chaos in the complex Ginzburg-Landau equation by a combined global and local time-delay feedback. Feedback terms are implemented as a control scheme, i.e., they are proportional to the difference between the time-delayed state of the system and its current state. We perform a linear stability analysis of uniform oscillations with respect to space-dependent perturbations and compare with numerical simulations. Similarly, for the fixed-point solution that corresponds to amplitude death in the spatially extended system, a linear stability analysis with respect to space-dependent perturbations is performed and complemented by numerical simulations.},
  language  = {en}
}
@article{StichBeta2011,
  author    = {Stich, Michael and Beta, Carsten},
  title     = {Standing waves in a complex Ginzburg-Landau equation with time-delay feedback},
  series = {Discrete and continuous dynamical systems : a journal bridging mathematics and sciences},
  journal   = {Discrete and continuous dynamical systems : a journal bridging mathematics and sciences},
  number    = {1},
  publisher = {American Institute of Mathematical Sciences},
  address   = {Springfield},
  issn      = {1078-0947},
  pages     = {1329 -- 1334},
  year      = {2011},
  abstract  = {Standing waves are studied as solutions of a complex Ginsburg-Landau equation subjected to local and global time-delay feedback terms. The onset of standing waves is studied at the instability of the homogeneous periodic solution with respect to spatially periodic perturbations. The solution of this spatiotemporal wave pattern is given and is compared to the homogeneous periodic solution.},
  language  = {en}
}