(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 13.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 17800684, 295422] NotebookOptionsPosition[ 17794195, 295322] NotebookOutlinePosition[ 17794617, 295339] CellTagsIndexPosition[ 17794574, 295336] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["C & Q Chaos IV - Phenomenology of broken integrability", "Title", CellChangeTimes->{{3.9514065972488813`*^9, 3.951406693566392*^9}},ExpressionUUID->"1175be48-f6c6-4551-858d-\ 66badba89f7c"], Cell[TextData[{ "Last time we explored the concept of what is Liouville integrability, and \ we started discussing what is the route to it being broken. The key element \ were the homoclinic/heteroclinic orbits and the tangles that they created \ upon perturbation. This explained the weird phenomenon that we saw around the \ unstable equilibrium in the phase space of the periodically driven \ mathematical pendulum. \n", Cell[BoxData[ GraphicsBox[{{}, { TagBox[ {RGBColor[0.24, 0.6, 0.8], PointSize[ NCache[ Rational[1, 360], 0.002777777777777778]], AbsoluteThickness[2], PointBox[CompressedData[" 1:eJwd13c8lu8XB3CEr0SlpDTsqKyUhMZjF1JIVMhKWYnsvbP33h6eQaFUqKhj ZGREJCPKaAjJqFDR79y/v3q9X59zznXd1z08CVre1LdmYmBgmFzHwED8y8Dw X7a5JicUNV88E59ZQmJgEBWf190DGaeczEMIB35ke7u2Fz5f1C1xI6zkqP/I QBLOxZmIXCNsvq7W68IhcG0W6zD4v9WWcxnkQOzpRLHy//sTXjveVQAJW8vH EoRHE1lyDE+A6C/q4rb/u4R7KFAJJhQuXl3NIBwXfshQBWLaqWwThO+XcXeJ q0FX4d7+ZsLmgjoGjBrQY/TlXTHhuZQy37engElMZnsk4YTMlBOlmrDy8L37 dcKBvaI1Qdqw4U7LJjXCm/UlRgx1IHnp0Cjf/+sd1qVLnINrkqy9S+noCD8u 1iBdCHsvwdBF+OKRwGR7PVD49cKaQrj4l5ODoT54yrya9iB839RCSPk8zNWe KtMkPHCj7664AUAJTzEv4dFrFwN5LoAY07X6L2nEvDhVGUZDcN2nsf0R4Yzj hlXThvCv2oDuT3hUfTnsrRH8Yfl9/hRhAW9PmfqLICly/uRGwvK/GiruXgLD 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{"DisplayFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& )}}, PlotRange->{{-3.141006546186542, 3.141464300494466}, {-2.055693845721765, 2.50001334209923}}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], GeneratedCell->False, CellAutoOverwrite->False, CellChangeTimes->{3.9501813447853203`*^9, 3.9501813755634117`*^9, 3.950181420210991*^9, 3.9501815225765953`*^9, 3.950181572496559*^9, 3.950181623216316*^9},ExpressionUUID-> "3600187e-8c3f-41a8-9c9e-e5a18c3700b5"], "\n\nIn this lesson, we will wander into similar structures that appear even \ near resonances. 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