### Abstract

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_{n}38}; 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, {x_{0}|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, g_{0}, is {x_{0}|1-g_{0}-1/c<x_{0}<g_{0}+1/c} for al l allowed g_{0}, (c-1)/2c< g_{0}<(1+c)/2c. There are bounded but non-entrained solutions if (2+c)/2c > g_{0} > (1+c)/2c, with the above b asin, or if (c-1)/2c > ;g_{0} > (c-4)/2c, with the basin {x > _{0}|g_{0}<x_{0}< 1-g_{0}}. The complimentary x_{0} region yields unbounded dynamics. Period-two governing values have two basins, depending on the initial value used for g_{0}. 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 x_{0} and g_{0}, is discussed. Other periodic and non-periodic responses to these periodic G_{n} are also studied.

Original language | English (US) |
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Pages (from-to) | 407-420 |

Number of pages | 14 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 44 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 1990 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics

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## Cite this

*Physica D: Nonlinear Phenomena*,

*44*(3), 407-420. https://doi.org/10.1016/0167-2789(90)90155-I