(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 25619718, 420849] NotebookOptionsPosition[ 25614119, 420684] NotebookOutlinePosition[ 25614793, 420708] CellTagsIndexPosition[ 25614750, 420705] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["The Duffing Equation", "Title"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell["\<\ We have already seen that chaotic behavior can emerge in a system as simple \ as the logistic map. In that case the \"route to chaos\" is called \ period-doubling. In practice one would like to understand the route to chaos \ in systems described by partial differential equations, such as flow in a \ randomly stirred fluid. This is, however, very complicated and difficult to \ treat either analytically or numerically. Here we consider an intermediate \ situation where the dynamics is described by a single ordinary differential \ equation, called the Duffing equation. In order to get chaos in such a simple system, we will need to add both a \ driving force and friction. First of all though we just consider the basic \ equation without these extra features. The Duffing equation describes the \ motion of a classical particle in a double well potential. We choose the \ units of length so that the minima are at x = \[PlusMinus] 1, and the units \ of energy so that the depth of each well is at -1/4. 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In fact this must \ be so since energy is conserved, i.e. ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ SuperscriptBox["v", "2"], "2"], " ", "-", FractionBox[ SuperscriptBox["x", "2"], "2"], "+", " ", FractionBox[ SuperscriptBox["x", "4"], "4"], " "}], TraditionalForm]]], "is constant, so if x is given then so is v, up to a sign. You can \ experiment by changing the time up to which you integrate in order to \ determine the time to go round the orbit once (the period) and to verify that \ at longer times than this the motion just repeats. " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Friction and Damping; periodic orbits", "Section"], Cell["\<\ Because of energy conservation one can clearly never get chaos from the \ motion of a single degree of freedom. We therefore add both a driving force \ and damping, in order to remove energy conservation. The equations of motion \ are then\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ FractionBox["dx", "dt"], " ", "=", " ", "v"}], " ", ",", " ", RowBox[{ FractionBox["dv", "dt"], " ", "=", " ", RowBox[{"x", " ", "-", " ", SuperscriptBox["x", "3"], "-", " ", "\[Gamma]v", " ", "+", " ", RowBox[{"d", " ", "cos", RowBox[{"(", "\[Omega]t", ")"}]}]}]}]}]], "DisplayFormula"], Cell["\<\ where \[Gamma] is the friction coefficient, and d is the strength of the \ driving force which oscillates at a frequency \[Omega]. We will see that a \ transition to chaos now occurs as the strength of the driving force is \ increased. 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The period of the orbit is \ just 2 \[Pi] / \[Omega] = 4.48799..., the period of the driving force. One \ can verify this by integrating for different periods of time. The figure \ below integrates over a time range of 4, and shows that the orbit is not \ quite completed, as expected." }], "Text", CellChangeTimes->{{3.452521510635829*^9, 3.4525215471793737`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", "[", "t", "]"}], ",", RowBox[{"v", "[", "t", "]"}]}], "}"}], "/.", "\[InvisibleSpace]", "sol2"}], ",", RowBox[{"{", RowBox[{"t", ",", "150", ",", "154"}], "}"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.4205105916474457`*^9, 3.420510593162703*^9}, { 3.420510984880333*^9, 3.420510989905943*^9}}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], PointSize[Large], Thickness[Large], LineBox[CompressedData[" 1:eJwV13c8VW0cAPBUpKJEKqOsEg0iW/oRCS0akpUyK69kp8xIIdlSpOyVUCGV h7JH9rznFrpk3eXaFe/TP3y+n3Puecb5jeeIXbt1znb1qlWr5vCff/8Zy9dP 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It is convenient to define a function, which we \ call \"", StyleBox["solution[d, tmax]", FontWeight->"Bold"], "\" , which will integrate the equations for a given value of the forcing, \ ", StyleBox["d", FontWeight->"Bold"], ", up to a time ", StyleBox["tmax", FontWeight->"Bold"], ". 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In \ fact the ", StyleBox["period has doubled", FontWeight->"Bold"], " to 4\[Pi]/\[Omega] as can be checked by trial and error. Depending on the \ exact value of d and the initial conditions the particle could have gone into \ a period doubled orbit near the x = +1 minimum. \n\nNext let's increase the \ driving force to 0.338." }], "Text", CellChangeTimes->{{3.4525215652129374`*^9, 3.452521583348371*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"sol3", " ", "=", " ", RowBox[{"solution", " ", "[", " ", RowBox[{"0.338", ",", " ", "2000"}], " ", "]"}]}], ";"}]], "Input"], Cell[BoxData[ RowBox[{"graph", "[", RowBox[{"1900", ",", " ", "2000"}], "]"}]], "Input", CellChangeTimes->{3.42051111381024*^9}], Cell[BoxData[ RowBox[{ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], PointSize[Large], Thickness[Large], LineBox[CompressedData[" 1:eJwUmnc4l98bx0syslrITEJlRiHCrVJKMstIUbLqG0nJJmUTRdmEQiJZCeHY e+/ts+3xGTIqfp/fX891rnM953mu89znfr3e1/Ucs35iZMuwa9euIbZdu/5/ HXXeSJD6QAMnDXyl75km6NMq4yZ+ocHT8asH/mzWwoU3hncel9Hg7MGrQ856 v0DanHvzaz0NdqkvqQX6FYGHfNKvyDYaNJBYLBSPfQKGQ172+zpp4P4m6ITx 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One \ might expect that the ", StyleBox["period has doubled again", FontWeight->"Bold"], ", and this is indeed the case. We have started a period doubling sequence \ of \"bifurcations\" that we first met in the logistic map. 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