Periodic entrainment of chaotic logistic map dynamics

E. Atlee Jackson, Alfred Hübler

Research output: Contribution to journalArticlepeer-review


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<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 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.

Original languageEnglish (US)
Pages (from-to)407-420
Number of pages14
JournalPhysica D: Nonlinear Phenomena
Issue number3
StatePublished - Sep 1 1990

ASJC Scopus subject areas

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


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