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First, using ", StyleBox["Mathematica", FontSlant->"Italic"], " define the coefficient matrix: \n\nx1'[t] = 13 x1[t] +4 x2[t] \nx2'[t] \ = 4 x1[t] + 7 x2[t] \nx1[0] = 11, x2[0] = 23\n\nUsing ", StyleBox["Mathematica", FontSlant->"Italic"], " we define the coefficient matrix:\n\nA = { {13,4}, {4,7} };\n\nand the \ initial vector: \n\nX0 = { {11}, {23} };\n\n2. Then a fundamental matrix for \ the system: x' = Ax is given by: \n\nfundamental = MatrixExp[ At ]\n\nHence \ the particular solution can be calculated as follows:\n\nsolution = \ fundamental . x0;\nSimplify [ solution ]\n\nb) Plot the particular solution \ in the phase space (x1,x2)\n\nc) Solve the initial value problem" }], "Text"], Cell[BoxData[{ \(\(x1'\)[t]\ = \ \(-149\)\ x1[t]\ - 50\ x2[t]\ - \ 154\ x3[t]\), \(\(x2'\)[t]\ = \ \ \ \ 537\ x1[t]\ + \ 180 x2[t]\ + \ 546\ x3[t]\), \(\(x3'\)[t]\ = \ \ \ \ \(-27\)\ x1[t]\ - 9 x2[t]\ - 25\ x3[t], \n x1[0]\ = \ 17, \ \ x2[0]\ = \ 43, \ \ x3[0]\ = \ 79. \)}], "Input"], Cell["Solution", "Subsubsection"], Cell["************************************************************", "Input"], Cell[CellGroupData[{ Cell["Problem 2", "Subsubsection"], Cell["\<\ Consider a first order differential equation of the form: dy/dx = (2x[t] y[t] -y^2[t]) / (x^2[t] -2x[t] y[t]) / which is difficult to solve explicitly. Its solution curves can be plotted as \ trajectories of the corresponding autonomous two-dimensional system: x'[t] = x^2[t] -2x[t] y[t] y'[t] = 2x[t] y[t] -y^2[t]. a) Plot some solution curves for the equation (1). b) Construct an example of your own: dy/dx = G(x,y)/F(x,y) and plot solution \ curves.\ \>", "Text"] }, Open ]], Cell["Solution", "Subsubsection"], Cell["************************************************************", "Input"], Cell[CellGroupData[{ Cell["Problem 3", "Subsubsection"], Cell[TextData[{ "The term \"bifurcation\" generally refers to something \"splitting \ apart\". With regard to differential equations or\nsystems involving \ parameter, it refers to abrupt changes in the character of the solutions as \ the parameter \nis changed continuously. The following problem illustrates a \ sensitive case in which a small perturbation\nin the coefficients of a \ linear system can change the type or stability of a critical point.\n\na) \ Consider the linear system:\n\n(2) x'[t] = e x[t] - y[t] \n \ y'[t] = x[t] + e y[t].\n\nShow that the critical point (0,0) is\n\na) a \ stable spiral point if e<0\n\nb) a center if e=0\n\nc) an unstable spiral \ point if e>0\n\nThus small perturbations of the system\n\nx'[t] = - y[t]\n\ y'[t] = x[t]\n\ncan change both the type and stability of the critical point.\ \n\nd) Plot phase portraits of the system (2) with e= -0.2, -0.05, 0, 0.05, \ 0.2\n and indicate stability in each case. You can use ", StyleBox["Mathematica", FontSlant->"Italic"], " code from Problem 5.\n " }], "Text"] }, Open ]], Cell["Solution", "Subsubsection"], Cell["************************************************************", "Input"], Cell[CellGroupData[{ Cell["Problem 4", "Subsubsection"], Cell[TextData[{ "Interesting and complicated phase portraits often result from simple \ nonlinear perturbations of linear systems.\nConsider the following nonlinear \ system:\n\nx'[t] = -x[t] - y^2[t] Cos[x+y]\ny'[t] = y[t] + x^2[t] Cos[x-y].\n\ \nThis system has a lot of critical points. Suppose that we want to locate \ accurately the apparent spiral point near (-1,1).\nOur problem is to solve \ numerically two equations in two unknown. \nUsing ", StyleBox["Mathematica", FontSlant->"Italic"], " we could proceed as follows.\n\na)\n" }], "Text"] }, Open ]], Cell[BoxData[ \(\(xp\ = \ \(-x\)\ - \ y^2\ Cos[x + y]; \n yp\ = \ y\ + \ x^2\ Cos[x - y]; \n soln\ = \ FindRoot[\ {xp\ == 0, \ yp\ == 0}, \ {x, \(-1\)}, \ {y, 1}\ ]; \n a\ = \ x /. \ soln; \ b\ = \ y\ /. \ soln; \n{a, b}\n\)\)], "Input"], Cell[TextData[{ "This should result in an approximate value of the critival point.\n\nb) To \ investigate the precise nature of this critical point, we need to substitute \ u=x-a, v=y-b\nso that x=a, y=b correspond to u=0, v=0. The right-hand sides \ of the resulting equations for du/dt and dv/dt will\nbe given by", StyleBox[ "\n subs = {x -> u + a, y -> v + b};\n up = xp /. subs\n vp = yp \ /. subs\n\n", FontFamily->"Courier", FontWeight->"Bold"], "We can extract the coefficients of the u- and v-terms by computing partial \ derivatives of up and vp at the origin\n", StyleBox[ "\n A = {{D[up,u], D[up,v]},\n {D[vp,u], D[vp,v]}} /. {u -> 0, v -> \ 0}\n\n", FontFamily->"Courier", FontWeight->"Bold"], "Calculate the eigenvalues of this matrix A by using the following command \ and determine the type and stability\nof the critical point." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Eigenvalues[A]\)], "Input"], Cell["Solution", "Subsubsection"], Cell["Problem 5", "Subsubsection"], Cell[TextData[{ "a) Plot phase ", "portrait", " of the predator-prey system:\n\ny1'[t] = 2 y1[t] - a y1[t] y2[t]\n\ y2'[t] = - y2[t] + a y1[t] y2[t]\n\nusing the following ", StyleBox["Mathematica", FontSlant->"Italic"], " code:\n" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[y1, y2]\), \(tmin = 0; \ntmax = 30; \nl1 = 1; \nl2 = 2; \ng1 = 2; \ng2 = 1; \n a = \ 0.01; \ny10 = \ 100; \ny20 = 50; \n\ \ Sol\ = \ \ \ \ \ NDSolve[{\(y1'\)[t] == \ 2\ \ y1[t]\ - \ a\ y1[t]\ \ y2[t], \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(y2'\)[t] == \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(-\ \ y2[t]\)\ + \ a\ y1[t]\ y2[t], \ny1[0] == y10, \n \ \ \ \ \ \ y2[0]\ == y20}, {y1[t], y2[t]}, {t, tmin, tmax}]; \n y1[t_]\ = \ y1[t]\ /. \ Sol; \ny2[t_]\ = \ y2[t]\ /. \ Sol; \n p8 = \ ParametricPlot[Evaluate[{y1[t], y2[t]} /. 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particular solutions (y1[t],y2[t]) of the \ predator-prey system as functions of time t using the following ", StyleBox["Mathematica", FontSlant->"Italic"], " code and choosing different initial conditions y10 and y20:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[y1, y2]\), \(tmin = 0; \ntmax = 30; \nl1 = 1; \nl2 = 2; \ng1 = 2; \ng2 = 1; \n a1 = 1; \na2 = 1; \ny10 = \ 2; \ny20 = 2; \n\ \ ss\ = \ \ \ \ \ NDSolve[{\(y1'\)[t] == \ \((l1\ - \ g1)\)\ \ y1[t]\ + \ a1\ y1[t]\ \ y2[t], \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(y2'\)[t] == \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((l2\ - \ g2)\)\ \ y2[t]\ - \ a2\ y1[t]\ y2[t], \n y1[0] == y10, \n\ \ \ \ \ \ y2[0]\ == y20}, {y1[t], y2[t]}, {t, tmin, tmax}]; \n p1\ = \ Plot[Evaluate[\ y1[t]\ /. \ ss\ ], {t, tmin, \n\ \ \ \ tmax}]; \np2\ = \ Plot[Evaluate[\ y2[t]\ /. \ ss\ ], {t, tmin, \n\ \ \ \ tmax}]; \n Show[p1, p2]; \)}], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { 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