Stochastic control stabilizing unstable or chaotic maps |
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Authors: | Elena Braverman Alexandra Rodkina |
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Institution: | 1. Department of Mathematics and Statistics, University of Calgary, Calgary, Canada;2. Department of Mathematics, The University of the West Indies, Mona Campus, Kingston, Jamaica |
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Abstract: | This paper considers a stabilizing stochastic control which can be applied to a variety of unstable and even chaotic maps. Compared to previous methods introducing control by noise, we relax assumptions on the class of maps, as well as consider a wider range of parameters for the same maps. This approach allows to stabilize unstable and chaotic maps by noise. The interplay between the map properties and the allowed neighbourhood where a solution can start to be stabilized is explored: as instability of the original map increases, the interval of allowed initial conditions narrows. A directed stochastic control aiming at getting to the target neighbourhood almost sure is combined with a controlling noise. Simulations illustrate that for a variety of problems, an appropriate bounded noise can stabilize an unstable positive equilibrium, without a limitation on the initial value. |
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Keywords: | stochastic difference equations stabilization control Kolmogorov's law of large numbers multiplicative noise |
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