(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 16252234, 331378] NotebookOptionsPosition[ 16239945, 331022] NotebookOutlinePosition[ 16240639, 331047] CellTagsIndexPosition[ 16240596, 331044] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["The Logistic Map", "Subtitle"], Cell[CellGroupData[{ Cell[" Introduction", "Section"], Cell[TextData[{ StyleBox["One of the most challenging topics in science is the study of \ chaos. As an example of chaos, consider fluid flowing round an object. If \ the velocity of the fluid is not very large the fluid flows in a smooth \ steady way, called \"laminar flow\", which can be calculated for simple \ geometries. However, if the velocity is increased the flow becomes more \ complicated, and depends on time and space in a non-trivial way which is \ essentially impossible to calculate. This \"turbulent\" behavior is called \ \"chaos\". \n\nNote that chaos does not mean that the behavior is completely \ random; one does see recognizable structures, such as little whirlpools, \ called \"vortices\", but it is very complicated. The reason that chaotic \ behavior is essentially impossible to calculate is that the future behavior \ of the fluid depends in a very sensitive way on what the fluid is doing now. \ If we consider two configurations of the velocity field, which differ only by \ a very small amount, then, in the chaotic regime, those differences grow \ exponentially time, (the coefficient in the exponential is called the \ \"Lyapunov exponent\" which we will calculate in this notebook for a \ particular model) so that eventually the two velocity fields become \ completely different. This \"sensitivity to initial conditions\" is \ generally taken as the definition of chaos. Because the flow of air in the \ atmosphere is chaotic, it is impossible to predict the weather in detail for \ more than about a week because we cannot integrate the equations with \ sufficient accuracy, even assuming that we had sufficiently detailed data on \ the state of the weather now (which we don't).\n\nWhile a lot of chaos theory \ was developed by Poincar\[EAcute] around the start of the 20th century, \ there were few subsequent developments (because the field is so difficult) \ until, starting in the 1980s, the mass availability of computers meant that \ the equations could easily be integrated numerically, and perhaps most \ importantly, the solutions visualized by computer graphics. \n\nAs the \ velocity of the fluid increases, rather than there being a ", "Text"], StyleBox["discontinuous", "Text", FontSlant->"Italic"], StyleBox[" change from laminar to turbulent flow at a critical velocity, the \ motion changes only ", "Text"], StyleBox["gradually", "Text", FontSlant->"Italic"], StyleBox[" as it passes the instability where it first becomes non-laminar, \ and it is only at larger values of the velocity that fully developed \ turbulence appears. An important question, then, is the nature of this ", "Text"], StyleBox["\"transition to chaos\"", "Text", FontWeight->"Bold"], StyleBox[". He we will use the graphical ability of ", "Text"], StyleBox["Mathematica", "Text", FontSlant->"Italic"], StyleBox[" to consider a very simple looking model which has a transtion to \ chaos. There does not seem to be a single route to chaos, but the model that \ we study here, shows one of them, known as ", "Text"], StyleBox["\"period doubling\"", "Text", FontWeight->"Bold"], StyleBox[". It is, in fact, the world's simplest model with a transition to \ chaos.", "Text"] }], "Text", CellChangeTimes->{3.514564915320327*^9}] }, Open ]], Cell[CellGroupData[{ Cell["The Model", "Section"], Cell[TextData[{ "Most realistic systems with chaotic behavior, such as fluid flow around an \ obstacle, are described by a non-linear partial differential equation, which \ determines how the velocity changes in space and time. We will see later in \ the course that chaos also occurs in many ordinary (non-linear) differential \ equations. Here we will consider something even simpler, an ", StyleBox["\"iterative map\"", FontWeight->"Bold"], ". This means that starting from an initial value of a variable, ", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], " say, we generate a sequence of values, ", Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["x", "2"], TraditionalForm]]], ", etc. from the map (i.e. function), ", Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox["x", RowBox[{"n", "+", "1"}]]}], TraditionalForm]]], " = f(", Cell[BoxData[ FormBox[ SubscriptBox["x", "n"], TraditionalForm]]], ") where we here make a simple choice f(x) = 4 \[Lambda] x (1 - x), i.e.\n\n\ \t", Cell[BoxData[ FormBox[ SubscriptBox["x", RowBox[{"n", "+", "1"}]], TraditionalForm]]], " = 4\[Lambda] ", Cell[BoxData[ FormBox[ SubscriptBox["x", "n"], TraditionalForm]]], "( 1-", Cell[BoxData[ FormBox[ SubscriptBox["x", "n"], TraditionalForm]]], "), \n\t\nwhere \[Lambda] is parameter. In other words, ", Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], " = 4\[Lambda] ", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], "(1-", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], "), ", Cell[BoxData[ FormBox[ SubscriptBox["x", "2"], TraditionalForm]]], " = 4\[Lambda] ", Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], "(1-", Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], "), etc. We will be interested in the behavior of successive iterations of \ this map, as a function of the parameter \[Lambda]. In particular we will \ study the behavior of the ", Cell[BoxData[ FormBox[ SubscriptBox["x", "n"], TraditionalForm]]], " for large n.\n\nWe consider \[Lambda] in the range from 0 to 1, so, if ", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], " is between 0 and 1, it is easy to see that all subsequent values of x \ also lie in this range. In fact the largest value of ", Cell[BoxData[ FormBox[ SubscriptBox["x", RowBox[{"n", "+", "1"}]], TraditionalForm]]], "(which occurs for ", Cell[BoxData[ FormBox[ SubscriptBox["x", "n"], TraditionalForm]]], "= 1/2) is equal to \[Lambda]. \n\nThis so-called ", StyleBox["\"logistic map\"", FontWeight->"Bold"], " has been used as model for population dynamics, but here we just treat it \ as a toy model which has a transition to chaos. 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We plot it for \[Lambda] = \ 0.4:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", "x", "]"}], "/.", "\[InvisibleSpace]", RowBox[{"\[Lambda]", "\[Rule]", "0.4"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{3.4202106803166103`*^9}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], PointSize[Large], Thickness[Large], LineBox[CompressedData[" 1:eJwd2Hk4VO/7B3CKiLKVfElpQSqlJFrwFkqFEBWSilJKsn1CJRTZqVAKofRp IVsoZRlFJCSFcxgzZ4wkS1QkSX7P5/fXXK/rmuuceZ77PvfznrPY6fTuo9ME BATmCAoI/Pe582jve9aXPfpT1iXuDWGH9YWMFDy48lq45S958fQcA6xLS0p5 Im8MfVEvzJSxhGlDW0GqvA0KPNOsPCUPYd+RsbJr8keQfyKncr+YBzo1woxi 5H1QcvrfGj2BIKTLWlyOlA9B2gyRBikqDsGiT5ZFyCfgvOPxji1habC1y8gd /paJR8m7KBmZPIhPBTuYLMzFFfUS5dzDT6Fi9bX3VVQhEhYPvtHSLYdv40Yk jZTAa1TP4PmDl5ArlOYIClaA90LHJqW+GsVdRvXvdF9i9e6eMpbwG7w8kTrf bqAK9YdmH9giWI9RVT3Jx6dqUJO4Mv1jWiOmjWdJzVepw5ejif/WZTZBe+pQ 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If we start with \[Lambda] < 0.25, then one can easily show \ analytically that ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", RowBox[{"n", "+", "1"}]], " ", "<", " ", SubscriptBox["x", "n"]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["lim", RowBox[{" ", RowBox[{"n", " ", "\[Rule]", " ", "\[Infinity]"}]}]], " ", RowBox[{"x", "(", "n", ")"}]}], " ", "=", " ", "0"}], TraditionalForm]]], ". To verify this in ", StyleBox["Mathematica", FontSlant->"Italic"], " we put \[Lambda] = 0.1 and generate a list of successive values with the \ ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["NestList[func, x0, n]", FontWeight->"Bold"], ", in which func(x) is the function to be iterated, x0 is the starting \ value of x, and n is the number of iterations" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"\[Lambda]", "=", "0.1"}], ";", " ", RowBox[{"NestList", "[", RowBox[{"f", ",", " ", "0.5", ",", " ", "10"}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ "0.5`", ",", "0.1`", ",", "0.036000000000000004`", ",", "0.013881600000000003`", ",", "0.005475560472576001`", ",", "0.0021782314840348657`", ",", "0.0008693947166547301`", ",", "0.0003474555477925532`", ",", "0.00013893392897394456`", ",", "0.00005556585053492977`", ",", "0.000022225105188473642`"}], "}"}]], "Output", CellChangeTimes->{{3.514719216583805*^9, 3.51471922643258*^9}}] }, Open ]], Cell[TextData[{ "Here we have set \[Lambda] = 0.1, and then iterated f(x) 10 times with the \ initial value 0.5. 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The first argument is the function and \ the second is the starting value for x. For example, still considering \ \[Lambda] = 0.4, we get" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FixedPoint", "[", RowBox[{"f", ",", " ", "0.5"}], "]"}]], "Input"], Cell[BoxData["0.37500000000000006`"], "Output", CellChangeTimes->{3.420210377389163*^9, 3.420212666546013*^9, 3.42021820316566*^9, 3.4202193443954277`*^9, 3.420365452849098*^9, 3.420508543833436*^9, 3.4205618513379593`*^9, 3.451832710538123*^9, 3.452352175965642*^9, 3.482969626116252*^9, 3.48314863984536*^9, 3.514719375586802*^9}] }, Open ]], Cell[TextData[{ StyleBox["FixedPoint", FontWeight->"Bold"], " gives a result when two successive values are ", StyleBox["exactly", FontSlant->"Italic"], " the same. If convergence is very slow, or if roundoff errors prevent exact \ agreement, it is best to put in a less stringent test for determining whether \ convergence has been reached. This is done with the option ", StyleBox["SameTest", FontWeight->"Bold"], " to the ", StyleBox["FixedPoint", FontWeight->"Bold"], " command. The following example shows how it works." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FixedPoint", "[", RowBox[{"f", ",", " ", "0.5", " ", ",", " ", RowBox[{"SameTest", " ", "\[Rule]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Abs", "[", RowBox[{"#1", " ", "-", " ", "#2"}], "]"}], " ", "<", " ", RowBox[{"10", "^", RowBox[{"-", "11"}]}]}], " ", "&"}], ")"}]}]}], "]"}]], "Input"], Cell[BoxData["0.37500000000596584`"], "Output", CellChangeTimes->{3.420210377538602*^9, 3.420212666602302*^9, 3.42021820325065*^9, 3.420219344432053*^9, 3.4203654529008837`*^9, 3.420508543884017*^9, 3.420561851389654*^9, 3.451832710707399*^9, 3.452352178276161*^9, 3.4829696261463614`*^9, 3.483148644517787*^9, 3.514719391027378*^9}] }, Open ]], Cell[TextData[{ "The notation is that of a pure function, in which ", StyleBox["#1", FontWeight->"Bold"], " and ", StyleBox["#2", FontWeight->"Bold"], " are the arguments and the function is terminated by ", StyleBox["&", FontWeight->"Bold"], ".\n\nWe can find the value of \[Lambda] where the fixed point at non-zero X \ becomes unstable quite easily using pencil and paper. We need to \ simultaneously solve" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"x", "=", " ", RowBox[{"f", RowBox[{"(", "x", ")"}]}]}], ",", " ", RowBox[{ RowBox[{"|", RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", "x", ")"}]}], "|"}], " ", "=", " ", "1", " "}]}]], "DisplayFormula"], Cell["The solution is ", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"\[Lambda]", " ", "=", " ", RowBox[{"3", "/", "4"}]}], ",", " ", RowBox[{"x", "=", " ", RowBox[{"2", "/", "3"}]}]}]], "DisplayFormula"], Cell[TextData[{ "(There is also a second solution \[Lambda] = 1/4, x = 0 which is the other \ limit of stability of the x \[NotEqual] 0 fixed point; as we have seen, for \ \[Lambda] < 0.25 the stable fixed point is at x = 0.) ", StyleBox["Mathematica", FontSlant->"Italic"], " will also give us the above result" }], "Text", CellChangeTimes->{3.4831486252650833`*^9}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"x", ",", " ", "\[Lambda]"}], "]"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"1", " ", "\[Equal]", " ", RowBox[{"4", " ", "\[Lambda]", " ", RowBox[{"(", RowBox[{"1", " ", "-", " ", "x"}], ")"}]}]}], ",", " ", RowBox[{"1", " ", "\[Equal]", " ", RowBox[{"Abs", "[", " ", RowBox[{"4", " ", "\[Lambda]", " ", RowBox[{"(", RowBox[{"1", " ", "-", " ", RowBox[{"2", " ", "x"}]}], ")"}]}], "]"}]}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "\[Lambda]"}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "0"}], ",", RowBox[{"\[Lambda]", "\[Rule]", FractionBox["1", "4"]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", FractionBox["2", "3"]}], ",", RowBox[{"\[Lambda]", "\[Rule]", FractionBox["3", "4"]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.514719423329768*^9, 3.514719433748006*^9}}] }, Open ]], Cell[TextData[{ "Let's plot ", StyleBox["f(x)", FontWeight->"Bold"], " and ", StyleBox["x", FontWeight->"Bold"], " at the borderline case \[Lambda] = 0.75. 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Since the magnitude \ of the derivative is 1, we have |f '(X)| = -1.\n\nWe have seen that for \ certain choices of \[Lambda] bigger that 0.75 successive values converge to a \ limit cycle of length 2. This is a fixed point of the twice iterated function \ f(f(x)). Lets call this f2(x). First we ", StyleBox["Clear", FontWeight->"Bold"], " \[Lambda] and x (we don't want to ", StyleBox["Remove", FontWeight->"Bold"], " \[Lambda] because ", StyleBox["Remove", FontWeight->"Bold"], " not only removes information about \[Lambda] but also removes \[Lambda] \ from the list of variables that ", StyleBox["Mathematica", FontSlant->"Italic"], " knows about; this gives problems when defining f2. ", StyleBox["Clear", FontWeight->"Bold"], " just removes the information about \[Lambda]):" }], "Text", CellChangeTimes->{{3.5147195575396833`*^9, 3.5147197094730253`*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", " ", RowBox[{"\[Lambda]", ",", " ", "x"}], "]"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"f2", "[", "x_", "]"}], " ", "=", " ", RowBox[{"f", "[", RowBox[{"f", "[", "x", "]"}], "]"}], " "}]], "Input"], Cell[BoxData[ RowBox[{"16", " ", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}], " ", "x", " ", SuperscriptBox["\[Lambda]", "2"], " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"4", " ", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}], " ", "x", " ", "\[Lambda]"}]}], ")"}]}]], "Output", CellChangeTimes->{3.4202103777542667`*^9, 3.420212667101564*^9, 3.4202182033863277`*^9, 3.420219344540124*^9, 3.4203642049614058`*^9, 3.420365453316988*^9, 3.420508544229303*^9, 3.420561851789742*^9, 3.451832711120932*^9, 3.4523521957051897`*^9, 3.4829696264972477`*^9, 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PlotRange->{{0, 1}, {0., 0.9999999795918367}}, PlotRangeClipping->True, PlotRangePadding->{ Scaled[0.02], Scaled[0.02]}]], "Output", CellChangeTimes->{3.420210377976571*^9, 3.420212667216199*^9, 3.420218203503072*^9, 3.420219344662705*^9, 3.420364219653759*^9, 3.420365453409607*^9, 3.420508544347499*^9, 3.4205618519936323`*^9, 3.451832711360836*^9, 3.452352205208291*^9, 3.4829696265944357`*^9, 3.483148946562498*^9, 3.514719910485388*^9}] }, Open ]], Cell[TextData[{ "We can locate this fixed point using ", StyleBox["FindRoot ", FontWeight->"Bold"], "(not ", StyleBox["FixedPoint ", FontWeight->"Bold"], "because it is untable)" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"unstableFP", " ", "=", " ", RowBox[{"FindRoot", "[", " ", RowBox[{ RowBox[{ RowBox[{"f", "[", "x", "]"}], " ", "\[Equal]", " ", "x"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "0.7"}], "}"}]}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"x", "\[Rule]", "0.6875`"}], "}"}]], "Output", CellChangeTimes->{3.4202103782311697`*^9, 3.420212668062454*^9, 3.420218203533617*^9, 3.420219344695691*^9, 3.420365453435093*^9, 3.420508544367936*^9, 3.4205618523734694`*^9, 3.4518327115606194`*^9, 3.452353898782865*^9, 3.4829696267289133`*^9, 3.483148950456429*^9, 3.514719913171723*^9}] }, Open ]], Cell["\<\ This fixed point is unstable because | f2'| (and also | f ' |) is greater \ than 1 at this value of x. Indeed, one can see from the figure above that \ f2[x] has a greater slope at the middle fixed point than the slope of x (i.e. \ its slope is greater than 1). We can also find the precise numerical values \ of the derivative of f and f2 at this fixed point,\ \>", "Text", CellChangeTimes->{{3.420508223640039*^9, 3.420508421645947*^9}, { 3.514719775649255*^9, 3.514719820474382*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"f", "'"}], "[", "x", "]"}], " ", "/.", " ", "unstableFP"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"f2", "'"}], "[", "x", "]"}], " ", "/.", " ", "unstableFP"}]}], "}"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", "1.2000000000000002`"}], ",", "1.4400000000000006`"}], "}"}]], "Output", CellChangeTimes->{3.420210378376895*^9, 3.420212668144043*^9, 3.420218203572328*^9, 3.420219344732636*^9, 3.420365453470132*^9, 3.42050854440416*^9, 3.420561852459578*^9, 3.451832711576118*^9, 3.482969626751316*^9, 3.4831489559863453`*^9, 3.5147199210959806`*^9}] }, Open ]], Cell["\<\ showing that they are greater in magnitude than 1, confirming that the fixed \ point at 0.6875 is unstable. The other two are fixed points of f2 (the two values of x correspond to the \ values of x in the period-2 cycle of f) and these are stable because |f2'| < \ 1 at these values of x, as is clear from the above figure and which we shall \ confirm quantitatively.\ \>", "Text", CellChangeTimes->{{3.420508424701892*^9, 3.420508472701926*^9}, { 3.451831875458398*^9, 3.451831939638607*^9}, {3.5147198666897793`*^9, 3.514719888033696*^9}}], Cell[TextData[{ "We can find the stable fixed points of f2 using the ", StyleBox["FixedPoint", FontWeight->"Bold"], " command" }], "Text", CellChangeTimes->{3.42050847952631*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"stableX", " ", "=", RowBox[{"FixedPoint", "[", RowBox[{"f2", ",", " ", "0.5", ",", RowBox[{"SameTest", " ", "\[Rule]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Abs", "[", RowBox[{"#1", " ", "-", " ", "#2"}], "]"}], " ", "<", " ", RowBox[{"10", "^", RowBox[{"-", "11"}]}]}], "&"}], ")"}]}]}], "]"}], " "}]], "Input"], Cell[BoxData["0.5130445095308785`"], "Output", CellChangeTimes->{3.4523540448365498`*^9, 3.482969626780572*^9, 3.483148960710148*^9, 3.514719923852387*^9}] }, Open ]], Cell["We verify that | f2' < 1 | at this value of X", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"f2", "'"}], "[", "stableX", "]"}]], "Input"], Cell[BoxData["0.15999999997859587`"], "Output", CellChangeTimes->{3.420210378406074*^9, 3.420212668258893*^9, 3.420218203672567*^9, 3.420219344800583*^9, 3.420365453538477*^9, 3.42050854450436*^9, 3.420561852544939*^9, 3.4518327119184437`*^9, 3.4829696268138638`*^9, 3.4831489633941307`*^9, 3.514719931012515*^9}] }, Open ]], Cell[TextData[{ "The value 0.513045 is the lower of the two stable fixed points of f2 in the \ above figure. Depending on the different initial value for ", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]]], "we would converge either to this one or the other one, which is close to \ x=0.8:" }], "Text", CellChangeTimes->{{3.4203644465619307`*^9, 3.420364486030292*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"stableX2", " ", "=", " ", RowBox[{"FixedPoint", "[", RowBox[{"f2", ",", " ", "0.8", ",", RowBox[{"SameTest", " ", "\[Rule]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Abs", "[", RowBox[{"#1", " ", "-", " ", "#2"}], "]"}], " ", "<", " ", RowBox[{"10", "^", RowBox[{"-", "11"}]}]}], "&"}], ")"}]}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.514719950295945*^9, 3.5147199576989317`*^9}, { 3.514719998159474*^9, 3.514719999091652*^9}}], Cell[BoxData["0.7994554904682839`"], "Output", CellChangeTimes->{ 3.4202103785517683`*^9, 3.42021266832584*^9, 3.420218203722989*^9, 3.42021934483423*^9, 3.420365453572913*^9, 3.420508544554934*^9, 3.420561852595244*^9, 3.451832711933419*^9, 3.452354214142643*^9, 3.482969626847301*^9, 3.483148966052709*^9, {3.5147199381405354`*^9, 3.514719958121417*^9}, 3.51471999988474*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"f2", "'"}], "[", "stableX2", "]"}]], "Input", CellChangeTimes->{3.514719967500079*^9}], Cell[BoxData["0.15999999997900582`"], "Output", CellChangeTimes->{3.514719968189313*^9, 3.514720002368682*^9}] }, Open ]], Cell[TextData[{ "These last results are for \[Lambda] = 0.8, where the fixed point of f2 \ (2-cycle of f) is stable. We now ask ", StyleBox["Mathematica", FontSlant->"Italic"], " to tell us where this attractor becomes unstable. We will help ", StyleBox["Mathematica", FontSlant->"Italic"], " by specifying that the derivative is -1 (rather than +1) at the \ instability point (We can see this, for example from the figure below, which \ is for \[Lambda] = 0.86, near the limit of stability. The stable fixed point \ values of f2 are the first and third fixed points (not counting the trivial \ one at 0), as discussed above, which are near 0.4 and 0.9. The derivative is \ clearly negative there. The intermediate fixed point, at about 0.7 is \ unstable since one can see from the figure that its derivative is greater \ than 1. 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To see which we want it is useful to compute the \ numerical values:\ \>", "Text", CellChangeTimes->{{3.4203645317708473`*^9, 3.4203645845188932`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Lambda]", " ", "/.", " ", RowBox[{"N", "[", "%", "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"0.`", "\[InvisibleSpace]", "-", RowBox[{"0.25`", " ", "\[ImaginaryI]"}]}], ",", RowBox[{"0.`", "\[InvisibleSpace]", "+", RowBox[{"0.25`", " ", "\[ImaginaryI]"}]}], ",", RowBox[{"0.5`", "\[InvisibleSpace]", "-", RowBox[{"0.25`", " ", "\[ImaginaryI]"}]}], ",", RowBox[{"0.5`", "\[InvisibleSpace]", "+", RowBox[{"0.25`", " ", "\[ImaginaryI]"}]}], ",", "0.8623724356957945`", ",", "0.8623724356957945`", ",", RowBox[{"-", "0.36237243569579447`"}], ",", RowBox[{"-", "0.36237243569579447`"}]}], "}"}]], "Output", CellChangeTimes->{3.420210378847293*^9, 3.420212668896574*^9, 3.420218203893577*^9, 3.420219344985199*^9, 3.420365453972316*^9, 3.420508545010447*^9, 3.420561853048484*^9, 3.451832712255879*^9, 3.4523542852677927`*^9, 3.482969627285385*^9, 3.4831490217539387`*^9, 3.514720216822928*^9}] }, Open ]], Cell[TextData[{ "The only solution with \[Lambda] real and positive is \[Lambda] = (1 + ", Cell[BoxData[ FormBox[ SqrtBox["6"], TraditionalForm]]], ")/4 = 0.86237. At this value of \[Lambda], the 2-cycle becomes unstable \ and gives way to a 4-cycle." }], "Text"], Cell["\<\ We can test for 4-cycles by looking at the fourth iterated function f4(x) = \ f2(f2(x)):\ \>", "Text"], Cell[BoxData[ RowBox[{"Clear", "[", " ", RowBox[{"\[Lambda]", ",", " ", "x"}], "]"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"f4", "[", "x_", "]"}], " ", "=", " ", RowBox[{"f2", "[", RowBox[{"f2", "[", "x", "]"}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"256", " ", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}], " ", "x", " ", SuperscriptBox["\[Lambda]", "4"], " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"4", " ", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}], " ", "x", " ", "\[Lambda]"}]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"16", " ", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}], " ", "x", " ", SuperscriptBox["\[Lambda]", "2"], " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"4", " ", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}], " ", "x", " ", "\[Lambda]"}]}], ")"}]}]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"64", " ", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}], " ", "x", " ", SuperscriptBox["\[Lambda]", "3"], " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"4", " ", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}], " ", "x", " ", "\[Lambda]"}]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"16", " ", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}], " ", "x", " ", SuperscriptBox["\[Lambda]", "2"], " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"4", " ", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}], " ", "x", " ", "\[Lambda]"}]}], ")"}]}]}], ")"}]}]}], ")"}]}]], "Output", CellChangeTimes->{3.4202103788706017`*^9, 3.420212668994617*^9, 3.420218203956439*^9, 3.42021934504807*^9, 3.420365454023613*^9, 3.42050854507263*^9, 3.4205618531142673`*^9, 3.451832712437501*^9, 3.452354335267782*^9, 3.482969627320619*^9, 3.483149030955504*^9, 3.514720242626834*^9}] }, Open ]], Cell["\<\ We take \[Lambda] = 0.88 where above we found a length-4 cycle:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"\[Lambda]", " ", "=", " ", "0.88"}], ";"}]], "Input"], Cell["\<\ This length-4 cycle is confirmed because we find a fixed point of f4:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FixedPoint", "[", RowBox[{"f4", ",", " ", "0.5", ",", " ", RowBox[{"SameTest", " ", "\[Rule]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Abs", "[", RowBox[{"#1", " ", "-", " ", "#2"}], " ", "]"}], " ", "<", " ", RowBox[{"10", "^", RowBox[{"-", "11"}]}]}], "&"}], ")"}]}]}], " ", "]"}]], "Input"], Cell[BoxData["0.5120763618418722`"], "Output", CellChangeTimes->{3.420210379030034*^9, 3.420212669077282*^9, 3.420218203999905*^9, 3.420219345098359*^9, 3.420365454072399*^9, 3.420508545117304*^9, 3.42056185317402*^9, 3.451832712626225*^9, 3.4523543765322723`*^9, 3.4829696273541193`*^9, 3.483149036363421*^9, 3.514720250705247*^9}] }, Open ]], Cell[TextData[{ "This value for x is one of the 4 values on the 4-cycle of f to which the \ trajectory converges. Starting with different values for ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "0"], ","}], TraditionalForm]]], "we could converge to the other values of x.\n\nUsing ", StyleBox["Mathematica", FontSlant->"Italic"], ", we can determine where the stable fixed point of f4 (4-cycle of f) \ becomes unstable. Since this is too complicated to solve analytically, and \ since ", StyleBox["NSolve", FontWeight->"Bold"], " gives a large number of roots, I prefer to use ", StyleBox["FindRoot", FontWeight->"Bold"], ", giving a reasonable initial guess based on the results from the \ iterations." }], "Text", CellChangeTimes->{{3.420364644383482*^9, 3.420364645031419*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Lambda]", " ", "/.", " ", RowBox[{"FindRoot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"f4", "[", "x", "]"}], " ", "\[Equal]", " ", "x"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"f4", "'"}], "[", "x", "]"}], " ", "\[Equal]", " ", RowBox[{"-", "1"}]}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "0.88"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"\[Lambda]", ",", " ", "0.9"}], "}"}]}], "]"}]}]], "Input"], Cell[BoxData["0.8860225898879807`"], "Output", CellChangeTimes->{3.4523545073983927`*^9, 3.4829696273976793`*^9, 3.483149061570644*^9, 3.514720300398951*^9}] }, Open ]], Cell["\<\ For \[Lambda] > 0.886023 the 4-cycle gives way to an 8-cycle. We can also \ find where the higher order cycles become unstable, though the calculations \ become increasingly complicated and require quite accurate initial guesses \ for the parameters \[Lambda] and x. For, example, to determine where the \ 8-cycle becomes unstable I find\ \>", "Text", CellChangeTimes->{{3.420210900739583*^9, 3.4202109018435373`*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "\[Lambda]", "]"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"f8", "[", "x_", "]"}], " ", "=", " ", RowBox[{"f4", "[", RowBox[{"f4", "[", "x", "]"}], "]"}]}], " ", ";"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Lambda]", " ", "/.", " ", RowBox[{"FindRoot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"f8", "[", "x", "]"}], " ", "\[Equal]", " ", "x"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"f8", "'"}], "[", "x", "]"}], " ", "\[Equal]", " ", RowBox[{"-", "1"}]}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", "0.891"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"\[Lambda]", ",", "0.891"}], "}"}]}], "]"}]}]], "Input"], Cell[BoxData["0.8911018165238584`"], "Output", CellChangeTimes->{ 3.420210379589829*^9, 3.42021266953198*^9, 3.420218204304762*^9, 3.42021934537079*^9, 3.420365454339336*^9, 3.420508545407011*^9, 3.4205618535477667`*^9, 3.451832713467658*^9, 3.452354540110784*^9, 3.482969627630975*^9, 3.483149082157771*^9, {3.5147203117650423`*^9, 3.514720318206451*^9}}] }, Open ]], Cell[TextData[{ "To conclude this section, we have seen that if the results of successive \ iterations of the map suggest that there is convergence to a fixed point, or \ limit cycle of a certain length, it is useful to confirm this hypothesis with \ the ", StyleBox["FixedPoint", FontWeight->"Bold"], " command. We can also use the stability criterion |fn'(X)| < 1 (where fn(x) \ is the n-th iterate of f(x)) to determine where cycles of length n become \ unstable, using ", StyleBox["Mathematica", FontSlant->"Italic"], "'s commands, ", StyleBox["Solve", FontWeight->"Bold"], ", ", StyleBox["NSolve", FontWeight->"Bold"], " and ", StyleBox["FindRoot", FontWeight->"Bold"], "." }], "Text", CellChangeTimes->{{3.451832043865953*^9, 3.451832078843685*^9}, { 3.514720346473761*^9, 3.5147203467495747`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["General Behavior", "Section"], Cell[TextData[{ "We now investigate systematically the behavior as a function of \[Lambda]. \ It is convenient to define a function which will iterate the map n times. \ Furthermore, the ultimate behavior, such as fixed point or limit cycle, only \ appears after an initial number of iterations during which the x values are \ in the process of \"converging\" to the limiting behavior. The length of this \ \"initial transient\" increases as one approaches a bifurcation point where \ the period doubles. We therefore want our function to drop the first m, say, \ iterations and only print a list of the remaining n-m values. The former is \ accomplished by the ", StyleBox["Mathematica", FontSlant->"Italic"], " commnd ", StyleBox["Drop", FontWeight->"Bold"], " so we define" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"iterate", " ", "[", " ", RowBox[{"m_", ",", " ", "n_"}], "]"}], " ", ":=", " ", RowBox[{"Drop", " ", "[", " ", RowBox[{ RowBox[{"NestList", " ", "[", RowBox[{"f", ",", " ", "0.5", ",", " ", "n"}], "]"}], ",", " ", "m"}], "]"}]}]], "Input"], Cell[TextData[{ "We will use ", StyleBox["Mathematica", FontSlant->"Italic"], " commands to display the points in the (x, \[Lambda]) plane. We display a \ point by the command ", StyleBox["Point", FontWeight->"Bold"], ", so we define the following function:" }], "Text", CellChangeTimes->{{3.4205077122847033`*^9, 3.420507725444132*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"drawpt", "[", "y_", "]"}], " ", ":=", " ", RowBox[{"Point", "[", RowBox[{"{", RowBox[{"\[Lambda]", ",", " ", "y"}], "}"}], "]"}]}]], "Input"], Cell[TextData[{ "The result of a the command ", StyleBox["Point,", FontWeight->"Bold"], " and related commands, is a \"graphics element\". Graphics elements can be \ combined together with the ", StyleBox["Graphics ", FontWeight->"Bold"], " command to produce a \"graphics object\" which can then be plotted with \ the command ", StyleBox["Show", FontWeight->"Bold"], ".\n\nHere, we want to plot a point for each of the values of x generated by \ our function ", StyleBox["iterate", FontWeight->"Bold"], ". 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