Consider the map dynamics xn+1=F(xn;c), with a control parameter c. Let the governing set {gn¦ n=0, 1, 2,…} be a desired periodic dynamic set (gN+n≡gn). It is noted that the non-autonomous system xn+1=F(xn;c)+Gn has such a solution, xn=gn, if Gn=gn+1−F(gn;c). In particular, the set of values {gn38}; might be obtained from the periodic solutions of gn+1=F(gn,c∗), using suitable values of c∗. This study explores the values of c∗ which yield entrained solutions, their basin of entrainment, {x0¦limn→∞¦xn−gn¦= 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 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 gn+1=F(gn,c∗). Other periodic sets, {gn&}; in Gn 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 Gn are also studied.