(*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 0, 0] NotebookDataLength[ 366486, 6948] NotebookOptionsPosition[ 356062, 6672] NotebookOutlinePosition[ 356403, 6687] CellTagsIndexPosition[ 356360, 6684] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Lyapunov Exponents Computation \nfor Various Non-Linear Dynamic \ Problems\n", "Subsubtitle", FontSize->18, FontVariations->{"CompatibilityType"->0}], StyleBox["Authors' Data", FontSize->14, FontWeight->"Bold"], StyleBox[": ", FontSize->14], StyleBox["Housam Binous & Nasri Zakia\nDepartment of Chemical Engineering\n\ National Institute of Applied Sciences and Technology\nTunis, TUNISIA\nEmail: \ binoushousam@yahoo.com ", FontSize->14, FontWeight->"Plain"] }], "Title", CellChangeTimes->{{3.4000354481602945`*^9, 3.400035458715472*^9}, 3.400035572909675*^9, {3.407590174329053*^9, 3.4075902150375886`*^9}, { 3.4130982131726866`*^9, 3.413098218971024*^9}, {3.4132653420031567`*^9, 3.413265386106574*^9}, 3.413288850406581*^9}, TextAlignment->Center, Background->RGBColor[0.605478, 0.996109, 0.605478]], Cell["\<\ Acknowlegment : 1- Sandri, M. \"Numerical Calculation of Lyapunov Exponents.\" Mathematica J. \ 6, 78-84, 1996. http://library.wolfram.com/infocenter/Articles/2902/. 2- Govorukhin V.N. E-Mail: vgov@math.rsu.ru, matds@nm.ru MATDS is a MATLAB-based program for dynamical system investigation. It works \ with version 6.5* of MATLAB. http://kvm.math.rsu.ru/matds/\ \>", "Subsubtitle", CellChangeTimes->{{3.4132872054011793`*^9, 3.413287212130856*^9}, { 3.4132874243259773`*^9, 3.413287452055851*^9}, 3.4132877204017134`*^9, { 3.4132877553519697`*^9, 3.4132878168904576`*^9}}, Background->RGBColor[0.7176470588235294, 1., 0.7176470588235294]], Cell["\<\ One can compare the results obtained in the present notebook (using \ Mathematica' s methods by M. Sandir) with the Matlab files in the compressed \ folder lyapunov2.zip (using the method by Govorukhin which is inpired from \ earlier work by A.Wolf, J.B.Swift, H.L.Swinney, and J.A.Vastano, \ \"Determining Lyapunov Exponents from a Time Series,\" Physica D, Vol .16, \ pp.285 - 317, 1985).\ \>", "Subsubtitle", CellChangeTimes->{{3.4132878283869886`*^9, 3.4132880511072445`*^9}, 3.413288863245042*^9}, Background->RGBColor[1., 0.7764705882352941, 0.6666666666666666]], Cell[CellGroupData[{ Cell["\<\ Test with Lorentz system. We get a chaotic behavior for this choice of \ parametres. 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K. Scott and A. S. Tomlin, Period doubling and other complex \ bifurcations in non-isothermal chemical systems, Phil. Trans. R. Soc. Lond. A \ (1990) 332, 51-68.", FontSize->16] }], "Subtitle", CellChangeTimes->{{3.4130982586981487`*^9, 3.413098272748352*^9}, { 3.413287148018667*^9, 3.413287158113182*^9}, 3.4132884128774447`*^9, { 3.413288904063736*^9, 3.4132889487880464`*^9}}, Background->RGBColor[1, 0, 1]], Cell[CellGroupData[{ Cell["\<\ For \[Mu] = 0.498. Estimate for the Max lyapunov exponent is almost zero \ (-0.008). Thus all exponents are less than zero. 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Max lyapunov exponent is positive (0.035). 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