| Abstract [eng] |
This paper is about discrete map stabilization. Said maps possess both stable and unstable attractors which is why stabilization in this work's context means a process during which a map's unstable trajectory is held as close as possible to the unstable orbit for as long as it is possible to do so. Due to not all systems yield the best results traversing just the stable orbits, there is a need to stabilize unstable trajectories for higher valued results. To realize such a temporary stabilization scheme first literature analysis is conducted, spanning topics like iterative maps and their applications, control problems and the principles behind solving them, and fractional logistic maps, their applications and stabilization attempts in literature. Another phase of said literature review includes comparisons between the classical, the invertible extended and fractional logistic maps to solidify the core differences of their properties. The final part of the literature review is the discussion of H-ranks and the methodology of using SVD to compute the aforementioned H-rank values. After the literature review a computational and analytical convergence property analysis of the fractional logistic map is conducted. After that is the application attempt of a naïve stabilization scheme to stabilize the fractional logistic map's trajectory. Finally, an extensive description of an impulse-based control scheme successfully applied to the fractional logistic map to temporarily yet periodically stabilize the fractional logistic map's trajectory. |
