(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-4, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R32768, L-4, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B32768, L-4, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; currentKernel; ] :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Math 325: Differential Equations Assignment 3 :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Name: :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Section: :[font = text; inactive; preserveAspect; startGroup] I affirm that the solutions presented in this assignment are entirely my own work. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] Signature: :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Initialization :[font = input; initialization; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] *) <(Log[50#+1]&) (this makes the arrows bigger near the critical points) and Frame->True. Use the vector field plot in b) to choose a representative sample of several initial points {x0,y0}. Use NDSolve[] to obtain numerical solutions through these points and plot the corresponding trajectories on a single graph. Indicate what happens to the trajectories for increasing t. ;[s] 20:0,0;33,1;87,2;89,0;310,2;312,0;342,1;359,0;376,1;405,0;466,1;477,0;508,2;510,0;571,1;578,0;584,1;593,0;761,1;762,0;764,-1; 3:10,13,9,Times,0,12,0,0,0;7,13,10,Courier,1,12,0,0,0;3,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = input; preserveAspect; endGroup] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 6 :[font = text; inactive; preserveAspect] In this problem we consider the equations describing the motion of a pendulum, both damped and undamped. For each situation below convert the equation into a system of equations in x[t] = theta and y[t] = theta'[t]. Plot the correspodining vector field. Use NDSolve[] to solve this system and plot on a single graph the trajectories corresponding to the given initial velocities b. Based on these plots, describe what the pendulum is doing in each case. Experiment to see if you can find a value for the initial velocity b such that the trajectory approaches the unstable equilibrium point {Pi,0} corresponding to the pendulum being stationary in an upright position. a) The undamped pendulum: theta''[t] + Sin[theta] == 0, theta[0] == 0, theta'[0] == b The initial velocities are b = 0.5 to 3.0 in steps of 0.5. b) The damped pendulum: theta''[t] + 0.5 theta'[t] + Sin[theta] == 0, theta[0] == 0, theta'[0] == b The initial velocities are b = 0.5 to 6.0 in steps of 0.5. ;[s] 33:0,0;181,1;193,0;198,1;214,0;258,1;267,0;379,1;380,0;521,1;522,0;590,1;596,0;669,2;671,0;695,1;765,0;792,1;799,0;803,1;806,0;819,1;822,0;824,2;827,0;850,1;936,0;963,1;970,0;974,1;977,0;990,1;993,0;995,-1; 3:17,13,9,Times,0,12,0,0,0;14,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = input; preserveAspect; endGroup] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 7 :[font = text; inactive; preserveAspect] a) Consider the linear system: x'[t] == y[t] + h*x[t], y'[t] == -x[t] + h*y[t] Show that {0,0} is the only critical point of this system for any value of h. For each of the values h==-0.5, 0.0, and 0.5, plot the vector field (using PlotVectorField[] with the options ScaleFunction->(Log[50#+1]&) and Frame->True) and several trajectories (using DSolve[] and ParametricPlot[]). What can you deduce about the stability of the equilibrium point {0,0} corresponding to relatively small perturbations of the system? b) Now consider the system x'[t] == y[t] + h*x[t](x[t]^2 + y[t]^2), y'[t] == -x[t] + h*y[t](x[t]^2 + y[t]^2) Show that {0,0} is the only critical point of this system and that the linear approximation of this system at {0,0} is always a center for any value of h. For each of the values h==-1 and 1 plot the vector field and several trajectories (using NDSolve[] this time); the case h==0 is the same as in a) and is a center. What can you deduce about the stability of the equilibrium point {0,0} corresponding to relatively small perturbations of the system? ;[s] 48:0,2;2,0;31,1;94,0;104,1;109,0;169,1;170,0;195,1;202,0;204,1;207,0;213,1;216,0;247,1;264,0;282,1;310,0;315,1;326,0;360,1;368,0;373,1;389,0;457,1;462,0;526,2;529,0;555,1;648,0;658,1;663,0;758,1;763,0;800,1;801,0;826,1;831,0;836,1;837,0;892,1;901,0;923,1;927,0;946,2;947,0;1031,1;1036,0;1100,-1; 3:24,13,9,Times,0,12,0,0,0;21,13,10,Courier,1,12,0,0,0;3,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = input; preserveAspect; endGroup] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 8 :[font = text; inactive; preserveAspect] Consider the competing species model: x'[t] == x[t]((5/4) - (3/4)x[t] - (1/4)y[t]) y'[t] == y[t]((7/4) - (3/2)x[t] - (1/2)y[t]) a) Find all critical points of this system. At each critical point, calculate the corresponding linear system and find the eigenvalues of the coefficient matrix. Then identify the type and stability of the critical point. b) Plot the vector field using PlotVectorField[] with the options ScaleFunction->(Log[50#+1]&) and Frame->True . Choose a representative sample of several initial points {x0,y0} in the first quadrant. Use NDSolve[] to obtain numerical solutions through these points and plot the corresponding trajectories on a single graph. Indicate what happens to the trajectories for increasing t. Explain why realistically there can be no "peaceful coexistence". ;[s] 18:0,0;39,1;138,2;140,0;361,2;363,0;393,1;410,0;427,1;456,0;461,1;472,0;532,1;539,0;567,1;576,0;744,1;745,0;813,-1; 3:9,13,9,Times,0,12,0,0,0;7,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = input; preserveAspect; endGroup] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 9 :[font = text; inactive; preserveAspect] Consider the predator-prey model: x'[t] == x[t]( 1 - y[t]) y'[t] == y[t](-2 + x[t]) where x[t] represents the population of the prey and y[t] represents the population of the predator. a) Find all critical points of this system. At each critical point, calculate the corresponding linear system and find the eigenvalues of the coefficient matrix. Then identify the type and stability of the critical point. b) Plot the vector field using PlotVectorField[] with the options ScaleFunction->(Log[50#+1]&) and Frame->True . Choose a representative sample of several initial points {x0,y0} in the first quadrant. Use NDSolve[] to obtain numerical solutions through these points and plot the corresponding trajectories on a single graph. Indicate what happens to the trajectories for increasing t. Explain how the populations vary for initial points close to the critical point inside the first quadrant. Superimpose the graphs of one pair x[t], y[t] as a functions of time t to see how the populations increa/decrease with respect to each other. ;[s] 30:0,0;35,1;94,0;100,1;104,0;147,1;151,0;194,1;195,2;197,0;418,2;420,0;450,1;467,0;484,1;513,0;518,1;529,0;589,1;596,0;624,1;633,0;801,1;802,0;946,1;950,0;952,1;956,0;980,1;981,0;1053,-1; 3:15,13,9,Times,0,12,0,0,0;13,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = input; preserveAspect; endGroup] ^*)