Parrondo’s paradox or chaos control in discrete two-dimensional dynamic systems

摘要

AbstractIn ecological modeling, seasonality can be represented as an alternation between environmental conditions. This concept of alternation holds common ground between ecologists and chemists, who design time-dependent settings for chemical reactors to influence the yield of a desired product. In this study and for a variety of maps, we consider a switching strategy that alternates between two undesirable dynamics that yields a stable desirable dynamic behavior. By comparing bifurcation diagrams of a map and its alternate version, we can easily find parameter values, which, on their own, yield chaotic orbits. When alternated, however, the parameter values yield a stable periodic orbit. Our analysis of the two-dimensional (2-D) maps is an extension of our previous work with one-dimensional (1-D) maps. In the case of 2-D maps, we consider the Beddington, Free, and Lawton and Udwadia and Raju maps. For these 2-D maps, we not only show that we can find “chaotic” parameters for the so-called “chaos” + “chaos” = “periodic” case, but we find two new “desirable” dynamic situations: “quasiperiodic” + “quasiperiodic” = “periodic” and “chaos” + “chaos” = “periodic coexistence.” In the former case, the alternation of chaotic dynamics yield two different periodic stable orbits implying the coexistence of attractors.]]>