Logistic MapDiscrete MapControl ParameterGraphical AnalysisGraphical Sequence1-CyclesPeriod DoublingBifurcation DiagramLogistic MapLogistic MapDiscrete MapDiscrete MapA map A map f f is defined on a is defined on a metric space metric space XX..Repeated application of Repeated application of ff forms a sequence.forms a sequence.•Discrete set of pointsDiscrete set of pointsA sequence of points forms A sequence of points forms an an orbitorbit..Xx110)( xxf fx2f21002)())(()( xxfxffxf 303)( xxf ffx0x3x4Control ParameterControl ParameterDefine a map based on a Define a map based on a quadratic function.quadratic function.•ff: : RR11 RR11•xx rxrx(1(1xx))The parameter r is the The parameter r is the control parameter.control parameter.•Range Range rr from 0 to 4 from 0 to 4Initial condition Initial condition xx00 = 0.5. = 0.5.•Range Range xx from 0 to 1 from 0 to 1)(xfy XYGraphical AnalysisGraphical Analysis)(xfy Fixed points occur when the Fixed points occur when the function intersects function intersects yy = = xx. . Solutions are Solutions are qq = 0, = 0, pp = 1–1/ = 1–1/rr..Stable Stable pp for for rr > 2> 2..XYxy 0)1()1(2rrxxrxrxxrxxrpfxrxf2)()21()(Graphical SequenceGraphical SequenceTreat each point in the Treat each point in the sequence as a pair (x, x). sequence as a pair (x, x). Find the next point.Find the next point.•Move vertically to the curveMove vertically to the curve•Move horizontally to the lineMove horizontally to the lineXY))(,(),( xfxxx 0x1x2x3x))(),(())(,( xfxfxfx 1-Cycles1-CyclesValues of Values of rr from 0 to 3 create from 0 to 3 create a single attracting fixed pointa single attracting fixed point0 < 0 < rr < 1: < 1:•pp is negative is negative•qq attracts, attracts, pp repels repels1 < 1 < rr < 3: < 3:•pp is positive and attracts is positive and attracts•qq repels repels•For For rr > 2 convergence > 2 convergence alternates around alternates around ppXY0x1x2xpPeriod DoublingPeriod DoublingAt At rr = 3 the derivative is -1 = 3 the derivative is -1•Neutral between attracting Neutral between attracting and repellingand repellingFor 3 < r < 3.4 there are two For 3 < r < 3.4 there are two new stable points.new stable points.•Limit cycle is period 2Limit cycle is period 2Period continues to double Period continues to double for higher for higher rr..•For For rr above 3.5699…, the above 3.5699…, the motion is chaotic.motion is chaotic.3/23/11 p3r132)(' pf))1(1)(1())((2xrxxxrxff 1)1(2))1(1)(1(2223322rxrrxrxrxrxxxrxBifurcation DiagramBifurcation DiagramFor each value of the control parameter there is a set For each value of the control parameter there is a set of points.of points.•Neglect the initial transient valuesNeglect the initial transient valuesnextdiagram from
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