Periodic entrainment of chaotic logistic map dynamics

Consider the map dynamics x_n+1=F(x_n;c), with a control parameter c. Let the governing set {g_n| n=0, 1, 2,...} be a desired periodic dynamic set (g_N+n≡g_n). It is noted that the non-autonomous system x_n+1=F(x_n;c)+G_n has such a solution, x_n=g_n, if G_n=g_n+1-F(g_n;c). In particular, the set of values {g_n38}; might be obtained from the periodic solutions of g_n+1=F(g_n,c^*), using suitable values of c^*. This study explores the values of c^* which yield entrained solutions, their basin of entrainment, {x₀|lim_n→∞|x_n-g_n|= 0}, and more generally the basins of bounded solutio ns and their character, when F(x,c)=cx(1-x), the logistic map. Of particular interest is the entrainment of chaotic dynamics, c=4. Generally, the basin of entrainment for period-one governing, g₀, is {x₀|1-g₀-1/c<x₀<g₀+1/c} for al l allowed g₀, (c-1)/2c< g₀<(1+c)/2c. There are bounded but non-entrained solutions if (2+c)/2c > g₀ > (1+c)/2c, with the above b asin, or if (c-1)/2c > ;g₀ > (c-4)/2c, with the basin {x > ₀|g₀<x₀< 1-g₀}. The complimentary x₀ region yields unbounded dynamics. Period-two governing values have two basins, depending on the initial value used for g₀. These basins are investigated for the sets g_n+1=F(g_n,c^*). Other periodic sets, {g_n&}; in G_n do not become disjoint i n space. Period-four entrainment, for suitable c^*, can have even larger basins. The unwanted occurrence of other disjoint basins of attraction, which interlace the basins of entrainment, as a function of both x₀ and g₀, is discussed. Other periodic and non-periodic responses to these periodic G_n are also studied.