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

In 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.