(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 937358, 50451]*) (*NotebookOutlinePosition[ 938543, 50487]*) (* CellTagsIndexPosition[ 938499, 50483]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Chaotic Dynamical Systems Tools", "Title"], Cell["Bifurcation diagrams with Mathematica", "Subtitle"], Cell["Jean-Marc GINOUX", "Author"], Cell["\<\ P.R.O.T.E.E. Laboratory, I.U.T. of Toulon, University of South, \ B.P. 20132, 83957, La Garde Cedex, FRANCE ginoux@univ-tln.fr\ \>", "TextAboutAuthor"], Cell["Bruno ROSSETTO", "Author"], Cell["\<\ P.R.O.T.E.E. Laboratory, I.U.T. of Toulon, University of South, \ B.P. 20132, 83957, La Garde Cedex, FRANCE rossetto@univ-tln.fr\ \>", "TextAboutAuthor"], Cell[TextData[{ "The aim of this work is to present applets elaborated with ", StyleBox["Mathematica", FontSlant->"Italic"], " in order to study two or three-dimensional autonomous dynamical systems \ depending on parameters. As the parameters vary, the phase portrait of such \ systems also varies. So, the appearance of a topologically nonequivalent \ phase portrait under the variation of parameters is called a bifurcation. \ More precisely, the Hopf-bifurcation plays a very important role in the study \ of dynamical systems. Hence, the first applet allows to compute the Hopf \ bifurcation parameter value of two or three-dimensional autonomous dynamical \ systems with a high accuracy. As pointed out by Glass and Mackey, the \ construction of a bifurcation diagram is a good means of locating the \ signature of chaos in a dynamical system. The second applet makes it possible \ to construct the bifurcation diagram of such systems. Thus, chaotic \ behaviours such as period doubling cascade may be highlighted. Various \ examples of applications are proposed to illustrate the usefulness of these \ applets. " }], "Abstract"], Cell[CellGroupData[{ Cell["Introduction ", "SectionFirst"], Cell[TextData[{ "Characterizing the evolution of a given physical phenomenon such as \ population dynamics, for example, may be done thanks to a mathematical model \ which may consist of ", StyleBox["system of differential equations", FontSlant->"Italic"], " involving ", StyleBox["state", FontSlant->"Italic"], " variables and parameters. If the variables of such system do not depend \ explicitly on time, the system is said to be ", StyleBox["autonomous", FontSlant->"Italic"], ". Under certain assumptions defined below such system may be considered as \ ", StyleBox["autonomous dynamical system", FontSlant->"Italic"], ". As time does not occur explicitly in equations, solution of a system of \ differential equations may be projected in a space called ", StyleBox["phase-space", FontSlant->"Italic"], " in which the behaviour of the state variables is described. The plot of \ the solution in such space is called ", StyleBox["phase portrait", FontSlant->"Italic"], ". The ", StyleBox["bifurcation", FontSlant->"Italic"], " of a system of differential equation, i.e., of autonomous dynamical \ system is concerned with changes in the qualitative behaviour of its phase \ portrait as parameters vary and more precisely, when such a ", StyleBox["bifurcation parameter", FontSlant->"Italic"], " reaches a certain value, called ", StyleBox["critical value", FontSlant->"Italic"], ". Thus, bifurcation theory is of great importance in dynamical systems \ study because it indicates stability changes, structural changes in a system \ etc... So, plotting the solution of autonomous dynamical system according to \ the", StyleBox[" ", FontSlant->"Italic"], "bifurcation parameter leads to the construction of a ", StyleBox["bifurcation diagram.", FontSlant->"Italic"], " Such diagram provides knowledge on the behaviour of the solution: \ constant, periodic, nonperiodic or even chaotic as pointed out by Glass and \ Mackey. As there is many kinds of behaviour of solutions there is many kinds \ of bifurcations. In this work, we focus on ", StyleBox["Hopf bifurcation", FontSlant->"Italic"], " corresponding to periodic solutions and ", StyleBox["period doubling bifurcation ", FontSlant->"Italic"], "or ", StyleBox["period doubling cascade ", FontSlant->"Italic"], "which is one of the route to chaos for dynamical systems. " }], "Text"], Cell[TextData[{ StyleBox["In the following we consider a system of differential equations \ defined in a compact U included in", FormatType->StandardForm, FontFamily->"Times"], StyleBox[" ", FormatType->StandardForm], Cell[BoxData[ FormBox[ StyleBox[\(\[DoubleStruckCapitalR]\^n\), FontFamily->"Times New Roman"], TraditionalForm]]], StyleBox[":", FormatType->StandardForm, FontFamily->"Times"] }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(\(d X\&\[LongRightArrow]\)\/dt = \(\(\[ScriptCapitalF]\&\ \[LongRightArrow]\) \((X\&\[LongRightArrow], \[Mu])\)\(\ \)\)\)], \ "NumberedEquation", TextAlignment->Center, TextJustification->1], Cell[TextData[{ StyleBox["with", FormatType->StandardForm, FontFamily->"Times"], StyleBox["\:f35f", FormatType->StandardForm] }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ RowBox[{ RowBox[{\(X\&\[LongRightArrow]\), "=", RowBox[{\(\[InvisiblePrefixScriptBase]\(\([x\_1, x\_2, ... , x\_n]\)\^t\) \[Epsilon]\ U\), "\[Subset]", FormBox[ StyleBox[\(\[DoubleStruckCapitalR]\^n\), FontFamily->"Times New Roman"], "TraditionalForm"]}]}], ",", " ", RowBox[{\(\[Mu]\ \[Epsilon]\ V\), "\[Subset]", FormBox[ StyleBox[\(\[DoubleStruckCapitalR]\^p\), FontFamily->"Times New Roman"], "TraditionalForm"], " "}]}]], "Text", TextAlignment->Center, TextJustification->1], Cell[TextData[{ StyleBox["and", FormatType->StandardForm, FontFamily->"Times"], StyleBox["\:f35f", FormatType->StandardForm] }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ RowBox[{\(\(\[ScriptCapitalF]\&\[LongRightArrow]\) \ \((X\&\[LongRightArrow], \[Mu])\)\), "=", RowBox[{\(\(\([\(f\_1\) \((X\&\[LongRightArrow], \[Mu])\), \(f\_2\) \ \((X\&\[LongRightArrow], \[Mu])\), ... , \(f\_n\) \((X\&\[LongRightArrow], \ \[Mu])\)]\)\^t\) \[Epsilon]\ U\), "\[Subset]", FormBox[ StyleBox[\(\[DoubleStruckCapitalR]\^n\), FontFamily->"Times New Roman"], "TraditionalForm"]}]}]], "Text", TextAlignment->Center, TextJustification->1], Cell[TextData[{ StyleBox["The vector ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ \(\[ScriptCapitalF]\&\[LongRightArrow]\)]], " ", StyleBox["defines a velocity vector field in U whose components ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ \(TraditionalForm\`f\_i\)]], " which ", StyleBox["are supposed to be continuous and infinitely derivable with \ respect to all ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ StyleBox[ SubscriptBox["x", StyleBox["i", FontSlant->"Plain"]], FontFamily->"Times New Roman"]], TextAlignment->Center, FontSlant->"Italic"], " and ", StyleBox["t", FontSlant->"Italic"], ", i.e., ", StyleBox["are ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ \(TraditionalForm\`C\^\[Infinity]\)]], " functions ", StyleBox["in U and with values included in ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["\[DoubleStruckCapitalR]", FormatType->StandardForm], StyleBox[", check the assumptions of the Cauchy-Lipschitz theorem. For more \ details, see for example Coddington & Levinson, [1955]. A solution of this \ system is an integral curve ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ \(X\&\[LongRightArrow]\)], TextAlignment->Center, TextJustification->0], "(t)", StyleBox[" tangent to ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ \(\[ScriptCapitalF]\&\[LongRightArrow]\)], TextAlignment->Center, TextJustification->0], " ", StyleBox["whose values define the ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["states", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox[" of the ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["dynamical system", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox[" described by the Eq. (1). Since none of the components ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ \(TraditionalForm\`f\_i\)]], " ", StyleBox["of the velocity vector field depends here explicitly on time, the \ system is said to be ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["autonomous", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox[".", FormatType->StandardForm, FontFamily->"Times"] }], "Text", TextAlignment->Left, TextJustification->1] }, Open ]], Cell[CellGroupData[{ Cell["Hopf bifurcation", "SectionFirst"], Cell[TextData[{ "Let's consider a dynamical system defined under the same conditions as \ above and having a fixed point ", Cell[BoxData[ \(\(X\&\[LongRightArrow]\) = \(\(X\&\[LongRightArrow]\)\^\(\(\ \ \)\(*\)\)\) \((\[Mu])\)\)]], "and let's make the following assumptions. " }], "Text"], Cell[TextData[{ "The jacobian functional matrix associated with this system has a pair of \ complex-conjugate eigenvalues ", Cell[BoxData[ \(\[Sigma]\_\(1, 2\) = \[Alpha] \((\[Mu])\) + \(\(\[ImaginaryI]\)\(\ \)\ \(\[Omega]\) \((\[Mu])\)\(\ \)\)\)]], "such that:\n\n- for a critical value of the bifurcation parameter ", Cell[BoxData[ \(\[Mu] = \[Mu]\_C\)]], ", ", Cell[BoxData[ \(\[Alpha] \((\[Mu]\_C)\) = 0\)]], " and ", Cell[BoxData[ \(\((d\[Alpha]\/d\[Mu])\)\_\(\[Mu]\ = \ \[Mu]\_C\) \[NotEqual] 0\)]], "\n- the n - 2 others eigenvalues evaluated in ", Cell[BoxData[ \(\[Mu]\)]], " have their real parts strictly negative. \n\nIf these assumptions are \ checked, then ", Cell[BoxData[ \(\((\(X\&\[LongRightArrow]\^\(\(\ \)\(*\)\)\) \((\[Mu]\_C)\), \ \[Mu]\_C)\)\)]], "is a Hopf bifurcation point which leads to the appearance, from the \ equilibrium state ", Cell[BoxData[ \(\(X\&\[LongRightArrow]\^\(\(\ \)\(*\)\)\) \((\[Mu]\_C)\)\)]], ", of a ", StyleBox["limit cycle", FontSlant->"Italic"], ".", " " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Computation of the Hopf bifurcation parameter value", "Section"], Cell["\<\ Using the following procedure it is possible to compute the Hopf \ bifurcation parameter value of two or three-dimensional dynamical systems. \ Since the two-dimensional procedure may be obtained by a simple reduction, \ the three-dimensional procedure is only presented.\ \>", "Text"], Cell[CellGroupData[{ Cell["Dynamical system", "Subsection"], Cell["\<\ In dimension three the autonomous dynamical system (1) may be \ defined under the same conditions as:\ \>", "Text"], Cell[BoxData[ \(\(d X\&\[LongRightArrow]\)\/dt = \(\(\[ScriptCapitalF]\&\ \[LongRightArrow]\) \((X\&\[LongRightArrow])\)\(\ \)\)\)], "NumberedEquation",\ TextAlignment->Center, TextJustification->1], Cell[TextData[{ StyleBox["with", FormatType->StandardForm, FontFamily->"Times"], StyleBox["\:f35f", FormatType->StandardForm] }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ RowBox[{ RowBox[{\(X\&\[LongRightArrow]\), "=", RowBox[{\(\[InvisiblePrefixScriptBase]\(\([x, y, z]\)\^t\) \[Epsilon]\ U\), "\[Subset]", FormBox[ StyleBox[\(\[DoubleStruckCapitalR]\^3\), FontFamily->"Times New Roman"], "TraditionalForm"]}]}], ",", " ", RowBox[{\(\((\[Lambda], \[Mu], \[Rho])\)\ \[Epsilon]\ V\), "\[Subset]", FormBox[ StyleBox[\(\[DoubleStruckCapitalR]\^3\), FontFamily->"Times New Roman"], "TraditionalForm"]}]}]], "Text", TextAlignment->Center, TextJustification->1], Cell[TextData[{ StyleBox["and", FormatType->StandardForm, FontFamily->"Times"], StyleBox["\:f35f", FormatType->StandardForm] }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ RowBox[{\(\(\[ScriptCapitalF]\&\[LongRightArrow]\) \ \((X\&\[LongRightArrow])\)\), "=", RowBox[{\(\(\([f \((X\&\[LongRightArrow], \[Lambda])\), g \((X\&\[LongRightArrow], \[Mu])\), h \((X\&\[LongRightArrow], \[Rho])\)]\)\^t\) \[Epsilon]\ U\), "\[Subset]", FormBox[ StyleBox[\(\[DoubleStruckCapitalR]\^3\), FontFamily->"Times New Roman"], "TraditionalForm"]}]}]], "Text", TextAlignment->Center, TextJustification->1], Cell["\<\ where \[Lambda], \[Mu] and \[Rho] are real parameters.\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(\(d X\&\[LongRightArrow]\)\/dt\), RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(\(d\ x\)\/dt\)}, {\(\(d\ y\)\/dt\[IndentingNewLine] \(d\ z\)\/dt\)} }], "\[NegativeThinSpace]", ")"}]}], "=", RowBox[{\(\[ScriptCapitalF]\&\[LongRightArrow]\), RowBox[{"(", GridBox[{ {\(f \((x, y, z, \[Lambda])\)\)}, { RowBox[{\(g \((x, y, z, \[Mu])\)\), "\[IndentingNewLine]", RowBox[{"h", RowBox[{"(", RowBox[{"x", ",", "y", ",", "z", StyleBox[",", FontFamily->"Courier", FontSize->14, FontWeight->"Bold", FontSlant->"Plain"], StyleBox["\[Rho]", FontFamily->"Courier", FontSize->14, FontWeight->"Bold", FontSlant->"Plain"]}], StyleBox[")", FontFamily->"Courier", FontSize->14, FontWeight->"Bold", FontSlant->"Plain"]}]}]}]} }], ")"}]}]}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Bifurcation parameter", "Subsection"], Cell["\<\ Choose a bifurcation parameter between the set of parameters of the \ dynamical system (2). \[Lambda] for example.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Fixed points", "Subsection"], Cell["\<\ Choosing \[Lambda] as the Hopf bifurcation parameter and while \ \"freezing\" all the others, i.e., \[Mu] and \[Rho], each fixed point may be \ expressed according to the parameter \[Lambda]. \ \>", "Text"], Cell[BoxData[ \(Solve[{f \((x, y, z, \[Lambda])\) == 0, g \((x, y, z, \[Mu])\) == 0, h \((x, y, z, \[Rho])\) == 0}, {x, y, z}]\)], "Input"], Cell[TextData[{ "Choose the fixed point in the vicinity of which the Hopf bifurcation may \ occurs. Let's call ", StyleBox["I", FontSlant->"Italic"], " this fixed point expressed according to the parameter \[Lambda]." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Functional jacobian matrix", "Subsection"], Cell[TextData[{ StyleBox["Since the functions", FormatType->StandardForm, FontFamily->"Times"], StyleBox[" ", FormatType->StandardForm], Cell[BoxData[ \(TraditionalForm\`f\_i\)]], " ", StyleBox["are supposed to be ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ \(TraditionalForm\`\(\(C\^\[Infinity]\)\(\ \)\)\)]], "functions ", StyleBox["in a compact U included in ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^n\)]], StyleBox[", it is possible to compute", FormatType->StandardForm, FontFamily->"Times"], " the functional ", StyleBox["jacobian matrix ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["\[ScriptCapitalJ]", FormatType->StandardForm], StyleBox[" of system (1) or (2),", FormatType->StandardForm, FontFamily->"Times"], " ", StyleBox["defined by: ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["\[ScriptCapitalJ] ", FormatType->StandardForm], StyleBox["= ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ StyleBox[\(\(d \( \[ScriptCapitalF]\&\[LongRightArrow]\)\)\/\(d \( X\&\[LongRightArrow]\)\)\), FontFamily->"Times New Roman", FontWeight->"Plain"]]], ". The following procedure provides each elements of the functional \ jacobian matrix." }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(a\_11 = Simplify[D[f, x]]\)], "Input"], Cell[BoxData[ \(a\_12 = Simplify[D[f, y]]\)], "Input"], Cell[BoxData[ \(a\_13 = Simplify[D[f, z]]\)], "Input"], Cell[BoxData[ \(a\_21 = Simplify[D[g, x]]\)], "Input"], Cell[BoxData[ \(a\_22 = Simplify[D[g, y]]\)], "Input"], Cell[BoxData[ \(a\_23 = Simplify[D[g, z]]\)], "Input"], Cell[BoxData[ \(a\_31 = Simplify[D[h, x]]\)], "Input"], Cell[BoxData[ \(a\_32 = Simplify[D[h, y]]\)], "Input"], Cell[BoxData[ \(a\_33 = Simplify[D[h, z]]\)], "Input"], Cell[BoxData[ \(J = {{a\_11, a\_12, a\_13}, {a\_21, a\_22, a\_23}, {a\_31, a\_32, a\_33}}\)], "Input"], Cell[BoxData[ \(MatrixForm[J]\)], "Input"], Cell[TextData[{ "Plugging expression of the fixed point ", StyleBox["I", FontSlant->"Italic"], " in the functional jacobian matrix and while using the Mathematica \ function Eigenvalues, it is possible to compute the eigenvalues according to \ the chosen parameter ", "\[Lambda]", ". Let's denote Sp (for spectrum) these eigenvalues." }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox["Sp", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Plain", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], ":=", \(Eigenvalues[J]\)}], ";"}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Graphical estimation of the Hopf bifurcation parameter value", \ "Subsection"], Cell["\<\ In order to find the range of this parameter, the real part of one \ of the complex-conjugate eigenvalue is plotted (with Plot function) according \ to the parameter chosen (\[Lambda]). Then, the location of the points where \ this function vanishes provides an estimation of the Hopf bifurcation \ parameter value.\ \>", "Text"], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Re", "[", RowBox[{ StyleBox["Sp", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Plain", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], "[", \([2]\), "]"}], "]"}], ",", \({\[Lambda], x\_min, x\_max}\), ",", \(PlotRange \[Rule] {y\_min, y\_max}\)}], "]"}]], "Input"], Cell[TextData[{ "It provides the approximate value: \[Lambda]", Cell[BoxData[ \(TraditionalForm\`\_estimate\)]], " " }], "Text"], Cell["\<\ To confirm this, the other real part of the other complex-conjugate \ eigenvalue is also plotted according to the parameter chosen (\[Lambda]). \ \ \>", "Text"], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Re", "[", RowBox[{ StyleBox["Sp", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Plain", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], "[", \([3]\), "]"}], "]"}], ",", \({\[Lambda], x\_min, x\_max}\), ",", \(PlotRange \[Rule] {y\_min, y\_max}\)}], "]"}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Numerical computation of the Hopf bifurcation parameter value\ \>", \ "Subsection"], Cell["\<\ Then, the Mathematica function: FindRoot, which uses a damped \ Newton's method, the secant method and Brent's method, provides a numerical \ computation of the Hopf bifurcation parameter value with a high accuracy \ depending on the choice of the operator and on the WorkingPrecision.\ \>", \ "Text"], Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"Re", "[", RowBox[{ StyleBox["Sp", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Plain", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], "[", \([2]\), "]"}], "]"}], "\[Equal]", "0"}], ",", RowBox[{"{", RowBox[{"\[Lambda]", ",", RowBox[{"\[Lambda]", FormBox[\(\_estimate\), "TraditionalForm"]}]}], "}"}]}], "]"}]], "Input"], Cell[TextData[{ "This provides the Hopf bifurcation parameter value starting from the \ vicinity of its graphical estimation: ", Cell[BoxData[ RowBox[{" ", RowBox[{ StyleBox["\[Lambda]", FontWeight->"Plain"], FormBox[\(\_estimate\), "TraditionalForm"]}]}]]] }], "Text"], Cell["\<\ Then, it can be confirmed also for the other real part of the other \ complex conjugate eigenvalue. \ \>", "Text"], Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"Re", "[", RowBox[{ StyleBox["Sp", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Plain", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], "[", \([3]\), "]"}], "]"}], "\[Equal]", "0"}], ",", RowBox[{"{", RowBox[{"\[Lambda]", ",", RowBox[{"\[Lambda]", FormBox[\(\_estimate\), "TraditionalForm"]}]}], "}"}]}], "]"}]], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Example: Hopf bifurcation in the Lorenz attractor", "Section"], Cell["\<\ The purpose of the model established by Edward Lorenz [1963] was in \ the beginning to analyze the impredictible behaviour of weather. After having \ developed non-linear partial derivative equations starting from the thermal \ equation and Navier-Stokes equations, Lorenz truncated them to retain only \ three modes. The most widespread form of the Lorenz model is as \ follows:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(V\&\[RightArrow]\), RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(\(d\ x\)\/dt\)}, {\(\(d\ y\)\/dt\[IndentingNewLine] \(d\ z\)\/dt\)} }], "\[NegativeThinSpace]", ")"}]}], "=", RowBox[{ RowBox[{\(\[ScriptCapitalF]\&\[LongRightArrow]\), RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(f \((x, y, z, \[Sigma])\)\)}, {\(\(\ \)\(g \((x, y, z, r)\)\[IndentingNewLine] h \((x, y, z, \[Beta])\)\)\)} }], "\[NegativeThinSpace]", ")"}]}], "=", RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(\[Sigma] \((y - x)\)\)}, {\(\(-x\)\ z + r\ x - y\[IndentingNewLine] x\ y - \[Beta]\ z\)} }], "\[NegativeThinSpace]", ")"}]}]}]], "Input", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "with \[Sigma], r, and \[Beta] are ", StyleBox["real", FormatType->StandardForm, FontFamily->"Times"], " parameters: \[Sigma] = 10, ", Cell[BoxData[ RowBox[{ StyleBox["\[Beta]", FontFamily->"Courier", FontSize->12, FontSlant->"Plain", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], "=", \(8\/3\)}]], TextAlignment->Center, TextJustification->0], ", ", "r = 28" }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ StyleBox["where the functions ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["f, g ", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox["and", FormatType->StandardForm, FontFamily->"Times"], StyleBox[" h", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox[" are infinitely derivable with respect to all ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ StyleBox[\(x\_i\), FontFamily->"Times New Roman"]], TextAlignment->Center, TextJustification->0, FontSlant->"Italic"], StyleBox[", and ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["t", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox[", i.e., are ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ \(TraditionalForm\`C\^\[Infinity]\)]], StyleBox["functions in a compact U included in ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^3\)]], " ", StyleBox["and with values in ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["\[DoubleStruckCapitalR].", FormatType->StandardForm], StyleBox[" Lorenz model has been extensively studied and the Hopf \ bifurcation parameter value ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["r", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox[" in the vicinity of the fixed points labelled ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ \(\(C\^\[PlusMinus]\) \((\(\[PlusMinus]\@\(\[Beta] \((r - 1)\)\)\), \(\ \[PlusMinus]\@\(\[Beta] \((r - 1)\)\)\), r - 1)\)\)]], "is well-known: " }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(r = \(r\_C = \[Sigma] \( \[Sigma] + \[Beta] + 3\)\/\(\[Sigma] - \[Beta] \ - 1\) \[TildeEqual] 24.7368\)\)], "Input"], Cell["So, let's apply the above procedure.", "Text"], Cell[BoxData[ \(Solve[{\[Sigma] \((y - x)\) == 0, \(-x\)\ z + r\ x - y == 0, x\ y - \[Beta]\ z == 0}, {x, y, z}]\)], "Input"], Cell[BoxData[ \({{y \[Rule] 0, z \[Rule] 0, x \[Rule] 0}, {y \[Rule] \(-\@\(\(-\[Beta]\) + r\ \[Beta]\)\), z \[Rule] \(-1\) + r, x \[Rule] \(-\@\(\(-\[Beta]\) + r\ \[Beta]\)\)}, {y \[Rule] \@\(\(-\ \[Beta]\) + r\ \[Beta]\), z \[Rule] \(-1\) + r, x \[Rule] \@\(\(-\[Beta]\) + r\ \[Beta]\)}}\)], "Text"], Cell[TextData[{ "Fixing parameters \[Sigma] and \[Beta], one seeks the Hopf bifurcation \ parameter value ", StyleBox["r", FontSlant->"Italic"], " in the vicinity of any fixed point ", Cell[BoxData[ \(\(C\^\[PlusMinus]\)\)]], ". Let's choose arbitrary ", Cell[BoxData[ \(\(C\^-\)\)]], " and compute the functional jacobian matrix.", " " }], "Text"], Cell[BoxData[ RowBox[{"J", "=", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-10\), "10", "0"}, {\(r - z\), \(-1\), \(-x\)}, {"y", "x", \(\(-8\)/3\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}]], "Input"], Cell[TextData[{ "Evaluated at ", Cell[BoxData[ \(\(C\^-\)\)]], " the functional jacobian matrix takes the form:" }], "Text"], Cell[BoxData[ RowBox[{"J", "=", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-10\), "10", "0"}, {"1", \(-1\), \(2\ \@\(2\/3\)\ \@\(\(-1\) + r\)\)}, {\(\(-2\)\ \@\(2\/3\)\ \@\(\(-1\) + r\)\), \(\(-2\)\ \@\(2\/3\)\ \ \@\(\(-1\) + r\)\), \(-\(8\/3\)\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}]], "Input"], Cell[TextData[{ "Then, it is possible to compute its eigenvalues according to the chosen \ parameter ", 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Using FindRoot function provides a \ numerical computation of the Hopf bifurcation value ", StyleBox["r", FontSlant->"Italic"], "." }], "Text"], Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"Re", "[", RowBox[{ StyleBox["Sp", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Plain", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], "[", \([2]\), "]"}], "]"}], "\[Equal]", "0"}], ",", \({r, 24}\)}], "]"}]], "Input"], Cell[BoxData[ \({r \[Rule] 24.736842105263143`}\)], "Input"], Cell["Of course the other provides the same result.", "Text"], Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"Re", "[", RowBox[{ StyleBox["Sp", FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Plain", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], "[", \([3]\), "]"}], "]"}], "\[Equal]", "0"}], ",", \({r, 24}\)}], "]"}]], "Input"], Cell[BoxData[ \({r \[Rule] 24.736842105263143`}\)], "Input"], Cell[TextData[{ "Both values of ", StyleBox["r", FontSlant->"Italic"], " are corresponding to the one theorically computed." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Bifurcation diagram", "SectionFirst"], Cell[TextData[{ "Since it has been established [Ruelle ", StyleBox["et al.", FontSlant->"Italic"], ", 1971] that chaos may occur in three-dimensional dynamical systems, \ analytical methods such as the existence of Shil'nikov orbit [Shil'nikov, \ 1965] and numerical tools such as the construction of a bifurcation diagram \ [Glass ", StyleBox["et al.", FontSlant->"Italic"], ", 1988] have been ", "developed", ". Let's consider as previously, a three-dimensional autonomous dynamical \ system defined under the same conditions as above. " }], "Text"], Cell[BoxData[ \(\(d X\&\[LongRightArrow]\)\/dt = \(\(\[ScriptCapitalF]\&\ \[LongRightArrow]\) \((X\&\[LongRightArrow])\)\(\ \)\)\)], "NumberedEquation",\ TextAlignment->Center, TextJustification->1], Cell[TextData[{ StyleBox["with", FormatType->StandardForm, FontFamily->"Times"], StyleBox["\:f35f", FormatType->StandardForm] }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ RowBox[{ RowBox[{\(X\&\[LongRightArrow]\), "=", RowBox[{\(\[InvisiblePrefixScriptBase]\(\([x, y, z]\)\^t\) \[Epsilon]\ U\), "\[Subset]", FormBox[ StyleBox[\(\[DoubleStruckCapitalR]\^3\), FontFamily->"Times New Roman"], "TraditionalForm"]}]}], ",", " ", RowBox[{\(\((\[Lambda], \[Mu], \[Rho])\)\ \[Epsilon]\ V\), "\[Subset]", FormBox[ StyleBox[\(\[DoubleStruckCapitalR]\^3\), FontFamily->"Times New Roman"], "TraditionalForm"]}]}]], "Text", TextAlignment->Center, TextJustification->1], Cell[TextData[{ StyleBox["and", FormatType->StandardForm, FontFamily->"Times"], StyleBox["\:f35f", FormatType->StandardForm] }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ RowBox[{\(\(\[ScriptCapitalF]\&\[LongRightArrow]\) \ \((X\&\[LongRightArrow])\)\), "=", RowBox[{\(\(\([f \((X\&\[LongRightArrow], \[Lambda])\), g \((X\&\[LongRightArrow], \[Mu])\), h \((X\&\[LongRightArrow], \[Rho])\)]\)\^t\) \[Epsilon]\ U\), "\[Subset]", FormBox[ StyleBox[\(\[DoubleStruckCapitalR]\^3\), FontFamily->"Times New Roman"], "TraditionalForm"]}]}]], "Text", TextAlignment->Center, TextJustification->1], Cell["\<\ where \[Lambda], \[Mu] and \[Rho] are real parameters.\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(\(d X\&\[LongRightArrow]\)\/dt\), RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(\(d\ x\)\/dt\)}, {\(\(d\ y\)\/dt\[IndentingNewLine] \(d\ z\)\/dt\)} }], "\[NegativeThinSpace]", ")"}]}], "=", RowBox[{\(\[ScriptCapitalF]\&\[LongRightArrow]\), RowBox[{"(", GridBox[{ {\(f \((x, y, z, \[Lambda])\)\)}, { RowBox[{\(g \((x, y, z, \[Mu])\)\), "\[IndentingNewLine]", RowBox[{"h", RowBox[{"(", RowBox[{"x", ",", "y", ",", "z", StyleBox[",", FontFamily->"Courier", FontSize->14, FontWeight->"Bold", FontSlant->"Plain"], StyleBox["\[Rho]", FontFamily->"Courier", FontSize->14, FontWeight->"Bold", FontSlant->"Plain"]}], StyleBox[")", FontFamily->"Courier", FontSize->14, FontWeight->"Bold", FontSlant->"Plain"]}]}]}]} }], ")"}]}]}]], "Input"], Cell[TextData[{ "Choose the bifurcation parameter (\[Lambda] for example) between the set \ of parameters of the dynamical system and \"freeze\" all the others (\[Mu] \ and \[Rho]). In order to construct a bifurcation diagram, the system is \ numerically integrated (with NDSolve function) according to the bifurcation \ parameter \[Lambda]. Then, each variable x[t], y[t] and z[t], solution of \ this dynamical system (3) may be numerically computed according to the \ bifurcation parameter \[Lambda]. A bifurcation diagram is the plot of the \ successive maxima of one of these variables (z[t], for example) as a function \ of the bifurcation parameter \[Lambda]. The following applet ", "developed", " by Eric Javoy, computer scientist, is based on dichotomy principle and \ uses a follower which is \"tracking\" the curve until it reaches its maximum. \ This applet provides the bifurcation diagram of a three-dimensional dynamical \ system. It plots the maximum ", Cell[BoxData[ \(Z\_max\)]], " of the solution of the third variable Z of this dynamical system in \ function of the bifurcation parameter chosen ", Cell[BoxData[ \(\[Lambda]\)]], ". The bifurcation parameter \[Lambda] is in the range: ", Cell[BoxData[ \(\[Lambda]\_min\ \[LessEqual] \ \[Lambda]\ \[LessEqual] \ \ \[Lambda]\_max\)]], ". The step (= Pas\[Lambda]) of evaluation may be adjusted in function of \ the region of the diagram: ", Cell[BoxData[ \(\[Lambda] < \[Lambda]\_int1, 0.0001, \((\[Lambda] \[GreaterEqual] \[Lambda]\_int1)\) && \((\[Lambda] \ < \[Lambda]\_int2)\), 0.0001, \[Lambda] \[GreaterEqual] \[Lambda]\_int2, 0.0001\)]], ".\nThe dynamical system is first integrated with a classical method of \ NDSolve between the range of time: ", Cell[BoxData[ \({t, t\_0, t\_max}\)]], ".\nThen, the parameter is varying with the chosen step (= Pas\[Lambda]) \ and Z is evaluated in order to find its maximum between ", Cell[BoxData[ \(t\_min\)]], " and ", Cell[BoxData[ \(t\_max\)]], " after elimination of the transient behaviour." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Applet by ", StyleBox["Eric Javoy", FontSlant->"Italic"], ", computer scientist" }], "Section"], Cell[BoxData[{\(Clear["\"]\), "\n", RowBox[{"Off", StyleBox["[", "MR"], StyleBox[\(General::spell1\), "MR"], StyleBox["]", "MR"]}], "\[IndentingNewLine]", StyleBox[\($RecursionLimit = Infinity;\), "MR"]}], "Input"], Cell[BoxData[{\(\(\(\[Mu]\)\(=\)\);\), "\[IndentingNewLine]", RowBox[{ StyleBox[\(\(\(\[Rho]\)\(=\)\);\), FontFamily->"Courier", FontSize->14, FontWeight->"Bold", FontSlant->"Plain"], "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Pas\[Lambda] = 0.001;\), "\[IndentingNewLine]", \(TableauPlot = {};\)}], "Input"], Cell[BoxData[ RowBox[{"For", "[", RowBox[{\(\[Lambda] = \[Lambda]\_min\), ",", \(\[Lambda] \[LessEqual] \[Lambda]\_max\), ",", " ", \(\[Lambda] = \[Lambda] + Pas\[Lambda]\), ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"solution", "=", RowBox[{ "NDSolve", "[", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{\(\(x'\)[t]\), "==", RowBox[{ StyleBox["f", FontFamily->"Courier New", FontWeight->"Bold"], StyleBox[\((x, y, z, \[Lambda])\), FontFamily->"Courier New", FontSize->14, FontWeight->"Bold"]}]}], ",", \(x[0] \[Equal] x\_0\), ",", "\[IndentingNewLine]", RowBox[{\(\(y'\)[t]\), "==", StyleBox[\(g \((x, y, z, \[Mu])\)\), FontFamily->"Courier", FontSize->14, FontWeight->"Bold"]}], ",", \(y[0] \[Equal] y\_0\), ",", "\[IndentingNewLine]", RowBox[{\(\(z'\)[t]\), "==", RowBox[{ StyleBox["h", FontFamily->"Courier", FontSize->14, FontWeight->"Bold"], RowBox[{ StyleBox["(", FontFamily->"Courier", FontSize->14, FontWeight->"Bold"], StyleBox[ RowBox[{ StyleBox["x", FontFamily->"Courier", FontSize->14, FontWeight->"Bold"], StyleBox[",", FontFamily->"Courier", FontSize->14, FontWeight->"Bold"], StyleBox["y", FontFamily->"Courier", FontSize->14, FontWeight->"Bold"], StyleBox[",", FontFamily->"Courier", FontSize->14, FontWeight->"Bold", FontSlant->"Plain"], StyleBox["z", FontFamily->"Courier", FontSize->14, FontWeight->"Bold", FontSlant->"Plain"], StyleBox[",", FontFamily->"Courier", FontSize->14, FontWeight->"Bold", FontSlant->"Plain"], StyleBox["\[Rho]", FontFamily->"Courier", FontSize->14, FontWeight->"Bold", FontSlant->"Plain"]}], FontWeight->"Bold"], StyleBox[")", FontFamily->"Courier", FontSize->14, FontWeight->"Bold", FontSlant->"Plain"]}]}]}], ",", \(z[0] \[Equal] z\_0\)}], "}"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", \({x, y, z}\), ",", \({t, t\_0, t\_max}\), ",", \(Method \[Rule] ExplicitRungeKutta\)}], "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", \(Pas\[Lambda] = Which[\[Lambda] < \[Lambda]\_int1, 0.0001, \((\[Lambda] \[GreaterEqual] \[Lambda]\_int1)\) && \((\ \[Lambda] < \[Lambda]\_int2)\), 0.0001, \[Lambda] \[GreaterEqual] \[Lambda]\_int2, 0.0001]\), ";", "\[IndentingNewLine]", "\[IndentingNewLine]", \(Print["\<\[Lambda]=\>", \[Lambda]]\), ";", "\[IndentingNewLine]", \(Pas = .05\), ";", "\[IndentingNewLine]", \(Tendance = \ 0\), ";", "\[IndentingNewLine]", \(BorneBasse = t\_min\), ";", "\[IndentingNewLine]", \(BorneHaute = t\_max\), ";", "\[IndentingNewLine]", \(Precedent = \(\(Evaluate[{z[BorneBasse]} /. solution]\)[\([1]\)]\)[\([1]\)]\), ";", "\[IndentingNewLine]", \(Tendance = Which[Precedent >= \(\(Evaluate[{z[BorneBasse + Pas]} /. solution]\)[\([1]\)]\)[\([1]\)], \(-1\), Precedent < \(\(Evaluate[{z[BorneBasse + Pas]} /. solution]\)[\([1]\)]\)[\([1]\)], 1]\), ";", "\[IndentingNewLine]", \(For\ [Valeur = BorneBasse + Pas, \ Valeur \[LessEqual] BorneHaute, Valeur = Valeur + Pas, \[IndentingNewLine]\ \ \ \ \ \ \ \ Actuel = \ \(\(Evaluate[{z[Valeur]} /. solution]\)[\([1]\)]\)[\([1]\)]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ If[\((Actuel >= Precedent)\), Goto\ [NonMaximum]]; \[IndentingNewLine]\ \ \ \ \ \ \ \ If[ Actuel < Precedent\ && \ Tendance \[Equal] \(-1\), Goto[ActualisePrecedent]]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ TableauPlot = Append[TableauPlot, {N[FromDigits[RealDigits[\[Lambda]]], 6], N[FromDigits[ RealDigits[\(\(Evaluate[{z[Valeur - Pas]} /. solution]\)[\([1]\)]\)[\([1]\)]]], 6]}]; \[IndentingNewLine]Tendance = \(-1\); \ \[IndentingNewLine]\ \ \ \ \ \ \ \ Goto\ [ ActualisePrecedent]; \ \ \ \ \ \ \ \[IndentingNewLine]Label[ NonMaximum]; \[IndentingNewLine]\ \ \ \ \ \ \ \ Tendance = 1; \[IndentingNewLine]\ \ \ \ \ \ \ \ Goto[ ActualisePrecedent]; \ \ \ \ \[IndentingNewLine]Label[ RechercheFine]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ Print["\"; Valeur]; \ \ \ \[IndentingNewLine]Label[ PremieresValeurs]; \[IndentingNewLine]\ \ \ \ \ \ \ \ If[ Valeur != BorneBasse, Tendance = Which[Actuel < Precedent, \(-1\), Actuel\ \[GreaterEqual] Precedent, 1]]; \[IndentingNewLine]Label[ ActualisePrecedent]; \ \ \ \ \ \ \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ Precedent = Actuel; \[IndentingNewLine]Label[ FinTraitement];\[IndentingNewLine]\ \ \ \ \ \ \ ]\)}]}], "\[IndentingNewLine]", "]"}]], "Input"], Cell["The following function plots the bifurcation diagram.", "Text"], Cell[BoxData[ \(ListPlot[TableauPlot, PlotStyle \[Rule] {PointSize[0.005], Hue[ .6]}, ImageSize \[Rule] {400, 400}, PlotRange \[Rule] {{\[Lambda]\_min, \[Lambda]\_max}, {z\_a, z\_b}}, Frame \[Rule] True, FrameLabel -> {"\<\[Lambda] \>", \*"\"\<\!\(Z\_max\)\>\""}, RotateLabel \[Rule] False]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Example: bifurcation diagram for the Volterra-Gause model", "Section"], Cell[TextData[{ "This ", StyleBox["three-dimensional predator-prey model, ", TextAlignment->Left, TextJustification->1], StyleBox["elaborated by", FormatType->StandardForm, FontFamily->"Times"], " Ginoux ", StyleBox["et al.", FontSlant->"Italic"], " [2005], consisted of a prey, a predator and top-predator has been named \ Volterra-Gause because it combines the original model of V.Volterra [1926] \ incorporating a logisitic limitation of P.F.Verhulst [1838] type on the \ growth of the prey and a limitation of G.F.Gause [1935] type on the intensity \ of the predation of the predator on the prey and of top-predator on the \ predator." }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(V\&\[RightArrow]\), RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(\(d\ x\)\/dt\)}, {\(\(d\ y\)\/dt\[IndentingNewLine] \(d\ z\)\/dt\)} }], "\[NegativeThinSpace]", ")"}]}], "=", RowBox[{ RowBox[{\(\[ScriptCapitalF]\&\[LongRightArrow]\), RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(f \((x, y, z, \[Xi])\)\)}, {\(\(\ \)\(g \((x, y, z, \[Delta]\_1)\)\[IndentingNewLine] h \((x, y, z, \[CurlyEpsilon], \[Delta]\_2)\)\)\)} }], "\[NegativeThinSpace]", ")"}]}], "=", RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(\(1\/\[Xi]\) \((x \((1 - x)\) - x\^\(1\/2\)\ y)\)\)}, {\(\(-\[Delta]\_1\)\ y + \(x\^\(1\/2\)\) y - \(y\^\(1\/2\)\) z\[IndentingNewLine] \[CurlyEpsilon]\ z \((y\^\(1\/2\) - \[Delta]\_2)\)\)} }], "\[NegativeThinSpace]", ")"}], StyleBox[" ", ShowAutoStyles->True]}]}]], "Input", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "with \[Xi], \[CurlyEpsilon], ", Cell[BoxData[ \(\[Delta]\_1\)]], " and ", Cell[BoxData[ \(\[Delta]\_2\)]], " are ", StyleBox["real", FormatType->StandardForm, FontFamily->"Times"], " parameters: ", Cell[BoxData[ \(\[Xi] = 0.866\)]], ", ", Cell[BoxData[ \(\[CurlyEpsilon] = 1.428\)]], ", ", Cell[BoxData[ \(\[Delta]\_1 = 0.577\)]], ", ", Cell[BoxData[ \(\[Delta]\_2 = 0.376\)]], StyleBox[" and where the functions ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["f, g ", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox["and", FormatType->StandardForm, FontFamily->"Times"], StyleBox[" h", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox[" are infinitely derivable with respect to all ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ StyleBox[\(x\_i\), FontFamily->"Times New Roman"]], TextAlignment->Center, TextJustification->0, FontSlant->"Italic"], ", and ", StyleBox["t", FontSlant->"Italic"], StyleBox[", i.e., are", FormatType->StandardForm, FontFamily->"Times"], StyleBox[" ", FormatType->StandardForm], Cell[BoxData[ \(TraditionalForm\`C\^\[Infinity]\)]], StyleBox["functions in a compact U included in ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^3\)]], " ", StyleBox["and with values in ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["\[DoubleStruckCapitalR].", FormatType->StandardForm], StyleBox[" Variations of the parameter ", FormatType->StandardForm, FontFamily->"Times"], Cell[BoxData[ \(\[Delta]\_1\)]], " induce a period doubling cascade highlighted in the bifurcation diagram \ and which transcribes the chaotic feature of this model. Thus, choosing ", Cell[BoxData[ \(\[Delta]\_1\)]], " as the bifurcation parameter and \"freezing\" all the others with the \ corresponding values: \[Xi] = 0.866, \[CurlyEpsilon] = 1.428, ", Cell[BoxData[ \(\[Delta]\_2 = 0.376\)]], ", the bifurcation diagram of this system may be built with the above \ applet. " }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[{ \(\(\[Xi] = 0.8660254037844386`;\)\), "\[IndentingNewLine]", \(\(\[CurlyEpsilon] = 1.4288690166235205`;\)\), "\[IndentingNewLine]", \(\(\[Delta]\_2 = 0.37602114655785496`;\)\), "\[IndentingNewLine]", \(\(k = 0.5;\)\), "\[IndentingNewLine]", \(\(p = 0.5;\)\), "\[IndentingNewLine]", \(\(Pas\[Lambda] = 0.001;\)\), "\[IndentingNewLine]", \(\(TableauPlot = {};\)\)}], "Input"], Cell[BoxData[ \(For[\[Lambda] = 0.6, \[Lambda] \[LessEqual] 0.67, \ \[Lambda] = \[Lambda] + Pas\[Lambda], \[IndentingNewLine]solution = NDSolve[{\[Xi]\ \(x'\)[t] == \ x[t] - x[t]^2 - 0.25\ x[t]^k\ y[t], x[0] \[Equal] 0.15, \(y'\)[t] == \(-\[Lambda]\)\ y[t] + x[t]^k\ y[t] - 0.25^0.5\ y[t]^p\ z[t], y[0] \[Equal] 0.5, \(z'\)[ t] \[Equal] \[CurlyEpsilon]\ \((0.25^0.5\ y[t]^ p - \[Delta]\_2)\)\ z[t], z[0] \[Equal] 0.5}, \[IndentingNewLine]{x, y, z}, {t, 0, 700}, Method \[Rule] ExplicitRungeKutta]; \[IndentingNewLine]Pas\[Lambda] = Which[\[Lambda] < 0.628, 0.0001, \((\[Lambda] \[GreaterEqual] 0.628)\) && \((\[Lambda] < 0.634)\), 0.0001, \[Lambda] \[GreaterEqual] 0.634, 0.0001]; \[IndentingNewLine]Print["\<\[Lambda]=\>", \[Lambda]]; \ \[IndentingNewLine]Pas = .05; \[IndentingNewLine]Tendance = \ 0; \[IndentingNewLine]BorneBasse = 600; \[IndentingNewLine]BorneHaute = 700; \[IndentingNewLine]Precedent = \(\(Evaluate[{z[BorneBasse]} /. solution]\)[\([1]\)]\)[\([1]\)]; \[IndentingNewLine]Tendance \ = Which[Precedent >= \(\(Evaluate[{z[BorneBasse + Pas]} /. solution]\)[\([1]\)]\)[\([1]\)], \(-1\), Precedent < \(\(Evaluate[{z[BorneBasse + Pas]} /. solution]\)[\([1]\)]\)[\([1]\)], 1]; \[IndentingNewLine]For\ [Valeur = BorneBasse + Pas, \ Valeur \[LessEqual] BorneHaute, Valeur = Valeur + Pas, \[IndentingNewLine]\ \ \ \ \ \ \ \ Actuel = \(\(Evaluate[{z[ Valeur]} /. solution]\)[\([1]\)]\)[\([1]\)]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ If[\((Actuel >= Precedent)\), Goto\ [NonMaximum]]; \[IndentingNewLine]\ \ \ \ \ \ \ \ If[ Actuel < Precedent\ && \ Tendance \[Equal] \(-1\), Goto[ActualisePrecedent]]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ TableauPlot = Append[TableauPlot, {N[FromDigits[RealDigits[\[Lambda]]], 6], N[FromDigits[ RealDigits[\(\(Evaluate[{z[Valeur - Pas]} /. solution]\)[\([1]\)]\)[\([1]\)]]], 6]}]; \[IndentingNewLine]Tendance = \(-1\); \ \[IndentingNewLine]\ \ \ \ \ \ \ \ Goto\ [ ActualisePrecedent]; \ \ \ \ \ \ \ \[IndentingNewLine]Label[ NonMaximum]; \[IndentingNewLine]\ \ \ \ \ \ \ \ Tendance = 1; \[IndentingNewLine]\ \ \ \ \ \ \ \ Goto[ ActualisePrecedent]; \ \ \ \ \[IndentingNewLine]Label[ RechercheFine]; \[IndentingNewLine]\ \ \ \ \ \ \ \ Print["\"; Valeur]; \ \ \ \[IndentingNewLine]Label[ PremieresValeurs]; \[IndentingNewLine]\ \ \ \ \ \ \ \ If[ Valeur != BorneBasse, Tendance = Which[Actuel < Precedent, \(-1\), Actuel\ \[GreaterEqual] Precedent, 1]]; \[IndentingNewLine]Label[ ActualisePrecedent]; \ \ \ \ \ \ \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ Precedent = Actuel; \[IndentingNewLine]Label[ 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Mdot .28714 .37128 Mdot .28857 .38646 Mdot .28857 .34293 Mdot .28857 .40153 Mdot .28857 .28745 Mdot .29 .25121 Mdot .29 .45705 Mdot .29 .14422 Mdot .29 .35932 Mdot .29143 .16473 Mdot .29143 .38901 Mdot .29143 .33609 Mdot .29143 .40976 Mdot .29286 .43576 Mdot .29286 .19615 Mdot .29286 .42494 Mdot .29286 .24402 Mdot .29429 .38865 Mdot .29429 .31522 Mdot .29429 .43199 Mdot .29429 .206 Mdot .29571 .3891 Mdot .29571 .31366 Mdot .29571 .43292 Mdot .29571 .20302 Mdot .29714 .30694 Mdot .29714 .43808 Mdot .29714 .18852 Mdot .29714 .41647 Mdot .29857 .15799 Mdot .29857 .37867 Mdot .29857 .35457 Mdot .29857 .37978 Mdot .3 .32948 Mdot .3 .41481 Mdot .3 .24977 Mdot .3 .45399 Mdot .30143 .44803 Mdot .30143 .1607 Mdot .30143 .382 Mdot .30143 .34655 Mdot .30286 .21011 Mdot .30286 .4347 Mdot .30286 .21068 Mdot .30286 .4351 Mdot .30429 .41599 Mdot .30429 .24512 Mdot .30429 .45181 Mdot .30429 .15541 Mdot .30571 .36523 Mdot .30571 .35843 Mdot .30571 .3705 Mdot .30571 .34849 Mdot .30714 .40944 Mdot .30714 .26081 Mdot .30714 .45279 Mdot .30714 .14918 Mdot .30857 .36833 Mdot .30857 .36968 Mdot .30857 .34894 Mdot .30857 .3849 Mdot .31 .44892 Mdot .31 .15659 Mdot .31 .37499 Mdot .31 .3567 Mdot .31143 .38591 Mdot .31143 .31456 Mdot .31143 .42644 Mdot .31143 .21518 Mdot .31286 .378 Mdot .31286 .33075 Mdot .31286 .40844 Mdot .31286 .26112 Mdot .31429 .35124 Mdot .31429 .37905 Mdot .31429 .32801 Mdot .31429 .41121 Mdot .31571 .25919 Mdot .31571 .45028 Mdot .31571 .15375 Mdot .31571 .37014 Mdot .31714 .2099 Mdot .31714 .43131 Mdot .31714 .21438 Mdot .31714 .43446 Mdot .31857 .28821 Mdot .31857 .44254 Mdot .31857 .17052 Mdot .31857 .39204 Mdot .32 .23999 Mdot .32 .44615 Mdot .32 .16684 Mdot .32 .38719 Mdot .32143 .34437 Mdot .32143 .3868 Mdot .32143 .30868 Mdot .32143 .42805 Mdot .32286 .23045 Mdot .32286 .44193 Mdot .32286 .17924 Mdot .32286 .40157 Mdot .32429 .18303 Mdot .32429 .40546 Mdot .32429 .28234 Mdot .32429 .44321 Mdot .32571 .25196 Mdot .32571 .44699 Mdot .32571 .16072 Mdot .32571 .37831 Mdot .32714 .30823 Mdot .32714 .42643 Mdot .32714 .20995 Mdot .32714 .42908 Mdot .32857 .23008 Mdot .32857 .44032 Mdot .32857 .18173 Mdot .32857 .40328 Mdot .33 .2168 Mdot .33 .433 Mdot .33 .20358 Mdot .33 .42359 Mdot .33143 .23014 Mdot .33143 .43962 Mdot .33143 .18263 Mdot .33143 .40371 Mdot .33286 .18596 Mdot .33286 .40693 Mdot .33286 .27479 Mdot .33286 .44294 Mdot .33429 .30133 Mdot .33429 .42931 Mdot .33429 .20019 Mdot .33429 .4199 Mdot .33571 .17572 Mdot .33571 .39529 Mdot .33571 .30199 Mdot .33571 .42833 Mdot .33714 .23873 Mdot .33714 .44119 Mdot .33714 .17484 Mdot .33714 .39403 Mdot .33857 .27259 Mdot .33857 .44168 Mdot .33857 .16802 Mdot .33857 .38562 Mdot .34 .26993 Mdot .34 .44186 Mdot .34 .16749 Mdot .34 .38472 Mdot .34143 .26427 Mdot .34143 .44227 Mdot .34143 .16655 Mdot .34143 .38331 Mdot .34286 .25314 Mdot .34286 .44215 Mdot .34286 .16805 Mdot .34286 .38496 Mdot .34429 .22941 Mdot .34429 .43601 Mdot .34429 .18792 Mdot .34429 .40667 Mdot .34571 .19125 Mdot .34571 .40957 Mdot .34571 .26216 Mdot .34571 .44118 Mdot .34714 .16953 Mdot .34714 .38604 Mdot .34714 .31774 Mdot .34714 .4104 Mdot .34857 .23154 Mdot .34857 .43574 Mdot .34857 .18672 Mdot .34857 .40465 Mdot .35 .31178 Mdot .35 .41517 Mdot .35 .23094 Mdot .35 .43513 Mdot .35143 .27922 Mdot .35143 .43557 Mdot .35143 .17947 Mdot .35143 .3966 Mdot .35286 .20368 Mdot .35286 .41868 Mdot .35286 .23455 Mdot .35286 .4357 Mdot .35429 .19289 Mdot .35429 .40935 Mdot .35429 .25889 Mdot .35429 .43884 Mdot .35571 .2512 Mdot .35571 .43828 Mdot .35571 .17433 Mdot .35571 .39016 Mdot .35714 .3064 Mdot .35714 .41734 Mdot .35714 .22305 Mdot .35714 .42986 Mdot .35857 .22373 Mdot .35857 .42981 Mdot .35857 .20005 Mdot .35857 .41457 Mdot .36 .17741 Mdot .36 .39279 Mdot .36 .29692 Mdot .36 .42347 Mdot .36143 .21165 Mdot .36143 .42234 Mdot .36143 .22047 Mdot .36143 .42749 Mdot .36286 .20598 Mdot .36286 .41813 Mdot .36286 .23151 Mdot .36286 .43201 Mdot .36429 .18344 Mdot .36429 .39832 Mdot .36429 .28201 Mdot .36429 .43027 Mdot .36571 .17759 Mdot .36571 .39195 Mdot .36571 .29633 Mdot .36571 .42192 Mdot .36714 .18307 Mdot .36714 .39741 Mdot .36714 .28296 Mdot .36714 .42893 Mdot .36857 .18801 Mdot .36857 .40195 Mdot .36857 .27126 Mdot .36857 .4326 Mdot .37 .18883 Mdot .37 .40245 Mdot .37 .26941 Mdot .37 .43258 Mdot .37143 .18684 Mdot .37143 .40026 Mdot .37143 .27411 Mdot .37143 .43087 Mdot .37286 .18424 Mdot .37286 .39748 Mdot .37286 .28026 Mdot .37286 .4282 Mdot .37429 .18249 Mdot .37429 .39545 Mdot .37429 .2844 Mdot .37429 .42588 Mdot .37571 .18201 Mdot .37571 .39471 Mdot .37571 .28554 Mdot .37571 .42485 Mdot .37714 .18256 Mdot .37714 .39498 Mdot .37714 .28427 Mdot .37714 .425 Mdot .37857 .18368 Mdot .37857 .39581 Mdot .37857 .28167 Mdot .37857 .42573 Mdot .38 .18497 Mdot .38 .39679 Mdot .38 .27868 Mdot .38 .42651 Mdot .38143 .18622 Mdot .38143 .3977 Mdot .38143 .27589 Mdot .38143 .42709 Mdot .38286 .18728 Mdot .38286 .39842 Mdot .38286 .27352 Mdot .38286 .42742 Mdot .38429 .18816 Mdot .38429 .39895 Mdot .38429 .27159 Mdot .38429 .42754 Mdot .38571 .18888 Mdot .38571 .39931 Mdot .38571 .2701 Mdot .38571 .4275 Mdot .38714 .18945 Mdot .38714 .39954 Mdot .38714 .26889 Mdot .38714 .42736 Mdot .38857 .18992 Mdot .38857 .39968 Mdot .38857 .26792 Mdot .38857 .42714 Mdot .39 .19035 Mdot .39 .39976 Mdot .39 .26708 Mdot .39 .42687 Mdot .39143 .19074 Mdot .39143 .39981 Mdot .39143 .26634 Mdot .39143 .42657 Mdot .39286 .1911 Mdot .39286 .39983 Mdot .39286 .26566 Mdot .39286 .42629 Mdot .39429 .19146 Mdot .39429 .39985 Mdot .39429 .26498 Mdot .39429 .42598 Mdot .39571 .19182 Mdot .39571 .39986 Mdot .39571 .2643 Mdot .39571 .42567 Mdot .39714 .19219 Mdot .39714 .39988 Mdot .39714 .26364 Mdot .39714 .42534 Mdot .39857 .19257 Mdot .39857 .39991 Mdot .39857 .26295 Mdot .39857 .42501 Mdot .4 .19296 Mdot .4 .39994 Mdot .4 .26225 Mdot .4 .42467 Mdot .40143 .19336 Mdot .40143 .39997 Mdot .40143 .26152 Mdot .40143 .42435 Mdot .40286 .19378 Mdot .40286 .40002 Mdot .40286 .2608 Mdot .40286 .42403 Mdot .40429 .19421 Mdot .40429 .40006 Mdot .40429 .26006 Mdot .40429 .4237 Mdot .40571 .19464 Mdot .40571 .40012 Mdot .40571 .25929 Mdot .40571 .42336 Mdot .40714 .19509 Mdot .40714 .40018 Mdot .40714 .25852 Mdot .40714 .42301 Mdot .40857 .19557 Mdot .40857 .40025 Mdot .40857 .25774 Mdot .40857 .42265 Mdot .41 .19604 Mdot .41 .40032 Mdot .41 .25693 Mdot .41 .42228 Mdot .41143 .19653 Mdot .41143 .4004 Mdot .41143 .25611 Mdot .41143 .42192 Mdot .41286 .19705 Mdot .41286 .40048 Mdot .41286 .25529 Mdot .41286 .42156 Mdot .41429 .19758 Mdot .41429 .40057 Mdot .41429 .25444 Mdot .41429 .42119 Mdot .41571 .19811 Mdot .41571 .40067 Mdot .41571 .25356 Mdot .41571 .42081 Mdot .41714 .19868 Mdot .41714 .40078 Mdot .41714 .25268 Mdot .41714 .42042 Mdot .41857 .19927 Mdot .41857 .40089 Mdot .41857 .25177 Mdot .41857 .42001 Mdot .42 .19987 Mdot .42 .40102 Mdot .42 .25083 Mdot .42 .41959 Mdot .42143 .20051 Mdot .42143 .40115 Mdot .42143 .24988 Mdot .42143 .41916 Mdot .42286 .20117 Mdot .42286 .40129 Mdot .42286 .2489 Mdot .42286 .41875 Mdot .42429 .20186 Mdot .42429 .40145 Mdot .42429 .24787 Mdot .42429 .41831 Mdot .42571 .20259 Mdot .42571 .40162 Mdot .42571 .24683 Mdot .42571 .41786 Mdot .42714 .20334 Mdot .42714 .4018 Mdot .42714 .24574 Mdot .42714 .41739 Mdot .42857 .20415 Mdot .42857 .40199 Mdot .42857 .24461 Mdot .42857 .4169 Mdot .43 .20499 Mdot .43 .4022 Mdot .43 .24346 Mdot .43 .41638 Mdot .43143 .20588 Mdot .43143 .40242 Mdot .43143 .24225 Mdot .43143 .41584 Mdot .43286 .20682 Mdot .43286 .40266 Mdot .43286 .24102 Mdot .43286 .41531 Mdot .43429 .20779 Mdot .43429 .40291 Mdot .43429 .23975 Mdot .43429 .41475 Mdot .43571 .20882 Mdot .43571 .40317 Mdot .43571 .23847 Mdot .43571 .41416 Mdot .43714 .20987 Mdot .43714 .40344 Mdot .43714 .23717 Mdot .43714 .41354 Mdot .43857 .21097 Mdot .43857 .4037 Mdot .43857 .23589 Mdot .43857 .4129 Mdot .44 .21206 Mdot .44 .40394 Mdot .44 .23461 Mdot .44 .41226 Mdot .44143 .2132 Mdot .44143 .40417 Mdot .44143 .2334 Mdot .44143 .41162 Mdot .44286 .2143 Mdot .44286 .40437 Mdot .44286 .23222 Mdot .44286 .41096 Mdot .44429 .2154 Mdot .44429 .40455 Mdot .44429 .23114 Mdot .44429 .41028 Mdot .44571 .21645 Mdot .44571 .40469 Mdot .44571 .23012 Mdot .44571 .40966 Mdot .44714 .21746 Mdot .44714 .4048 Mdot .44714 .22923 Mdot .44714 .40904 Mdot .44857 .21843 Mdot .44857 .40486 Mdot .44857 .22842 Mdot .44857 .40841 Mdot .45 .21933 Mdot .45 .40489 Mdot .45 .22774 Mdot .45 .40784 Mdot .45143 .22016 Mdot .45143 .40486 Mdot .45143 .22715 Mdot .45143 .4073 Mdot .45286 .22093 Mdot .45286 .4048 Mdot .45286 .22665 Mdot .45286 .40676 Mdot .45429 .22163 Mdot .45429 .40469 Mdot .45429 .22627 Mdot .45429 .40627 Mdot .45571 .22226 Mdot .45571 .40454 Mdot .45571 .22596 Mdot .45571 .40581 Mdot .45714 .22281 Mdot .45714 .40437 Mdot .45714 .22572 Mdot .45714 .40534 Mdot .45857 .22332 Mdot .45857 .40419 Mdot .45857 .22557 Mdot .45857 .40493 Mdot .46 .22378 Mdot .46 .40398 Mdot .46 .22547 Mdot .46 .40453 Mdot .46143 .22419 Mdot .46143 .40375 Mdot .46143 .22542 Mdot .46143 .40412 Mdot .46286 .22455 Mdot .46286 .40348 Mdot .46286 .22541 Mdot .46286 .40376 Mdot .46429 .22488 Mdot .46429 .4032 Mdot .46429 .22546 Mdot .46429 .40339 Mdot .46571 .22517 Mdot .46571 .40292 Mdot .46571 .22553 Mdot .46571 .40303 Mdot .46714 .22544 Mdot .46714 .40264 Mdot .46714 .22563 Mdot .46714 .4027 Mdot .46857 .22569 Mdot .46857 .40234 Mdot .46857 .22574 Mdot .46857 .40235 Mdot .47 .22592 Mdot .47 .40203 Mdot .47 .22587 Mdot .47 .40201 Mdot .47143 .22613 Mdot .47143 .4017 Mdot .47143 .22602 Mdot .47143 .40169 Mdot .47286 .22633 Mdot .47286 .40137 Mdot .47286 .22618 Mdot .47286 .40134 Mdot .47429 .22653 Mdot .47429 .40106 Mdot .47429 .22635 Mdot .47429 .40101 Mdot .47571 .22672 Mdot .47571 .40074 Mdot .47571 .22653 Mdot .47571 .40069 Mdot .47714 .2269 Mdot .47714 .40041 Mdot .47714 .22671 Mdot .47714 .40034 Mdot .47857 .22708 Mdot .47857 .40008 Mdot .47857 .22689 Mdot .47857 .40002 Mdot .48 .22725 Mdot .48 .39973 Mdot .48 .22707 Mdot .48 .3997 Mdot .48143 .22743 Mdot .48143 .39939 Mdot .48143 .22726 Mdot .48143 .39935 Mdot .48286 .2276 Mdot .48286 .39906 Mdot .48286 .22745 Mdot .48286 .39903 Mdot .48429 .22778 Mdot .48429 .39873 Mdot .48429 .22765 Mdot .48571 .22796 Mdot .48571 .39839 Mdot .48571 .22785 Mdot .48714 .22813 Mdot .48714 .39805 Mdot .48714 .22804 Mdot .48857 .22832 Mdot .48857 .39769 Mdot .48857 .22823 Mdot .49 .2285 Mdot .49 .39734 Mdot .49 .22843 Mdot .49143 .22869 Mdot .49143 .39701 Mdot .49143 .22862 Mdot .49286 .22888 Mdot .49286 .39667 Mdot .49286 .22881 Mdot .49429 .22907 Mdot .49429 .39633 Mdot .49429 .22901 Mdot .49571 .22926 Mdot .49571 .39598 Mdot .49571 .22922 Mdot .49714 .22945 Mdot .49714 .39563 Mdot .49714 .22942 Mdot .49857 .22964 Mdot .49857 .39527 Mdot .49857 .22962 Mdot .5 .22984 Mdot .5 .39492 Mdot .5 .22982 Mdot .50143 .23004 Mdot .50143 .39458 Mdot .50143 .23002 Mdot .50286 .23023 Mdot .50286 .39423 Mdot .50286 .23022 Mdot .50429 .23043 Mdot .50429 .39388 Mdot .50429 .23042 Mdot .50571 .23063 Mdot .50571 .39352 Mdot .50571 .23062 Mdot .50714 .23083 Mdot .50714 .39316 Mdot .50714 .23082 Mdot .50857 .23103 Mdot .50857 .39279 Mdot .50857 .23103 Mdot .51 .23124 Mdot .51 .39243 Mdot .51 .23124 Mdot .51143 .23146 Mdot .51143 .39208 Mdot .51143 .23145 Mdot .51286 .23167 Mdot .51286 .39172 Mdot .51286 .23166 Mdot .51429 .23188 Mdot .51429 .39137 Mdot .51429 .23188 Mdot .51571 .23209 Mdot .51571 .391 Mdot .51571 .23209 Mdot .51714 .23231 Mdot .51714 .39063 Mdot .51714 .23231 Mdot .51857 .23252 Mdot .51857 .39026 Mdot .51857 .23252 Mdot .52 .23273 Mdot .52 .38989 Mdot .52 .23274 Mdot .52143 .23295 Mdot .52143 .38951 Mdot .52143 .23296 Mdot .52286 .23317 Mdot .52286 .38914 Mdot .52286 .23318 Mdot .52429 .23339 Mdot .52429 .38877 Mdot .52429 .2334 Mdot .52571 .23362 Mdot .52571 .38841 Mdot .52571 .23362 Mdot .52714 .23385 Mdot .52714 .38803 Mdot .52714 .23384 Mdot .52857 .23407 Mdot .52857 .38766 Mdot .52857 .23407 Mdot .53 .2343 Mdot .53 .38727 Mdot .53 .23429 Mdot .53143 .23452 Mdot .53143 .38689 Mdot .53143 .23452 Mdot .53286 .23475 Mdot .53286 .3865 Mdot .53286 .23475 Mdot .53429 .23498 Mdot .53429 .38611 Mdot .53429 .23498 Mdot .53571 .23522 Mdot .53571 .38571 Mdot .53571 .23521 Mdot .53714 .23546 Mdot .53714 .38533 Mdot .53714 .23545 Mdot .53857 .2357 Mdot .53857 .38495 Mdot .53857 .23568 Mdot .54 .23593 Mdot .54 .38457 Mdot .54 .23592 Mdot .54143 .23617 Mdot .54143 .38418 Mdot .54143 .23616 Mdot .54286 .2364 Mdot .54286 .38378 Mdot .54286 .2364 Mdot .54429 .23665 Mdot .54429 .38339 Mdot .54429 .23665 Mdot .54571 .38299 Mdot .54571 .23691 Mdot .54571 .38299 Mdot .54571 .23689 Mdot .54714 .38258 Mdot .54714 .23716 Mdot .54714 .38258 Mdot .54714 .23714 Mdot .54857 .38217 Mdot .54857 .2374 Mdot .54857 .38218 Mdot .54857 .23739 Mdot .55 .38176 Mdot .55 .23765 Mdot .55 .38177 Mdot .55 .23764 Mdot .55143 .38135 Mdot .55143 .23789 Mdot .55143 .38136 Mdot .55143 .2379 Mdot .55286 .38094 Mdot .55286 .23816 Mdot .55286 .38094 Mdot .55286 .23815 Mdot .55429 .38054 Mdot .55429 .23842 Mdot .55429 .38053 Mdot .55429 .23841 Mdot .55571 .38014 Mdot .55571 .23868 Mdot .55571 .38013 Mdot .55571 .23867 Mdot .55714 .37972 Mdot .55714 .23894 Mdot .55714 .37972 Mdot .55714 .23893 Mdot .55857 .37931 Mdot .55857 .23919 Mdot .55857 .37931 Mdot .55857 .2392 Mdot .56 .37889 Mdot .56 .23946 Mdot .56 .37889 Mdot .56 .23946 Mdot .56143 .37847 Mdot .56143 .23973 Mdot .56143 .37847 Mdot .56143 .23973 Mdot .56286 .37804 Mdot .56286 .24001 Mdot .56286 .37805 Mdot .56286 .24 Mdot .56429 .37761 Mdot .56429 .24027 Mdot .56429 .37762 Mdot .56429 .24028 Mdot .56571 .37718 Mdot .56571 .24054 Mdot .56571 .3772 Mdot .56571 .24055 Mdot .56714 .37674 Mdot .56714 .24081 Mdot .56714 .37677 Mdot .56714 .24083 Mdot .56857 .37632 Mdot .56857 .2411 Mdot .56857 .37633 Mdot .56857 .2411 Mdot .57 .37589 Mdot .57 .24138 Mdot .57 .3759 Mdot .57 .24138 Mdot .57143 .37546 Mdot .57143 .24166 Mdot .57143 .37546 Mdot .57143 .24167 Mdot .57286 .37502 Mdot .57286 .24194 Mdot .57286 .37501 Mdot .57286 .24195 Mdot .57429 .37458 Mdot .57429 .24223 Mdot .57429 .37457 Mdot .57429 .24224 Mdot .57571 .37414 Mdot .57571 .24253 Mdot .57571 .37412 Mdot .57571 .24252 Mdot .57714 .37368 Mdot .57714 .24282 Mdot .57714 .37367 Mdot .57714 .24281 Mdot .57857 .37323 Mdot .57857 .24311 Mdot .57857 .37322 Mdot .57857 .2431 Mdot .58 .37276 Mdot .58 .2434 Mdot .58 .37276 Mdot .58 .2434 Mdot .58143 .37232 Mdot .58143 .24371 Mdot .58143 .37231 Mdot .58143 .24371 Mdot .58286 .37187 Mdot .58286 .24402 Mdot .58286 .37186 Mdot .58286 .24401 Mdot .58429 .37141 Mdot .58429 .24432 Mdot .58429 .3714 Mdot .58429 .24432 Mdot .58571 .37095 Mdot .58571 .24462 Mdot .58571 .37094 Mdot .58571 .24463 Mdot .58714 .37049 Mdot .58714 .24494 Mdot .58714 .37048 Mdot .58714 .24494 Mdot .58857 .37001 Mdot .58857 .24526 Mdot .58857 .37001 Mdot .58857 .24526 Mdot .59 .36953 Mdot .59 .24557 Mdot .59 .36954 Mdot .59 .24557 Mdot .59143 .36905 Mdot .59143 .24588 Mdot .59143 .36906 Mdot .59143 .24589 Mdot .59286 .36857 Mdot .59286 .24622 Mdot .59286 .36859 Mdot .59286 .2462 Mdot .59429 .3681 Mdot .59429 .24654 Mdot .59429 .36811 Mdot .59429 .24653 Mdot .59571 .36762 Mdot .59571 .24687 Mdot .59571 .36762 Mdot .59571 .24687 Mdot .59714 .36714 Mdot .59714 .2472 Mdot .59714 .36713 Mdot .59714 .2472 Mdot .59857 .36664 Mdot .59857 .24754 Mdot .59857 .36664 Mdot .59857 .24754 Mdot .6 .36614 Mdot .6 .24788 Mdot .6 .36615 Mdot .6 .24788 Mdot .60143 .36563 Mdot .60143 .24821 Mdot .60143 .36565 Mdot .60143 .24822 Mdot .60286 .36512 Mdot .60286 .24857 Mdot .60286 .36515 Mdot .60286 .24857 Mdot .60429 .36463 Mdot .60429 .24893 Mdot .60429 .36464 Mdot .60429 .24891 Mdot .60571 .36412 Mdot .60571 .24927 Mdot .60571 .36413 Mdot .60571 .24927 Mdot .60714 .36361 Mdot .60714 .24963 Mdot .60714 .36362 Mdot .60714 .24963 Mdot .60857 .36309 Mdot .60857 .25001 Mdot .60857 .3631 Mdot .60857 .25 Mdot .61 .36256 Mdot .61 .25037 Mdot .61 .36258 Mdot .61 .25037 Mdot .61143 .36203 Mdot .61143 .25074 Mdot .61143 .36205 Mdot .61143 .25074 Mdot .61286 .36148 Mdot .61286 .25114 Mdot .61286 .36152 Mdot .61286 .25111 Mdot .61429 .36095 Mdot .61429 .25152 Mdot .61429 .36098 Mdot .61429 .25148 Mdot .61571 .36041 Mdot .61571 .2519 Mdot .61571 .36044 Mdot .61571 .25188 Mdot .61714 .35986 Mdot .61714 .25232 Mdot .61714 .35989 Mdot .61714 .25228 Mdot .61857 .3593 Mdot .61857 .25273 Mdot .61857 .35933 Mdot .61857 .25268 Mdot .62 .35872 Mdot .62 .25314 Mdot .62 .35877 Mdot .62 .25308 Mdot .62143 .35814 Mdot .62143 .25359 Mdot .62143 .3582 Mdot .62143 .25349 Mdot .62286 .35754 Mdot .62286 .25403 Mdot .62286 .35763 Mdot .62286 .25392 Mdot .62429 .35695 Mdot .62429 .25448 Mdot .62429 .35704 Mdot .62429 .25437 Mdot .62571 .35635 Mdot .62571 .25497 Mdot .62571 .35645 Mdot .62571 .25481 Mdot .62714 .35572 Mdot .62714 .25545 Mdot .62714 .35585 Mdot .62714 .25527 Mdot .62857 .35508 Mdot .62857 .25597 Mdot .62857 .35523 Mdot .62857 .25573 Mdot .63 .35442 Mdot .63 .25651 Mdot .63 .3546 Mdot .63 .25624 Mdot .63143 .35373 Mdot .63143 .25706 Mdot .63143 .35396 Mdot .63143 .25675 Mdot .63286 .35302 Mdot .63286 .25767 Mdot .63286 .3533 Mdot .63286 .25728 Mdot .63429 .35231 Mdot .63429 .25827 Mdot .63429 .35262 Mdot .63429 .25782 Mdot .63571 .35156 Mdot .63571 .25895 Mdot .63571 .35192 Mdot .63571 .25842 Mdot .63714 .35079 Mdot .63714 .25964 Mdot .63714 .3512 Mdot .63714 .25903 Mdot .63857 .34998 Mdot .63857 .26039 Mdot .63857 .35045 Mdot .63857 .25967 Mdot .64 .34913 Mdot .64 .26116 Mdot .64 .34967 Mdot .64 .26037 Mdot .64143 .34823 Mdot .64143 .26201 Mdot .64143 .34886 Mdot .64143 .2611 Mdot .64286 .3473 Mdot .64286 .26289 Mdot .64286 .34801 Mdot .64286 .26186 Mdot .64429 .34633 Mdot .64429 .26385 Mdot .64429 .34712 Mdot .64429 .26271 Mdot .64571 .34533 Mdot .64571 .26485 Mdot .64571 .3462 Mdot .64571 .26359 Mdot .64714 .34428 Mdot .64714 .2659 Mdot .64714 .34522 Mdot .64714 .26453 Mdot .64857 .34318 Mdot .64857 .26703 Mdot .64857 .3442 Mdot .64857 .26553 Mdot .65 .34203 Mdot .65 .2682 Mdot .65 .34313 Mdot .65 .2666 Mdot .65143 .34083 Mdot .65143 .26942 Mdot .65143 .34201 Mdot .65143 .26773 Mdot .65286 .33958 Mdot .65286 .27071 Mdot .65286 .34084 Mdot .65286 .26892 Mdot .65429 .3383 Mdot .65429 .27203 Mdot .65429 .33962 Mdot .65429 .27016 Mdot .65571 .33697 Mdot .65571 .27338 Mdot .65571 .33834 Mdot .65571 .27147 Mdot .65714 .33563 Mdot .65714 .27475 Mdot .65714 .33703 Mdot .65714 .27282 Mdot .65857 .33426 Mdot .65857 .27615 Mdot .65857 .33567 Mdot .65857 .27421 Mdot .66 .33286 Mdot .66 .27757 Mdot .66 .33427 Mdot .66 .27564 Mdot .66143 .33145 Mdot .66143 .27899 Mdot .66143 .33284 Mdot .66143 .27709 Mdot .66286 .33002 Mdot .66286 .28039 Mdot .66286 .33139 Mdot .66286 .27853 Mdot .66429 .32858 Mdot .66429 .28178 Mdot .66429 .32993 Mdot .66429 .27999 Mdot .66571 .32716 Mdot .66571 .28313 Mdot .66571 .32846 Mdot .66571 .28144 Mdot .66714 .32576 Mdot .66714 .28444 Mdot .66714 .32699 Mdot .66714 .28286 Mdot .66857 .32437 Mdot .66857 .28571 Mdot .66857 .32552 Mdot .66857 .28423 Mdot .67 .323 Mdot .67 .28692 Mdot .67 .32408 Mdot .67 .28556 Mdot .67143 .32164 Mdot .67143 .28808 Mdot .67143 .32266 Mdot .67143 .28685 Mdot .67286 .32035 Mdot .67286 .28919 Mdot .67286 .32126 Mdot .67286 .28808 Mdot .67429 .31908 Mdot .67429 .29024 Mdot .67429 .3199 Mdot .67429 .28924 Mdot .67571 .31784 Mdot .67571 .29122 Mdot .67571 .31858 Mdot .67571 .29032 Mdot .67714 .31665 Mdot .67714 .29212 Mdot .67714 .31729 Mdot .67714 .29134 Mdot .67857 .31551 Mdot .67857 .29296 Mdot .67857 .31607 Mdot .67857 .2923 Mdot .68 .3144 Mdot .68 .29372 Mdot .68 .3149 Mdot .68 .29316 Mdot .68143 .31334 Mdot .68143 .29443 Mdot .68143 .31377 Mdot .68143 .29395 Mdot .68286 .31234 Mdot .68286 .29509 Mdot .68286 .31268 Mdot .68286 .29468 Mdot .68429 .31135 Mdot .68429 .29567 Mdot .68429 .31165 Mdot .68429 .29534 Mdot .68571 .31045 Mdot .68571 .29619 Mdot .68571 .31069 Mdot .68571 .29592 Mdot .68714 .30956 Mdot .68714 .29664 Mdot .68714 .30975 Mdot .68714 .29645 Mdot .68857 .30873 Mdot .68857 .29707 Mdot .68857 .30886 Mdot .68857 .29691 Mdot .69 .30792 Mdot .69 .29743 Mdot .69 .30804 Mdot .69 .2973 Mdot .69143 .30717 Mdot .69143 .29774 Mdot .69143 .30724 Mdot .69143 .29766 Mdot .69286 .30644 Mdot .69286 .29799 Mdot .69286 .30648 Mdot .69286 .29796 Mdot .69429 .30576 Mdot .69429 .29823 Mdot .69429 .30578 Mdot .69429 .29819 Mdot .69571 .3051 Mdot .69571 .29842 Mdot .69571 .30509 Mdot .69571 .29842 Mdot .69714 .30448 Mdot .69714 .29856 Mdot .69714 .30447 Mdot .69714 .29857 Mdot .69857 .30388 Mdot .69857 .29867 Mdot .69857 .30386 Mdot .69857 .2987 Mdot .7 .30333 Mdot .7 .29876 Mdot .7 .30329 Mdot .7 .29879 Mdot .70143 .30277 Mdot .70143 .2988 Mdot .70143 .30275 Mdot .70143 .29884 Mdot .70286 .30227 Mdot .70286 .29882 Mdot .70286 .30222 Mdot .70286 .29887 Mdot .70429 .30177 Mdot .70429 .29883 Mdot .70429 .30174 Mdot .70429 .29885 Mdot .70571 .30131 Mdot .70571 .29879 Mdot .70571 .30125 Mdot .70571 .29884 Mdot .70714 .30086 Mdot .70714 .29874 Mdot .70714 .30082 Mdot .70714 .29878 Mdot .70857 .30041 Mdot .70857 .29868 Mdot .70857 .30037 Mdot .70857 .29872 Mdot .71 .30002 Mdot .71 .29859 Mdot .71 .29998 Mdot .71 .29862 Mdot .71143 .29961 Mdot .71143 .29848 Mdot .71143 .29957 Mdot .71143 .29852 Mdot .71286 .29923 Mdot .71286 .29837 Mdot .71286 .29921 Mdot .71286 .29839 Mdot .71429 .29886 Mdot .71429 .29822 Mdot .71429 .29883 Mdot .71429 .29825 Mdot .71571 .29849 Mdot .71571 .29808 Mdot .71571 .29849 Mdot .71571 .2981 Mdot .71714 .29815 Mdot .71714 .29793 Mdot .71714 .29814 Mdot .71714 .29793 Mdot .71857 .29782 Mdot .71857 .29774 Mdot .71857 .29781 Mdot .71857 .29776 Mdot .72 .29747 Mdot .72 .29758 Mdot .72 .29749 Mdot .72 .29756 Mdot .72143 .29717 Mdot .72143 .29738 Mdot .72143 .29718 Mdot .72143 .29738 Mdot .72286 .29686 Mdot .72286 .29718 Mdot .72286 .29687 Mdot .72286 .29716 Mdot .72429 .29654 Mdot .72429 .29698 Mdot .72429 .29657 Mdot .72429 .29696 Mdot .72571 .29626 Mdot .72571 .29676 Mdot .72571 .29629 Mdot .72571 .29673 Mdot .72714 .29596 Mdot .72714 .29654 Mdot .72714 .29599 Mdot .72714 .29652 Mdot .72857 .29567 Mdot .72857 .29632 Mdot .72857 .29572 Mdot .72857 .29627 Mdot .73 .2954 Mdot .73 .29607 Mdot .73 .29543 Mdot .73 .29606 Mdot .73143 .29512 Mdot .73143 .29586 Mdot .73143 .29517 Mdot .73143 .2958 Mdot .73286 .29484 Mdot .73286 .29562 Mdot .73286 .2949 Mdot .73286 .29557 Mdot .73429 .29458 Mdot .73429 .29537 Mdot .73429 .29464 Mdot .73429 .29531 Mdot .73571 .29431 Mdot .73571 .29513 Mdot .73571 .29437 Mdot .73571 .29508 Mdot .73714 .29404 Mdot .73714 .29487 Mdot .73714 .29411 Mdot .73714 .29481 Mdot .73857 .29379 Mdot .73857 .29463 Mdot .73857 .29386 Mdot .73857 .29457 Mdot .74 .29353 Mdot .74 .29438 Mdot .74 .29358 Mdot .74 .2943 Mdot .74143 .29326 Mdot .74143 .2941 Mdot .74143 .29335 Mdot .74143 .29406 Mdot .74286 .29302 Mdot .74286 .29386 Mdot .74286 .29309 Mdot .74286 .29379 Mdot .74429 .29277 Mdot .74429 .2936 Mdot .74429 .29284 Mdot .74429 .29354 Mdot .74571 .2925 Mdot .74571 .29333 Mdot .74571 .29259 Mdot .74571 .29327 Mdot .74714 .29226 Mdot .74714 .29308 Mdot .74714 .29233 Mdot .74714 .29301 Mdot .74857 .29202 Mdot .74857 .29281 Mdot .74857 .29209 Mdot .74857 .29274 Mdot .75 .29175 Mdot .75 .29255 Mdot .75 .29183 Mdot .75 .29248 Mdot .75143 .29151 Mdot .75143 .29229 Mdot .75143 .29159 Mdot .75143 .29221 Mdot .75286 .29127 Mdot .75286 .292 Mdot .75286 .29134 Mdot .75286 .29194 Mdot .75429 .29101 Mdot .75429 .29175 Mdot .75429 .29109 Mdot .75429 .29168 Mdot .75571 .29076 Mdot .75571 .29148 Mdot .75571 .29085 Mdot .75571 .29141 Mdot .75714 .29052 Mdot .75714 .2912 Mdot .75714 .29058 Mdot .75714 .29115 Mdot .75857 .29027 Mdot .75857 .29095 Mdot .75857 .29035 Mdot .75857 .29087 Mdot .76 .29001 Mdot .76 .29067 Mdot .76 .29009 Mdot .76 .29061 Mdot .76143 .28978 Mdot .76143 .2904 Mdot .76143 .28985 Mdot .76143 .29033 Mdot .76286 .28953 Mdot .76286 .29014 Mdot .76286 .2896 Mdot .76286 .29007 Mdot .76429 .28926 Mdot .76429 .28986 Mdot .76429 .28934 Mdot .76429 .28979 Mdot .76571 .28903 Mdot .76571 .28959 Mdot .76571 .28911 Mdot .76571 .28954 Mdot .76714 .28879 Mdot .76714 .28933 Mdot .76714 .28884 Mdot .76714 .28925 Mdot .76857 .28853 Mdot .76857 .28904 Mdot .76857 .2886 Mdot .76857 .289 Mdot .77 .28828 Mdot .77 .28879 Mdot .77 .28835 Mdot .77 .28872 Mdot .77143 .28804 Mdot .77143 .28852 Mdot .77143 .28809 Mdot .77143 .28846 Mdot .77286 .28778 Mdot .77286 .28823 Mdot .77286 .28785 Mdot .77286 .28819 Mdot .77429 .28752 Mdot .77429 .28798 Mdot .77429 .28758 Mdot .77429 .28792 Mdot .77571 .28728 Mdot .77571 .2877 Mdot .77571 .28734 Mdot .77571 .28765 Mdot .77714 .28703 Mdot .77714 .28742 Mdot .77714 .28709 Mdot .77714 .28738 Mdot .77857 .28676 Mdot .77857 .28717 Mdot .77857 .28683 Mdot .77857 .28712 Mdot .78 .28652 Mdot .78 .28689 Mdot .78 .28659 Mdot .78 .28684 Mdot .78143 .28628 Mdot .78143 .28662 Mdot .78143 .28632 Mdot .78143 .28658 Mdot .78286 .28601 Mdot .78286 .28636 Mdot .78286 .28607 Mdot .78286 .28629 Mdot .78429 .28576 Mdot .78429 .28608 Mdot .78429 .28582 Mdot .78429 .28604 Mdot .78571 .28551 Mdot .78571 .28581 Mdot .78571 .28555 Mdot .78571 .28576 Mdot .78714 .28526 Mdot .78714 .28555 Mdot .78714 .28531 Mdot .78714 .28551 Mdot .78857 .28498 Mdot .78857 .28527 Mdot .78857 .28504 Mdot .78857 .28523 Mdot .79 .28474 Mdot .79 .285 Mdot .79 .28479 Mdot .79 .28497 Mdot .79143 .28449 Mdot .79143 .28474 Mdot .79143 .28453 Mdot .79143 .2847 Mdot .79286 .28422 Mdot .79286 .28446 Mdot .79286 .28426 Mdot .79286 .28442 Mdot .79429 .28396 Mdot .79429 .28419 Mdot .79429 .28401 Mdot .79429 .28417 Mdot .79571 .28372 Mdot .79571 .28393 Mdot .79571 .28375 Mdot .79571 .28388 Mdot .79714 .28346 Mdot .79714 .28365 Mdot .79714 .28349 Mdot .79714 .28363 Mdot .79857 .28318 Mdot .79857 .28338 Mdot .79857 .28323 Mdot .79857 .28335 Mdot .8 .28293 Mdot .8 .28312 Mdot .8 .28295 Mdot .8 .28309 Mdot .80143 .28268 Mdot .80143 .28284 Mdot .80143 .28271 Mdot .80143 .28282 Mdot .80286 .28241 Mdot .80286 .28257 Mdot .80286 .28244 Mdot .80286 .28255 Mdot .80429 .28214 Mdot .80429 .28231 Mdot .80429 .28218 Mdot .80429 .28228 Mdot .80571 .28189 Mdot .80571 .28203 Mdot .80571 .28192 Mdot .80571 .282 Mdot .80714 .28163 Mdot .80714 .28176 Mdot .80714 .28165 Mdot .80714 .28175 Mdot .80857 .28136 Mdot .80857 .2815 Mdot .80857 .28139 Mdot .80857 .28146 Mdot .81 .2811 Mdot .81 .28122 Mdot .81 .28113 Mdot .81 .28121 Mdot .81143 .28085 Mdot .81143 .28095 Mdot .81143 .28085 Mdot .81143 .28093 Mdot .81286 .28058 Mdot .81286 .28069 Mdot .81286 .2806 Mdot .81286 .28067 Mdot .81429 .2803 Mdot .81429 .28041 Mdot .81429 .28033 Mdot .81429 .2804 Mdot .81571 .28005 Mdot .81571 .28014 Mdot .81571 .28007 Mdot .81571 .28012 Mdot .81714 .27979 Mdot .81714 .27988 Mdot .81714 .27981 Mdot .81714 .27986 Mdot .81857 .27952 Mdot .81857 .2796 Mdot .81857 .27953 Mdot .81857 .27958 Mdot .82 .27924 Mdot .82 .27933 Mdot .82 .27927 Mdot .82 .27932 Mdot .82143 .27899 Mdot .82143 .27907 Mdot .82143 .27901 Mdot .82143 .27905 Mdot .82286 .27873 Mdot .82286 .27879 Mdot .82286 .27872 Mdot .82286 .27878 Mdot .82429 .27845 Mdot .82429 .27851 Mdot .82429 .27847 Mdot .82429 .27851 Mdot .82571 .27818 Mdot .82571 .27825 Mdot .82571 .2782 Mdot .82571 .27823 Mdot .82714 .27792 Mdot .82714 .27798 Mdot .82714 .27793 Mdot .82714 .27797 Mdot .82857 .27766 Mdot .82857 .2777 Mdot .82857 .27767 Mdot .82857 .27769 Mdot .83 .27738 Mdot .83 .27744 Mdot .83 .27739 Mdot .83 .27743 Mdot .83143 .27711 Mdot .83143 .27717 Mdot .83143 .27713 Mdot .83143 .27716 Mdot .83286 .27685 Mdot .83286 .27688 Mdot .83286 .27686 Mdot .83286 .27688 Mdot .83429 .27658 Mdot .83429 .27662 Mdot .83429 .27658 Mdot .83429 .27662 Mdot .83571 .2763 Mdot .83571 .27635 Mdot .83571 .27632 Mdot .83571 .27633 Mdot .83714 .27603 Mdot .83714 .27607 Mdot .83714 .27605 Mdot .83714 .27607 Mdot .83857 .27577 Mdot .83857 .2758 Mdot .83857 .27576 Mdot .83857 .2758 Mdot .84 .2755 Mdot .84 .27554 Mdot .84 .27551 Mdot .84 .27552 Mdot .84143 .27522 Mdot .84143 .27526 Mdot .84143 .27524 Mdot .84143 .27526 Mdot .84286 .27495 Mdot .84286 .27498 Mdot .84286 .27495 Mdot .84286 .27496 Mdot .84429 .27469 Mdot .84429 .27472 Mdot .84429 .2747 Mdot .84429 .27471 Mdot .84571 .27442 Mdot .84571 .27444 Mdot .84571 .27442 Mdot .84571 .27443 Mdot .84714 .27413 Mdot .84714 .27415 Mdot .84714 .27414 Mdot .84714 .27416 Mdot .84857 .27387 Mdot .84857 .27389 Mdot .84857 .27388 Mdot .84857 .27389 Mdot .85 .2736 Mdot .85 .27362 Mdot .85 .2736 Mdot .85 .2736 Mdot .85143 .27333 Mdot .85143 .27333 Mdot .85143 .27332 Mdot .85143 .27334 Mdot .85286 .27304 Mdot .85286 .27307 Mdot .85286 .27306 Mdot .85286 .27306 Mdot .85429 .27277 Mdot .85429 .2728 Mdot .85429 .27278 Mdot .85429 .27279 Mdot .85571 .27251 Mdot .85571 .27252 Mdot .85571 .27251 Mdot .85571 .27252 Mdot .85714 .27223 Mdot .85714 .27223 Mdot .85714 .27224 Mdot .85714 .27223 Mdot .85857 .27195 Mdot .85857 .27197 Mdot .85857 .27195 Mdot .85857 .27197 Mdot .86 .27167 Mdot .86 .2717 Mdot .86 .27168 Mdot .86 .27169 Mdot .86143 .27141 Mdot .86143 .27141 Mdot .86143 .27141 Mdot .86143 .27141 Mdot .86286 .27114 Mdot .86286 .27114 Mdot .86286 .27113 Mdot .86286 .27114 Mdot .86429 .27085 Mdot .86429 .27087 Mdot .86429 .27086 Mdot .86429 .27085 Mdot .86571 .27057 Mdot .86571 .27059 Mdot .86571 .27059 Mdot .86571 .27059 Mdot .86714 .27031 Mdot .86714 .2703 Mdot .86714 .2703 Mdot .86714 .27031 Mdot .86857 .27003 Mdot .86857 .27004 Mdot .86857 .27003 Mdot .86857 .27002 Mdot .87 .26975 Mdot .87 .26976 Mdot .87 .26976 Mdot .87 .26976 Mdot .87143 .26946 Mdot .87143 .26947 Mdot .87143 .26947 Mdot .87143 .26948 Mdot .87286 .2692 Mdot .87286 .2692 Mdot .87286 .2692 Mdot .87286 .2692 Mdot .87429 .26892 Mdot .87429 .26893 Mdot .87429 .26893 Mdot .87429 .26893 Mdot .87571 .26864 Mdot .87571 .26865 Mdot .87571 .26864 Mdot .87571 .26863 Mdot .87714 .26836 Mdot .87714 .26836 Mdot .87714 .26836 Mdot .87714 .26837 Mdot .87857 .26808 Mdot .87857 .26809 Mdot .87857 .26809 Mdot .87857 .26809 Mdot .88 .26781 Mdot .88 .26782 Mdot .88 .2678 Mdot .88 .2678 Mdot .88143 .26753 Mdot .88143 .26753 Mdot .88143 .26753 Mdot .88143 .26754 Mdot .88286 .26725 Mdot .88286 .26725 Mdot .88286 .26726 Mdot .88286 .26725 Mdot .88429 .26696 Mdot .88429 .26698 Mdot .88429 .26697 Mdot .88429 .26697 Mdot .88571 .26669 Mdot .88571 .2667 Mdot .88571 .26669 Mdot .88571 .2667 Mdot .88714 .26642 Mdot .88714 .2664 Mdot .88714 .26642 Mdot .88714 .2664 Mdot .88857 .26613 Mdot .88857 .26613 Mdot .88857 .26613 Mdot .88857 .26614 Mdot .89 .26584 Mdot .89 .26586 Mdot .89 .26584 Mdot .89 .26586 Mdot .89143 .26556 Mdot .89143 .26557 Mdot .89143 .26558 Mdot .89143 .26556 Mdot .89286 .26529 Mdot .89286 .26528 Mdot .89286 .26529 Mdot .89286 .2653 Mdot .89429 .26501 Mdot .89429 .26501 Mdot .89429 .265 Mdot .89429 .26501 Mdot .89571 .26473 Mdot .89571 .26473 Mdot .89571 .26473 Mdot .89571 .26472 Mdot .89714 .26443 Mdot .89714 .26445 Mdot .89714 .26445 Mdot .89714 .26445 Mdot .89857 .26416 Mdot .89857 .26416 Mdot .89857 .26415 Mdot .89857 .26416 Mdot .9 .26389 Mdot .9 .26389 Mdot .9 .26388 Mdot .9 .26388 Mdot .90143 .2636 Mdot .90143 .26361 Mdot .90143 .2636 Mdot .90143 .26361 Mdot .90286 .26331 Mdot .90286 .26331 Mdot .90286 .26331 Mdot .90286 .26331 Mdot .90429 .26302 Mdot .90429 .26303 Mdot .90429 .26303 Mdot .90571 .26275 Mdot .90571 .26276 Mdot .90571 .26276 Mdot .90714 .26247 Mdot .90714 .26247 Mdot .90714 .26247 Mdot .90857 .26219 Mdot .90857 .26218 Mdot .90857 .26218 Mdot .91 .2619 Mdot .91 .2619 Mdot .91 .2619 Mdot .91143 .26161 Mdot .91143 .26162 Mdot .91143 .26162 Mdot .91286 .26133 Mdot .91286 .26134 Mdot .91286 .26132 Mdot .91429 .26105 Mdot .91429 .26104 Mdot .91429 .26105 Mdot .91571 .26076 Mdot .91571 .26076 Mdot .91571 .26077 Mdot .91714 .26047 Mdot .91714 .26048 Mdot .91714 .26047 Mdot .91857 .26018 Mdot .91857 .2602 Mdot .91857 .26019 Mdot .92 .25991 Mdot .92 .2599 Mdot .92 .25991 Mdot .92143 .25962 Mdot .92143 .25961 Mdot .92143 .25962 Mdot .92286 .25934 Mdot .92286 .25934 Mdot .92286 .25932 Mdot .92429 .25904 Mdot .92429 .25905 Mdot .92429 .25905 Mdot .92571 .25875 Mdot .92571 .25876 Mdot .92571 .25876 Mdot .92714 .25847 Mdot .92714 .25846 Mdot .92714 .25846 Mdot .92857 .25819 Mdot .92857 .25819 Mdot .92857 .25818 Mdot .93 .2579 Mdot .93 .2579 Mdot .93 .2579 Mdot .93143 .25761 Mdot .93143 .25761 Mdot .93143 .25761 Mdot .93286 .25731 Mdot .93286 .25731 Mdot .93286 .25731 Mdot .93429 .25703 Mdot .93429 .25703 Mdot .93429 .25704 Mdot .93571 .25675 Mdot .93571 .25675 Mdot .93571 .25675 Mdot .93714 .25646 Mdot .93714 .25646 Mdot .93714 .25645 Mdot .93857 .25617 Mdot .93857 .25616 Mdot .93857 .25616 Mdot .94 .25587 Mdot .94 .25587 Mdot .94 .25588 Mdot .94143 .25558 Mdot .94143 .25559 Mdot .94143 .25559 Mdot .94286 .2553 Mdot .94286 .2553 Mdot .94286 .25528 Mdot .94429 .25501 Mdot .94429 .255 Mdot .94429 .25501 Mdot .94571 .25472 Mdot .94571 .25471 Mdot .94571 .25472 Mdot .94714 .25442 Mdot .94714 .25443 Mdot .94714 .25442 Mdot .94857 .25412 Mdot .94857 .25414 Mdot .94857 .25412 Mdot .95 .25384 Mdot .95 .25384 Mdot .95 .25384 Mdot .95143 .25355 Mdot .95143 .25354 Mdot .95143 .25355 Mdot .95286 .25326 Mdot .95286 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