From d5771a2fea57c0c37a8c9e2f70ff5367c0b36b36 Mon Sep 17 00:00:00 2001 From: Tianyi Wang <202015@houde.edu.cn> Date: Mon, 1 Jul 2019 19:06:16 -0400 Subject: [PATCH 1/5] edit --- .DS_Store | Bin 0 -> 8196 bytes Final Project/.DS_Store | Bin 0 -> 6148 bytes Homework/.DS_Store | Bin 0 -> 6148 bytes Homework/Drafts/ComputationalEssayTemplate.nb | 23 +++++++++++------- 4 files changed, 14 insertions(+), 9 deletions(-) create mode 100644 .DS_Store create mode 100644 Final Project/.DS_Store create mode 100644 Homework/.DS_Store diff --git a/.DS_Store b/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..976a46c91ad82d1b78dae327b29e07270d67a0cd GIT binary patch literal 8196 zcmeHMPiqrF6o1=9-BhYMi0wt#i{PQurVWNr3Nejh@luG`gG!q}YZKkujmva+pmf@p?_Y*dc% zi)e-vKF@2YOsJ7-KnwchZ6)$0R;!03&46LRFkl!k3>XFs1E+-nJhOSRM?ClCtZ5Ac zhJiE50M8E&D#wh%oyCwJY?ZNr-*!gl zQi*~?mhVD*(DIR*#dfo^FbkKHM51r zBmQfx5LAk)|9~;C-!Bv&R^9eS{K^XRgF@AJ8$!8Sb|Lb3z3P`)JIh*rxhdAk%L&_# 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b/Homework/Drafts/ComputationalEssayTemplate.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 2498, 79] -NotebookOptionsPosition[ 1424, 43] -NotebookOutlinePosition[ 2402, 77] -CellTagsIndexPosition[ 2336, 72] +NotebookDataLength[ 2737, 84] +NotebookOptionsPosition[ 1587, 48] +NotebookOutlinePosition[ 2549, 81] +CellTagsIndexPosition[ 2483, 76] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -38,7 +38,12 @@ Cell["< First and Last name>", "Text", 3.5498187170087957`*^9}, {3.549819049020514*^9, 3.5498190533440437`*^9}, { 3.77005561844785*^9, 3.7700556489651537`*^9}},ExpressionUUID->"94536f76-3fd2-4bf6-b228-\ -fa93ec9acae9"] +fa93ec9acae9"], + +Cell[BoxData["123456"], "Input", + CellChangeTimes->{{3.771010837572556*^9, + 3.7710108378604507`*^9}},ExpressionUUID->"5b259aa7-ff12-41d7-ae23-\ +7df2fb1a06ab"] }, Open ]] }, WindowSize->{720, 855}, @@ -55,8 +60,7 @@ DockedCells->{ RGBColor[0.97, 0.83, 0.35], FontFamily -> "Source Sans Pro", FontSize -> 40]}], CellFrameMargins -> 10, Background -> RGBColor[0.12, 0.14, 0.15]]}, -FrontEndVersion->"11.3 for Mac OS X x86 (32-bit, 64-bit Kernel) (March 5, \ -2018)", +FrontEndVersion->"12.0 for Mac OS X x86 (64-bit) (April 11, 2019)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) @@ -71,7 +75,7 @@ CellTagsIndex->{ *) (*CellTagsIndex CellTagsIndex->{ - {"Title", 2195, 66} + {"Title", 2342, 70} } *) (*NotebookFileOutline @@ -79,7 +83,8 @@ Notebook[{ Cell[CellGroupData[{ Cell[580, 22, 461, 10, 160, "Title",ExpressionUUID->"49ab34ee-bfd5-49f5-beef-400d3cc456dd", CellTags->"Title"], -Cell[1044, 34, 364, 6, 35, "Text",ExpressionUUID->"94536f76-3fd2-4bf6-b228-fa93ec9acae9"] +Cell[1044, 34, 364, 6, 35, "Text",ExpressionUUID->"94536f76-3fd2-4bf6-b228-fa93ec9acae9"], +Cell[1411, 42, 160, 3, 30, "Input",ExpressionUUID->"5b259aa7-ff12-41d7-ae23-7df2fb1a06ab"] }, Open ]] } ] From 97e244eada546373ef9d4c43dcf723f9288c930c Mon Sep 17 00:00:00 2001 From: Tianyi Wang <202015@houde.edu.cn> Date: Mon, 1 Jul 2019 19:09:53 -0400 Subject: [PATCH 2/5] 111 --- .DS_Store | Bin 8196 -> 8196 bytes Final Project/.DS_Store | Bin 6148 -> 6148 bytes .../FinalProjectNBTemplate.nb | 21 +++++++++++------- Homework/.DS_Store | Bin 6148 -> 6148 bytes 4 files changed, 13 insertions(+), 8 deletions(-) diff --git a/.DS_Store b/.DS_Store index 976a46c91ad82d1b78dae327b29e07270d67a0cd..14aba2635caeebbc296d942dd0fa1286ab6db91b 100644 GIT binary patch delta 55 zcmV-70LcG@K!iZB;SmAIljReD3Kx5OGB+$BG&C`ja1Normal*) (* Beginning of Notebook Content *) @@ -39,7 +39,12 @@ Cell["< Mentor\[CloseCurlyQuote]s First and Last name>", "Text", 3.5498187208956137`*^9, 3.5498187221413403`*^9}, {3.5498190558319674`*^9, 3.54981906151682*^9}, {3.770055624686027*^9, 3.770055659820487*^9}},ExpressionUUID->"b70c649b-a958-4202-9f22-\ -238ea1f9c7d4"] +238ea1f9c7d4"], + +Cell[BoxData["1111"], "Input", + CellChangeTimes->{{3.771011381835395*^9, + 3.77101138228284*^9}},ExpressionUUID->"c9b6398c-65a8-4dd6-9a38-\ +ad86b10c7319"] }, Open ]] }, WindowSize->{720, 855}, @@ -56,8 +61,7 @@ DockedCells->{ RGBColor[0.97, 0.83, 0.35], FontFamily -> "Source Sans Pro", FontSize -> 40]}], CellFrameMargins -> 10, Background -> RGBColor[0.12, 0.14, 0.15]]}, -FrontEndVersion->"11.3 for Mac OS X x86 (32-bit, 64-bit Kernel) (March 5, \ -2018)", +FrontEndVersion->"12.0 for Mac OS X x86 (64-bit) (April 11, 2019)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) @@ -74,7 +78,8 @@ Notebook[{ Cell[CellGroupData[{ Cell[580, 22, 213, 4, 98, "Title",ExpressionUUID->"49ab34ee-bfd5-49f5-beef-400d3cc456dd"], Cell[796, 28, 364, 6, 35, "Text",ExpressionUUID->"94536f76-3fd2-4bf6-b228-fa93ec9acae9"], -Cell[1163, 36, 341, 5, 35, "Text",ExpressionUUID->"b70c649b-a958-4202-9f22-238ea1f9c7d4"] +Cell[1163, 36, 341, 5, 35, "Text",ExpressionUUID->"b70c649b-a958-4202-9f22-238ea1f9c7d4"], +Cell[1507, 43, 155, 3, 30, "Input",ExpressionUUID->"c9b6398c-65a8-4dd6-9a38-ad86b10c7319"] }, Open ]] } ] diff --git a/Homework/.DS_Store b/Homework/.DS_Store index 69810b88a6b5baaf793edc1684b83339fd7eab4f..8be380123a6ae3ddf639106159ccb91a48ea71cf 100644 GIT binary patch delta 46 zcmZoMXffCj%))qLatKQUmw0uxk-3h7iHYIl`z*4IGbaCKk!PH_S(fz`(`Gh~fBXPm C8V?Zw delta 47 zcmZoMXffCj%))qbatKQUw?uWdp^34Mg0YF^ DYF7_Q From 47dc280211e8bf5180d4666b5de0be77e40ec14b Mon Sep 17 00:00:00 2001 From: Tianyi Wang <202015@houde.edu.cn> Date: Mon, 1 Jul 2019 19:22:27 -0400 Subject: [PATCH 3/5] Edit --- .DS_Store | Bin 8196 -> 8196 bytes Final Project/.DS_Store | Bin 6148 -> 8196 bytes .../FinalProjectNBTemplate.nb | 18 ++++++++++++------ 3 files changed, 12 insertions(+), 6 deletions(-) diff --git a/.DS_Store b/.DS_Store index 14aba2635caeebbc296d942dd0fa1286ab6db91b..739107bc8efb04c4f8be6334bd3997b74356f568 100644 GIT binary patch delta 47 zcmZp1XmQx^P;l~N;T8@PGYcIBBNNNX`67;!8wI5|pAfQR-OMiWjb-u`5#i1JqFgr1OfqWAmIu!d}HBv=E?jro*=6jm>|}IT*9z9o@Wj-0375XQ2+n{ diff --git a/Final Project/Final Submission/FinalProjectNBTemplate.nb b/Final Project/Final Submission/FinalProjectNBTemplate.nb index 8c97ba0c..354bc30d 100644 --- a/Final Project/Final Submission/FinalProjectNBTemplate.nb +++ b/Final Project/Final Submission/FinalProjectNBTemplate.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 2742, 79] -NotebookOptionsPosition[ 1678, 49] -NotebookOutlinePosition[ 2486, 76] -CellTagsIndexPosition[ 2443, 73] +NotebookDataLength[ 2994, 85] +NotebookOptionsPosition[ 1838, 54] +NotebookOutlinePosition[ 2646, 81] +CellTagsIndexPosition[ 2603, 78] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -44,7 +44,12 @@ Cell["< Mentor\[CloseCurlyQuote]s First and Last name>", "Text", Cell[BoxData["1111"], "Input", CellChangeTimes->{{3.771011381835395*^9, 3.77101138228284*^9}},ExpressionUUID->"c9b6398c-65a8-4dd6-9a38-\ -ad86b10c7319"] +ad86b10c7319"], + +Cell[BoxData["222"], "Input", + CellChangeTimes->{{3.771012134736993*^9, + 3.7710121350537558`*^9}},ExpressionUUID->"28c1ce50-92ef-4b74-b5da-\ +11d8066a3c29"] }, Open ]] }, WindowSize->{720, 855}, @@ -79,7 +84,8 @@ Cell[CellGroupData[{ Cell[580, 22, 213, 4, 98, "Title",ExpressionUUID->"49ab34ee-bfd5-49f5-beef-400d3cc456dd"], Cell[796, 28, 364, 6, 35, "Text",ExpressionUUID->"94536f76-3fd2-4bf6-b228-fa93ec9acae9"], Cell[1163, 36, 341, 5, 35, "Text",ExpressionUUID->"b70c649b-a958-4202-9f22-238ea1f9c7d4"], -Cell[1507, 43, 155, 3, 30, "Input",ExpressionUUID->"c9b6398c-65a8-4dd6-9a38-ad86b10c7319"] +Cell[1507, 43, 155, 3, 30, "Input",ExpressionUUID->"c9b6398c-65a8-4dd6-9a38-ad86b10c7319"], +Cell[1665, 48, 157, 3, 30, "Input",ExpressionUUID->"28c1ce50-92ef-4b74-b5da-11d8066a3c29"] }, Open ]] } ] From 27c51cbfed0c390da31926db278a9a0132c570f6 Mon Sep 17 00:00:00 2001 From: WangTianyi919 <51503088+WangTianyi919@users.noreply.github.com> Date: Thu, 11 Jul 2019 11:45:21 -0400 Subject: [PATCH 4/5] Add files via upload added my presentation nb --- presentation.nb | 28058 ++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 28058 insertions(+) create mode 100644 presentation.nb diff --git a/presentation.nb b/presentation.nb new file mode 100644 index 00000000..3a0739bb --- /dev/null +++ b/presentation.nb @@ -0,0 +1,28058 @@ +(* Content-type: application/vnd.wolfram.mathematica *) + +(*** Wolfram Notebook File ***) +(* http://www.wolfram.com/nb *) + +(* CreatedBy='Mathematica 12.0' *) + +(*CacheID: 234*) +(* Internal cache information: +NotebookFileLineBreakTest +NotebookFileLineBreakTest +NotebookDataPosition[ 158, 7] +NotebookDataLength[ 1544711, 28050] +NotebookOptionsPosition[ 1541814, 28002] +NotebookOutlinePosition[ 1542240, 28019] +CellTagsIndexPosition[ 1542197, 28016] +WindowFrame->Normal*) + +(* Beginning of Notebook Content *) +Notebook[{ +Cell[TextData[{ + StyleBox["Differential Equation and Chaos Theory", "Title"], + "\n", + StyleBox["Tianyi", + IgnoreSpellCheck->True], + " Wang, Wolfram High School Summer Camp 2019\n\nDifferential equations are \ +widely used in math, physics, and engineering. These equations are very \ +useful in modelling complex systems that are changing with time. ", + StyleBox["For this exploration, the differential equations we are looking at \ +have a particular form: x\[CloseCurlyQuote]\[CloseCurlyQuote][t]+", "CodeText", + FontColor->GrayLevel[0]], + StyleBox["x\[CloseCurlyQuote]", "Text", + IgnoreSpellCheck->True, + FontColor->GrayLevel[0]], + StyleBox["[t]==f[x[t]], where f is a real-valued function. The solutions \ +exhibit a variety of different behaviors, some of which are stable, others \ +chaotic. The goal of this project is to investigate different chaotic \ +behaviors of these equations and to be able to detect them. Using supervised \ +machine learning, we were able to classify different chaotic behaviors of \ +differential equations.", "CodeText", + FontColor->GrayLevel[0]] +}], "Text", + CellChangeTimes->{{3.771528128950418*^9, 3.771528186003169*^9}, { + 3.771528420719922*^9, 3.771528530690333*^9}, {3.7715285717612143`*^9, + 3.7715286904402647`*^9}, {3.771755036476959*^9, 3.771755067008974*^9}, { + 3.771755104757888*^9, 3.771755197369994*^9}, {3.771763412577745*^9, + 3.771763561350706*^9}, 3.771772096011525*^9, {3.7717802502779827`*^9, + 3.771780327811838*^9}},ExpressionUUID->"4aa1843d-4c86-4521-85f0-\ +5aa9bd552e59"], + +Cell[CellGroupData[{ + +Cell["1.Phase Portrait Visualization", "Section", + CellChangeTimes->{{3.7715293706844263`*^9, 3.7715293818254337`*^9}, { + 3.771844506293363*^9, + 3.771844506533245*^9}},ExpressionUUID->"c37bd754-db2c-4146-b582-\ +08c6547660bf"], + +Cell[BoxData[ + Graphics3DBox[{{{}, {}, + TagBox[ + {GrayLevel[0.5], AbsoluteThickness[2], Line3DBox[CompressedData[" 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For this \ +example, it is rotating around a specific point and finally converges, also \ +evidenced by the following plot:\ +\>", "Text", + CellChangeTimes->{{3.771529933013981*^9, 3.771529998249354*^9}, { + 3.771530062230709*^9, 3.771530097698976*^9}, {3.771755474392704*^9, + 3.771755505073258*^9}, {3.771780479424118*^9, + 3.7717804837070103`*^9}},ExpressionUUID->"fd92b9c9-05cd-40eb-924f-\ +5f71bc402756"], + +Cell[BoxData[ + GraphicsBox[{{{}, {}, + TagBox[ + {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[ + 1.], LineBox[CompressedData[" +1:eJwc23c81e8XAHB7z3u5NncYGQlRSHWMBhlJRjaFwtdONtllp1RCCpEoVCSU +VEgSaQqRhtFQZDX8zuf3V6/367r3fj7Pc855zvN8bjTPwD1eLExMTL2cTEzE +v45Cde15oUzg3XIl2SKDBMl9FTZjYUxwhPlyfC46ozz+KYsEMwRms1mmVZPA +TPsQSZTBDKy+my5r1pCA68EeW6W1zBBlJPrnVS2+f0Lh7S5DZnj2/Mhr6XoS +xNB7P508xAy6ZT/elDeRIKBU/J/ibWa4xjwis/MhCayL6tV2ObGAj+ykxd93 +JBBSOxfo4sUCVHcF+8BxEvS1JDcEBrKAP89U3tB7/P639ronk1gg94XKkfMf 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The \ +output includes a function that generates the phase portrait of some \ +differential equations, and a function that tells the interval(s) in which \ +the function is chaotic, for input functions (f[x]) that are continuous. \ +Finally, there is a machine learned classifier function that can classify \ +different behaviors of chaotic functions with accuracy about 99%-100%.", + FontColor->GrayLevel[0]]], "Text", + CellChangeTimes->{{3.771764633321381*^9, 3.7717647417734623`*^9}, { + 3.771771449846631*^9, 3.771771452104789*^9}, + 3.7717807752752666`*^9},ExpressionUUID->"bbdd345f-8044-4e41-8eac-\ +ea4bf27736fc"] +}, Open ]] +}, +WindowSize->{1428, 780}, +WindowMargins->{{Automatic, 0}, {Automatic, 0}}, +TaggingRules->{"SlideshowSettings" -> {"Toolbar" -> True}}, +Magnification:>1.5 Inherited, +FrontEndVersion->"12.0 for Mac OS X x86 (64-bit) (April 8, 2019)", +StyleDefinitions->"Default.nb" +] +(* End of Notebook Content *) + +(* Internal cache information *) +(*CellTagsOutline 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.../FinalProjectNBTemplate.nb | 61357 +++++++++++++++- 1 file changed, 61328 insertions(+), 29 deletions(-) diff --git a/Final Project/Final Submission/FinalProjectNBTemplate.nb b/Final Project/Final Submission/FinalProjectNBTemplate.nb index 354bc30d..72572132 100644 --- a/Final Project/Final Submission/FinalProjectNBTemplate.nb +++ b/Final Project/Final Submission/FinalProjectNBTemplate.nb @@ -10,50 +10,61184 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 2994, 85] -NotebookOptionsPosition[ 1838, 54] -NotebookOutlinePosition[ 2646, 81] -CellTagsIndexPosition[ 2603, 78] +NotebookDataLength[ 3349168, 61384] +NotebookOptionsPosition[ 3335580, 61188] +NotebookOutlinePosition[ 3336388, 61215] +CellTagsIndexPosition[ 3336345, 61212] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ -Cell["< Project Title >", "Title", +Cell["Differential Equation and Chaos Theory", "Title", CellChangeTimes->{{3.770055572438499*^9, 3.7700555729498453`*^9}, { - 3.770055604008165*^9, - 3.7700556066618843`*^9}},ExpressionUUID->"49ab34ee-bfd5-49f5-beef-\ + 3.770055604008165*^9, 3.7700556066618843`*^9}, {3.771885698835348*^9, + 3.771885710278338*^9}},ExpressionUUID->"49ab34ee-bfd5-49f5-beef-\ 400d3cc456dd"], -Cell["< First and Last name>", "Text", +Cell[TextData[{ + StyleBox["Tianyi", + IgnoreSpellCheck->True], + " Wang" +}], "Text", CellChangeTimes->{{3.549624168509506*^9, 3.549624171985794*^9}, { 3.549624508772977*^9, 3.549624510775043*^9}, {3.5498187159613867`*^9, 3.5498187170087957`*^9}, {3.549819049020514*^9, 3.5498190533440437`*^9}, { - 3.77005561844785*^9, - 3.7700556489651537`*^9}},ExpressionUUID->"94536f76-3fd2-4bf6-b228-\ + 3.77005561844785*^9, 3.7700556489651537`*^9}, {3.771885712718396*^9, + 3.771885714465294*^9}},ExpressionUUID->"94536f76-3fd2-4bf6-b228-\ fa93ec9acae9"], -Cell["< Mentor\[CloseCurlyQuote]s First and Last name>", "Text", +Cell[TextData[{ + StyleBox["Eryn", + IgnoreSpellCheck->True], + " ", + StyleBox["Gillam", + IgnoreSpellCheck->True], + ", Peter ", + StyleBox["Barendse", + IgnoreSpellCheck->True] +}], "Text", CellChangeTimes->{{3.549624515773694*^9, 3.5496245246892014`*^9}, { 3.5498187208956137`*^9, 3.5498187221413403`*^9}, {3.5498190558319674`*^9, - 3.54981906151682*^9}, {3.770055624686027*^9, - 3.770055659820487*^9}},ExpressionUUID->"b70c649b-a958-4202-9f22-\ + 3.54981906151682*^9}, {3.770055624686027*^9, 3.770055659820487*^9}, { + 3.771885717816828*^9, + 3.77188575754593*^9}},ExpressionUUID->"b70c649b-a958-4202-9f22-\ 238ea1f9c7d4"], -Cell[BoxData["1111"], "Input", - CellChangeTimes->{{3.771011381835395*^9, - 3.77101138228284*^9}},ExpressionUUID->"c9b6398c-65a8-4dd6-9a38-\ -ad86b10c7319"], +Cell[TextData[{ + "\nDifferential equations are widely used in math, physics, and engineering. \ +These equations are very useful in modelling complex systems that are \ +changing with time. ", + StyleBox["For this exploration, the differential equations we are looking at \ +have a particular form: x\[CloseCurlyQuote]\[CloseCurlyQuote][t]+", "CodeText", + FontColor->GrayLevel[0]], + StyleBox["x\[CloseCurlyQuote]", "Text", + IgnoreSpellCheck->True, + FontColor->GrayLevel[0]], + StyleBox["[t]==f[x[t]], where f is a real-valued function. The solutions \ +exhibit a variety of different behaviors, some of which are stable, others \ +chaotic. The goal of this project is to investigate different chaotic \ +behaviors of these equations and to be able to detect them. Using supervised \ +machine learning, we were able to classify different chaotic behaviors of \ +differential equations.", "CodeText", + FontColor->GrayLevel[0]] +}], "Text", + CellChangeTimes->{{3.771528128950418*^9, 3.771528186003169*^9}, { + 3.771528420719922*^9, 3.771528530690333*^9}, {3.7715285717612143`*^9, + 3.7715286904402647`*^9}, {3.771755036476959*^9, 3.771755067008974*^9}, { + 3.771755104757888*^9, 3.771755197369994*^9}, {3.771763412577745*^9, + 3.771763561350706*^9}, 3.771772096011525*^9, {3.7717802502779827`*^9, + 3.771780327811838*^9}, + 3.7718857943042316`*^9},ExpressionUUID->"1b7fc950-bb4b-4216-a8a0-\ +da01b9f94e43"], -Cell[BoxData["222"], "Input", - CellChangeTimes->{{3.771012134736993*^9, - 3.7710121350537558`*^9}},ExpressionUUID->"28c1ce50-92ef-4b74-b5da-\ -11d8066a3c29"] +Cell[CellGroupData[{ + +Cell["1.Visualization of Chaotic Differential Equation", "Section", + CellChangeTimes->{{3.7715314386568813`*^9, + 3.771531465515983*^9}},ExpressionUUID->"51322771-399a-4f55-aa83-\ +79ac5c451221"], + +Cell["\<\ +To get an idea of what a chaotic ODE looks like, we can generate plots for a \ +few examples. 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DisplayFunction->Identity, + Frame->{{False, False}, {False, False}}, + FrameLabel->{{None, None}, {None, None}}, + FrameTicks->{{Automatic, + Charting`ScaledFrameTicks[{Identity, Identity}]}, {Automatic, + Charting`ScaledFrameTicks[{Identity, Identity}]}}, + GridLines->{None, None}, + GridLinesStyle->Directive[ + GrayLevel[0.5, 0.4]], + ImagePadding->All, + Method->{ + "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> { + "Version" -> 1.2, "TrackMousePosition" -> {True, False}, + "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, + "Droplines" -> { + "freeformCursorMode" -> True, + "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, + "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ + (Identity[#]& )[ + Part[#, 1]], + (Identity[#]& )[ + Part[#, 2]]}& ), "CopiedValueFunction" -> ({ + (Identity[#]& )[ + Part[#, 1]], + (Identity[#]& )[ + Part[#, 2]]}& )}}, + 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Axes->{True, True}, + AxesLabel->{None, None}, + AxesOrigin->{Automatic, 0}, + DisplayFunction->Identity, + Frame->{{False, False}, {False, False}}, + FrameLabel->{{None, None}, {None, None}}, + FrameTicks->{{Automatic, + Charting`ScaledFrameTicks[{Identity, Identity}]}, {Automatic, + Charting`ScaledFrameTicks[{Identity, Identity}]}}, + GridLines->{None, None}, + GridLinesStyle->Directive[ + GrayLevel[0.5, 0.4]], + ImagePadding->All, + Method->{ + "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> { + "Version" -> 1.2, "TrackMousePosition" -> {True, False}, + "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, + "Droplines" -> { + "freeformCursorMode" -> True, + "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, + "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ + (Identity[#]& )[ + Part[#, 1]], + (Identity[#]& )[ + Part[#, 2]]}& ), "CopiedValueFunction" -> ({ 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AxesOrigin->{Automatic, Automatic}, + DisplayFunction->Identity, + Frame->{{False, False}, {False, False}}, + FrameLabel->{{None, None}, {None, None}}, + FrameTicks->{{Automatic, + Charting`ScaledFrameTicks[{Identity, Identity}]}, {Automatic, + Charting`ScaledFrameTicks[{Identity, Identity}]}}, + GridLines->{None, None}, + GridLinesStyle->Directive[ + GrayLevel[0.5, 0.4]], + ImagePadding->All, + Method->{ + "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> { + "Version" -> 1.2, "TrackMousePosition" -> {True, False}, + "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, + "Droplines" -> { + "freeformCursorMode" -> True, + "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, + "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ + (Identity[#]& )[ + Part[#, 1]], + (Identity[#]& )[ + Part[#, 2]]}& ), "CopiedValueFunction" -> ({ + (Identity[#]& )[ + Part[#, 1]], + 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-> { + "freeformCursorMode" -> True, + "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, + "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ + (Identity[#]& )[ + Part[#, 1]], + (Identity[#]& )[ + Part[#, 2]]}& ), "CopiedValueFunction" -> ({ + (Identity[#]& )[ + Part[#, 1]], + (Identity[#]& )[ + Part[#, 2]]}& )}}, + PlotRange->{All, All}, + PlotRangeClipping->True, + PlotRangePadding->{{ + Scaled[0.02], + Scaled[0.02]}, { + Scaled[0.05], + Scaled[0.05]}}, + Ticks->{Automatic, Automatic}], ",", + GraphicsBox[{{{}, {}, + TagBox[ + {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], + Opacity[1.], LineBox[CompressedData[" +1:eJwVV2c81u8XttdjrySkhyISki33MbPDF8VPsjKSWUSEKKOsZEXILCNKykgl +KXs830eilLKK7L39+786n/Pius51rvvFdZ9Djt7mF6koKCikqCko/l+zQx3p +ZJ3vqv8n53yAgoIbmsnCRmeF3VGRfHGYMx037H16GyksHITW+iOSKFm4Ibsx +/iCtcCwy6f/weJyfGy6f8vdcO/gACXszT1kqc8OLvp+Fy1oP0ZBXWE2YBjds 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"DefaultGraphicsInteraction" -> { + "Version" -> 1.2, "TrackMousePosition" -> {True, False}, + "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, + "Droplines" -> { + "freeformCursorMode" -> True, + "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, + "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ + (Identity[#]& )[ + Part[#, 1]], + (Identity[#]& )[ + Part[#, 2]]}& ), "CopiedValueFunction" -> ({ + (Identity[#]& )[ + Part[#, 1]], + (Identity[#]& )[ + Part[#, 2]]}& )}}, + PlotRange->{All, All}, + PlotRangeClipping->True, + PlotRangePadding->{{ + Scaled[0.02], + Scaled[0.02]}, { + Scaled[0.05], + Scaled[0.05]}}, + Ticks->{Automatic, Automatic}]}], "}"}]], "Input", + CellChangeTimes->{{3.771531539648981*^9, + 3.771531541911334*^9}},ExpressionUUID->"742314af-c123-4f78-81a3-\ +a1bfe6eef6fa"], + +Cell["\<\ + +As you can see, the 3rd, 4rd, 7th, and 9th graphs seem to exhibit chaotic \ +behavior. They are chaotic because they are oscillating wildly in certain \ +intervals. However, it doesn\[CloseCurlyQuote]t mean that the other ODEs are \ +not chaotic- they may exhibit chaos if we change our plot range or initial \ +conditions.\ +\>", "Text", + CellChangeTimes->{{3.7715315441558857`*^9, 3.771531602395949*^9}, { + 3.7715316424068737`*^9, 3.771531699316224*^9}, {3.7717803845895863`*^9, + 3.771780431310235*^9}},ExpressionUUID->"cbeb7331-7814-4a7e-ac7c-\ +4bda458ba1e9"] +}, Open ]], + +Cell[CellGroupData[{ + +Cell["2.Phase Portrait Visualization", "Section", + CellChangeTimes->{{3.7715293706844263`*^9, + 3.7715293818254337`*^9}},ExpressionUUID->"b20f8ecb-fc06-4d09-971f-\ +c6d57500cd56"], + +Cell[TextData[{ + StyleBox["Another way to visualize the chaotic behavior of an ODE is through \ +its phase portrait. A phase portrait shows ", + FontColor->GrayLevel[0]], + StyleBox["how a ", "CodeText", + FontColor->GrayLevel[0]], + StyleBox["solution x[t] to the ODE", + FontColor->GrayLevel[0]], + StyleBox[", its derivative, and its second derivative are changing", + "CodeText", + FontColor->GrayLevel[0]], + StyleBox[" as the independent variable (time, usually) changes", + FontColor->GrayLevel[0]], + StyleBox[". Here is an example that is ", "CodeText", + FontColor->GrayLevel[0]], + StyleBox["POSSIBLY", + FontColor->GrayLevel[0]], + StyleBox[" chaotic in range {t,0,15}:", "CodeText", + FontColor->GrayLevel[0]] +}], "Text", + CellChangeTimes->{{3.771529395351013*^9, 3.771529620384506*^9}, { + 3.771755334991972*^9, 3.771755422690702*^9}, {3.771772129059067*^9, + 3.771772138597995*^9}, + 3.77178046411742*^9},ExpressionUUID->"da69729d-e4f1-46e9-aed7-\ +16c84c9248b8"], + +Cell["\<\ +This piece of code first solves the differential equation numerically, then \ +generates a phase portrait in 3D space. y is the solution, Dy is its first \ +derivative, and DDy is the second derivative.\ +\>", "Text", + CellChangeTimes->{{3.7717554427539463`*^9, 3.771755449104808*^9}, { + 3.771763689848899*^9, 3.77176369512789*^9}, {3.771763735092689*^9, + 3.771763793056753*^9}},ExpressionUUID->"adec7b36-4708-426f-8ff3-\ +b1c7d719c4b8"], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{ + RowBox[{"y", "=", + 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+ "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, + "Droplines" -> { + "freeformCursorMode" -> True, + "placement" -> {"x" -> "All", "y" -> "None"}}}}}, + PlotRange->{{Automatic, Automatic}, {Automatic, Automatic}, { + Automatic, Automatic}}, + PlotRangePadding->{{ + Scaled[0.05], + Scaled[0.05]}, { + Scaled[0.05], + Scaled[0.05]}, { + Scaled[0.05], + Scaled[0.05]}}, + Ticks->{Automatic, Automatic, Automatic}, + ViewPoint->{1.3, -2.4, 2.}, + ViewVertical->{0., 0., 1.}]}], "}"}]], "Output", + CellChangeTimes->{{3.771530201221322*^9, 3.7715302103473253`*^9}}, + CellLabel->"Out[40]=",ExpressionUUID->"538cec8a-9c42-46a7-a80b-35445a8c675c"] +}, Open ]], + +Cell["\<\ +Many of these portraits don\[CloseCurlyQuote]t seem to have graphs in them. \ +The graphs are there, but since the volume of the rectangular space is very \ +large compared to the graph, they are not visible. We have to search through \ +these volumes to get visually distinguishable results.\ +\>", "Text", + CellChangeTimes->{{3.7715302321670103`*^9, 3.771530417705709*^9}, { + 3.771755675927786*^9, + 3.7717558326428003`*^9}},ExpressionUUID->"c657b96e-606c-4dca-b543-\ +ad3350e6bc97"], + +Cell["plot and rasterize, and then detect the amount of red:", "CodeText", + CellChangeTimes->{{3.7717723556176243`*^9, + 3.771772371967301*^9}},ExpressionUUID->"93ce6cbe-266f-4f30-9206-\ +248a574cff07"], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"p", "=", + RowBox[{"Rasterize", "@", + RowBox[{"ParametricPlot3D", "[", + RowBox[{ + RowBox[{"{", + RowBox[{"x", ",", + RowBox[{"x", "^", "3"}], ",", + RowBox[{"1", "-", "x"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"x", ",", "0", ",", "2"}], "}"}], ",", + RowBox[{"PlotStyle", "\[Rule]", + RowBox[{"{", + RowBox[{"Red", ",", "Thick"}], "}"}]}]}], + "]"}]}]}], "\[IndentingNewLine]", + RowBox[{"Total", "@", + RowBox[{"Flatten", "@", + RowBox[{"ImageData", 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These \ +cases need to be removed.\ +\>", "Text", + CellChangeTimes->{{3.77171205518353*^9, 3.771712086882423*^9}, { + 3.7717599985259247`*^9, + 3.771760029998274*^9}},ExpressionUUID->"6f97f8cc-c971-4922-ba33-\ +db5e4767e5a9"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"allclasses", "=", + RowBox[{ + RowBox[{"Select", "[", + RowBox[{ + RowBox[{"2", "<", + RowBox[{"ByteCount", "[", "#", "]"}], "<", "25000"}], "&"}], "]"}], "/@", + "allclasses"}]}], ";"}]], "Input", + CellLabel->"In[25]:=",ExpressionUUID->"08966ac6-c783-4648-aa89-edc69c927577"], + +Cell["\<\ +It is important to keep in mind that there may still be things we don\ +\[CloseCurlyQuote]t want inside our data. 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++3zNeK8k4nwgLra/BfNUsF2rsR51P2nEmJmAcWsXjFdUob9smVrL/IBzIy2G +Z08G5qXLX7p4EXXBvnHD4QBP1OdHgV5i6FMkz989NrOPDWknut73aqDe3/wQ +MS3IhgsdHgLm6zDPydrnWeiz4t5dpbkgzr16BTp5sE/b8jk/2WIdRz5+9LZy +RB+kKnmMB+uRnyLBJYx6U2tT0Pv//ckojsKxXgV5cG5xKo+J96Md2oFRFPRj +Cp8/zmLeXHTLTwLWyB/nC4vvOfG+xK5jHp8h4JVwU30T6uBWd/pgoQLeH0KV +Ey9iv39vDz4uiHNA4N5w5htb9FPfnPlO4j3Ev+pkxKgl8l9o6TsJnAdtL0ob +ZfG+NcTfuD4edSJt5d+/R07ifOCVTW/F/RYbmoce/H/v3y3vtxb5FmZzZ8v3 +APzd+5nIQ+RbkdWqMTExxLtuh9SYKvJGNnvKbgXy5KLuh8vaJNze1me/D/su +42rewI38/g8NGEFb + "], "Method" -> "Naive", "VectorLength" -> + 999, "Output" -> + Association[ + "f1" -> Association[ + "Type" -> "NumericalVector", "Weight" -> 1]], "Type" -> + "NumericalVector"]], "RandomImputer" -> + MachineLearning`MLProcessor["ImputeMissing", + Association[ + "Invertibility" -> "Perfect", "Missing" -> "Imputed", + "Input" -> + Association[ + "f1" -> Association[ + "Type" -> "NumericalVector", "Weight" -> 1]], + "Mean" -> CompressedData[" 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"NumericalVector", "Weight" -> 1]], + "Preprocessor" -> MachineLearning`MLProcessor["Sequence", + Association["Processors" -> { + MachineLearning`MLProcessor["List"], + MachineLearning`MLProcessor["WrapMLDataset", + Association[ + "FeatureTypes" -> {"NumericalVector"}, + "FeatureKeys" -> {"f1"}, "FeatureWeights" -> Automatic, + "ExampleWeights" -> Automatic, "RawExample" -> + Missing["KeyAbsent", "RawExample"]]]}]], "ScalarFeature" -> + True, "Invertibility" -> "Perfect", "Missing" -> + "Allowed"]], "Processor" -> + MachineLearning`MLProcessor["LogTransformNumericalVector", + Association[ + "Invertibility" -> "Perfect", "Missing" -> "Allowed", + "Input" -> + Association[ + "f1" -> Association[ + "Type" -> "NumericalVector", "Weight" -> 1]], + "Indices" -> CompressedData[" +1:eJwl12PXEIgWBtA327Zt27Zt27Zt27Zt27Zdk6eaaubu1v2w1zl/4DlI1LhD +5faBAgICrgcJCPhTA6MNCEowghOCkIQiNGEISzjCE4GIRCIyUYhKNKITg5jE +IjZxiEs84pOAhCQiMUlISjKSk4KUpCI1aUhLOtKTgYxkIjNZyEo2spODnOQi 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MachineLearning`MLProcessor["FromMLDataset", + Association[ + "DatasetFormat" -> Automatic, "Input" -> + Association[ + "f1" -> Association[ + "Type" -> "NumericalVector", "Weight" -> 1]], "Output" -> + Association[ + "f1" -> Association[ + "Type" -> "NumericalVector", "Length" -> 27]], + "InversePreprocessor" -> + MachineLearning`MLProcessor["Sequence", + Association["Processors" -> { + MachineLearning`MLProcessor["List"], + MachineLearning`MLProcessor["WrapMLDataset", + Association[ + "FeatureTypes" -> {"NumericalVector"}, + "FeatureKeys" -> {"f1"}, "FeatureWeights" -> {1}, + "ExampleWeights" -> 1]]}]], "ScalarFeature" -> True, + "Invertibility" -> "Perfect", "Missing" -> "Allowed"]], + "Model" -> Association["Matrix" -> CompressedData[" +1:eJwUmXc8V/8Xx0VKkpVCtibJSAnhoLILJUlWZWWUvWXv9bE3Hx8fe31sWYcQ ++aokSUKSUUiUikp+fo/7x/3z3nPe57yer9e9AnceXLWgpqKimqehouLavjvv +qmOPmrkLp66u2+mMpQGXdFnMHqk8+CJ7vcQ5ow879IpEyhem8Oant/Qup6aQ +JWlg/wP9Kbx9kE1zZS8FvzVkmGdSsmAyyk8zr6MTRQ1nPe551eD46lFBF+Fq 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+ "Type" -> "NumericalVector", "Weight" -> 1]], "Type" -> + "NumericalVector", "Version" -> {12., 0}, "ID" -> + 7263179337424669307]], + MachineLearning`MLProcessor["LogRescaleNumericalVector", + Association[ + "Invertibility" -> "Perfect", "Missing" -> "Allowed", "Input" -> + Association[ + "f1" -> Association[ + "Type" -> "NumericalVector", "Weight" -> 1]], + "Indices" -> CompressedData[" +1:eJwl19OyGIgSBdATTWzbtm3btm3btm3btm3bziB2clfqPqzq/oHdSNiofaV2 +gQICAp4FCQj4UwOjDQhKMIITgpCEIjRhCEs4whOBiEQiMlGISjSiE4OYxCI2 +cYhLPOKTgIQkIjFJSEoykpOClKQiNWlISzrSk4GMZCIzWchKNrKTg5zkIjd5 +yEs+8lOAghSiMEUoSjGKU4KSlKI0ZShLOcpTgYpUojJVqEo1qlODmtSiNnWo +Sz3q04CGNKIxTWhKM5rTgpa0ojVtaEs72tOBjnSiM13oSje604Oe9KI3fehL +P/ozgIEMYjBDGMowhjOCkYxiNGMYyzjGM4GJTGIyU5jKNKYzg5nMYjZzmMs8 +5rOAhSxiMUtYyjKWs4KVrGI1a1jLOtazgY1sYjNb2Mo2trODnexiN3vYyz72 +c4CDHOIwRzjKMY5zgpOc4jRnOMs5znOBi1ziMle4yjWuc4Ob3OI2d7jLPe7z +gIc84jFPeMoznvOCl7ziNW/4m3/4l/94yzve84GPfOIzX/jKN77zg5/84jd/ 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"Nominal"]], + "Output" -> + Association["f1" -> Association["Type" -> "Nominal", "Weight" -> 1]], + "Preprocessor" -> MachineLearning`MLProcessor["Sequence", + Association["Processors" -> { + MachineLearning`MLProcessor["List"], + MachineLearning`MLProcessor["WrapMLDataset", + Association[ + "FeatureTypes" -> {"Nominal"}, "FeatureKeys" -> {"f1"}, + "FeatureWeights" -> Automatic, "ExampleWeights" -> Automatic, + "RawExample" -> Missing["KeyAbsent", "RawExample"]]]}]], + "ScalarFeature" -> True, "Invertibility" -> "Perfect", "Missing" -> + "Allowed"]], "Processor" -> MachineLearning`MLProcessor["Sequence", + Association[ + "Input" -> Association[ + "f1" -> Association["Type" -> "Nominal", "Weight" -> 1]], "Output" -> + Association["f1" -> Association["Type" -> "Nominal", "Weight" -> 1]], + "Processors" -> { + MachineLearning`MLProcessor["ToVector", + Association[ + "Invertibility" -> "Perfect", "Missing" -> "Allowed", "Input" -> + Association[ + "f1" -> Association["Type" -> "Nominal", "Weight" -> 1]], + "Output" -> + Association[ + "f1" -> Association["Type" -> "NominalVector", "Weight" -> 1]], + "Version" -> {12., 0}, "ID" -> 1594110942329843211]], + MachineLearning`MLProcessor["IntegerEncodeNominalVector", + Association[ + "Invertibility" -> "Perfect", "Missing" -> "Allowed", "Input" -> + Association[ + "f1" -> Association["Type" -> "NominalVector", "Weight" -> 1]], + "Index" -> { + Association[ + 1 -> 1, 2 -> 2, 3 -> 3, 4 -> 4, 6 -> 5, 7 -> 6, 8 -> 7, 9 -> + 8]}, "MissingCode" -> 0, "Version" -> {12., 0}, "ID" -> + 2148322281829064388, "Output" -> + Association[ + "f1" -> Association[ + "Type" -> "NominalVector", "Weight" -> 1]]]], + MachineLearning`MLProcessor["FromVector", + Association[ + "Invertibility" -> "Perfect", "Missing" -> "Allowed", "Input" -> + Association[ + "f1" -> Association[ + "Type" -> "NominalVector", "Weight" -> 1, "SetSize" -> {8}]], + "Output" -> + Association[ + "f1" -> Association["Type" -> "Nominal", "Weight" -> 1]], + "Version" -> {12., 0}, "ID" -> 1567982190013531112]], + MachineLearning`MLProcessor["FirstValues", + Association[ + "Info" -> Association[ + "Type" -> "Nominal", "Weight" -> 1, "SetSize" -> 8], "Key" -> + "f1", "Invertibility" -> "Perfect", "Missing" -> "Allowed"]]}, + "Invertibility" -> "Perfect", "Missing" -> "Allowed"]], + "ProbabilityPostprocessor" -> Identity, "Name" -> "class", "Marginal" -> + Association[ + 1 -> 0.1288265306122449, 2 -> 0.09821428571428571, 3 -> + 0.1288265306122449, 4 -> 0.1288265306122449, 6 -> 0.1288265306122449, + 7 -> 0.1288265306122449, 8 -> 0.1288265306122449, 9 -> + 0.1288265306122449]], "Prior" -> Automatic, "Utility" -> + SparseArray[ + Automatic, {8, 9}, 0., { + 1, {{0, 1, 2, 3, 4, 5, 6, 7, 8}, {{2}, {3}, {4}, {5}, {6}, {7}, {8}, { + 9}}}, {1., 1., 1., 1., 1., 1., 1., 1.}}], "Threshold" -> 0, + "TieBreaker" -> RandomChoice, "PerformanceGoal" -> Automatic, + "BatchProcessing" -> Automatic, "Model" -> + Association["LogProbabilitiesFunction" -> LinearLayer[ + Association[ 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This shows that ", "Text"], + StyleBox[Cell[BoxData[ + FormBox[ + RowBox[{ + SubscriptBox["\[CapitalDelta]", "i"], "=", + RowBox[{ + SubscriptBox["d", + RowBox[{"i", "+", "1"}]], "-", + SubscriptBox["d", "i"]}]}], TraditionalForm]], "Text",ExpressionUUID-> + "28dddef4-8bf1-45fc-9b96-a0bfd69d8c9b"], "Text"], + StyleBox["< \[Epsilon]. \[EmptySquare]", "Text"] +}], "Text", + CellChangeTimes->{{3.771808759687923*^9, + 3.771808770640636*^9}},ExpressionUUID->"de1f7675-3e36-4f0c-b403-\ +d614bffc60fa"] +}, Open ]], + +Cell[CellGroupData[{ + +Cell["8.Conclusions", "Section", + CellChangeTimes->{{3.771764547513405*^9, + 3.771764551054288*^9}},ExpressionUUID->"95fb5d62-db7a-4fca-9988-\ +4a6fd4eff07c"], + +Cell[TextData[StyleBox["Throughout this project, I successfully detected the \ +chaotic behavior of different differential equations and classified them. The \ +output includes a function that generates the phase portrait of some \ +differential equations, and a function that tells the interval(s) in which \ +the function is chaotic, for input functions (f[x]) that are continuous. \ +Finally, there is a machine learned classifier function that can classify \ +different behaviors of chaotic functions with accuracy about 99%-100%.", + FontColor->GrayLevel[0]]], "Text", + CellChangeTimes->{{3.771764633321381*^9, 3.7717647417734623`*^9}, { + 3.771771449846631*^9, 3.771771452104789*^9}, + 3.7717807752752666`*^9},ExpressionUUID->"55e0b427-016a-4737-afc5-\ +255616223f5a"] +}, Open ]] }, Open ]] }, -WindowSize->{720, 855}, -WindowMargins->{{360, Automatic}, {Automatic, 0}}, +WindowSize->{1215, 781}, +WindowMargins->{{40, Automatic}, {Automatic, 21}}, DockedCells->{ Cell[ TextData[{ @@ -66,7 +61200,7 @@ DockedCells->{ RGBColor[0.97, 0.83, 0.35], FontFamily -> "Source Sans Pro", FontSize -> 40]}], CellFrameMargins -> 10, Background -> 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