ME 406 A Graphical View of the Transition from Local to GlobalsysidMathematica 6.0.3, DynPac 11.01, 1ê13ê2009intreset;plotreset; imsize = 250;üINTRODUCTIONIn this notebook, we study the transition between a local and global description of a saddle point. We construct a sequence of graphs which can be animated to give a movie of the transition. This is done for a particular system, but the code could be applied to any phase portrait.üDEFINING THE SYSTEMThe system we use here was given as an example in Nonlinear Ordinary Differential Equations, D.W. Jordan and P. Smith, second edition, Oxford Press, 1987, pp. 51-52.setstate@8x, y<D;setparm@8<D;parmval = 8<;slopevec = 8x - y, 1 - x * y<;sysname = "";üEQUILIBRIUM POINTSWe use findpolyeq to find the equilibrium points:eqpoints = findpolyeq88-1, -1<, 81, 1<<eq1 = eqpoints@@1DD8-1, -1<locglob.nb 1eq2 = eqpoints@@2DD81, 1<We now classify these:classify2D@eq1Dunstable - spiralclassify2D@eq2Dunstable - saddleThus we have an unstable spiral and an unstable saddle. We use saddleportrait to get a quick look at the phase portrait.plrange = 88-5, 7<, 8-5, 7<<;quickgraph = saddleportrait@eq2, plrangeD-4-2246x-4-2246yThis shows the basic topology of the orbits and the connected unstable spiral and saddle. We will now construct more carefully a phase portrait which will then be the basis of our movie. We could of course focus on either the spiral or the saddle point. We choose to focus on the saddle point here. We start with the stable and unstable manifolds of the saddle point, with the curves being shown in red. We construct these manually so that we have full control over the time step and starting points. We start by getting the eigenvectors and eigenvalues at the saddle point. eiger = Eigensystem@dermatval@[email protected], 1.41421<, 880.382683, 0.92388<, 80.92388, -0.382683<<<locglob.nb 2vec1 = First@[email protected], 0.92388<vec2 = Last@[email protected], -0.382683<Now we will construct the integral curves entering and leaving the saddle. We do this by choosing initial conditions displaced from the equilibrium a small distance along the eigenvectors. We choose the time direction of integration to move away from the point on all four curves. We also use range checking on the integration to prevent overflow. We turn arrows on, and put an arrow at the midpoint of each curve. We set the color to Red. We turn the axes off.eps = 0.001;rangeflag = True;ranger = plrange;arrowflag = True;arrowvec = 81 ê 2<;setcolor@8Red<D;axon = False;sol1 = integrate@eq2 + eps * vec2, 0.0, 0.005, 2000D;man1 = phaser@sol1Dsol2 = integrate@eq2 - eps * vec2, 0.0, 0.005, 2000D;locglob.nb 3man2 = phaser@sol2Dsol3 = integrate@eq2 + eps * vec1, 0.0, -0.005, 2000D;man3 = phaser@sol3Dsol4 = integrate@eq2 - eps * vec1, 0.0, -0.005, 2000D;locglob.nb 4man4 = phaser@sol4Dmangraph = show@man1, man2, man3, man4DNow we create a second piece of the portrait with some initial conditions near the saddlepoint.initlist = 882, 1.5<, 80.75, 2<, 81.5, 0<, 80, 0.5<, 80, 3.2<, 83, 2<, 82.0, -1.5<<;We show the curves in blue.locglob.nb 5We show the curves in blue.setcolor@8Blue<D;We integrate in both directions, with range checking.bothdirflag = True;rangeflag = True;ranger = 88-6, 8<, 8-6, 8<<;We ask for a suggested time step, and we will use half the suggested value.h = [email protected] = portrait@initlist, 0.0, h ê 2, 2000, 1, 2Dlocglob.nb 6show@mangraph, orbgraph1DWe add just a few more curves coming from the spiral.initset = 88-1, 0<, 8-2, -1<, 8-3, 0<<;orbgraph2 = portrait@initset, 0.0, h ê 2, 2000, 1, 2Dlocglob.nb 7fingraph = show@orbgraph1, orbgraph2, mangraphDThis is the graph that we will use to make the movie. The movie is made by constructing a sequence of graphs with an increasing plot range in the sequence. The sequence is constructed by the routine zoom[graph,smallwin,bigwin,steps] defined below. The arguments are graph, the name of the graph, smallwin, the smallest plotting window in the sequence, bigwin, the largest plotting window in the sequence, and steps, one less than the number of graphs in the sequence.zoom@graph_, smallwin_, bigwin_, steps_D := ModuleA8i, tempplrange<,tempplrange = plrange;DoAplrange = IHsteps - iL ë stepsM * smallwin + Ii ë stepsM * bigwin;Print@show@graphDD, 8i, 0, steps<E; plrange = tempplrangeEWe try this with our present graph, with an initial window smallwin = 880.9, 1.1<, 80.9, 1.1<<;an a final window ofbigwin = 88-5, 7<, 8-5, 7<<;We now create a 51-graph sequence. At the beginning of the sequence our view is so close to the saddle point that the stable and unstable manifolds appear to be straight lines, which is the local approximation. As we rise above the phase plane, we first see some curvature, and then later we see the spiral and its connection with the saddle point. In the printed version of this notebook, only the first frame of the movie is shown. To see the movie, animate the graphs grouped together below.zoom@fingraph, smallwin, bigwin, 50Dlocglob.nb 888-5, 7<, 8-5, 7<<locglob.nb