ME 406Bifurcations IIITranscritical BifurcationsysidMathematica 4.1.2, DynPac 10.66, 3ê5ê2002intreset; plotreset;‡ 1. IntroductionIn this notebook, the third in a series of notebooks on bifurcations, we look at a simple example of atranscritical bifurcation. We construct a movie showing the changes of a selected set of orbits with the bifurcationparameter. ‡ 2. Definition of the SystemWe consider the following system, depending on one parameter a:x° = mx - x2 , y° = - y .This system has two equilibria for any m π 0, and one equilibrium for m = 0. The bifurcation is m = 0, for whichthe two equilibria coalesce at the origin. We begin our analysis by defining the system for DynPac.setstate@8x, y<D; setparm@8m<D; slopevec = 8m x - x2, - y<;sysname = "Transcritical Bifurcation";eq1 = 80, 0<; eq2 = 8m,0<;Let's look at the nature of the equilibria.eigsys@eq1D88-1, m<, 880, 1<, 81, 0<<<eigsys@eq2D88-1, -m<, 880, 1<, 81, 0<<<Thus for m < 0, eq1 is a stable node and eq2 is a saddle. For m > 0, eq1 is a saddle and eq2 is a stable node. For m= 0, linearization is inconclusive, although in this case it is easy to integrate the equations directly and show thatthe equilibrium is unstable. This kind of bifurcation is often called an exchange of stabilities. We may visualizethis with a bifurcation diagram, showing the equilibria as functions of m, with stable in solid, unstable in dashed.plot1 = Plot@80, m<, 8m, -1, 0<,PlotRange Ø 88-1, 1<, 8-1, 1<<,PlotLabel Ø "Transcritical Bifurcation",FrameLabel Ø 8"m", "x"<, Axes Ø False,ImageSize Ø imsize, Frame Ø True, DisplayFunction Ø Identity,PlotStyle Ø [email protected], 0<D, [email protected], 0.03<D<D;plot2 = Plot@8m,0<, 8m,0,1<,PlotRange Ø 88-1, 1<, 8-1, 1<<,PlotLabel Ø "Transcritical Bifurcation", Frame Ø True,FrameLabel Ø 8"m", "x"<, Axes Ø False,DisplayFunction Ø Identity, ImageSize Ø imsize,PlotStyle Ø [email protected], 0<D, [email protected], 0.03<D<D;Show@8plot1, plot2<, DisplayFunction Ø $DisplayFunctionD;-0.75 -0.5 -0.25 0 0.25 0.5 0.75 1m-0.75-0.5-0.2500.250.50.751xTranscritical BifurcationNow we will construct phase plots for various values of the parmeter m. We first construct a shortsequence which will show the essential features, and then we construct a long sequence suitable for a movie. Wechoose a plotting window of {{-2,2},{-2,2}}, and a set of initial conditions attached to points on the window, plusseveral initial conditions near the equilibria. The function f(m) defined below gives the position of the unstableequilibrium as a function of m., and g(m) gives the position of the stable node. f@m_D := If@m>0, 0, mD;g@m_D := If@m>0, m,0D;∂∂∂∂= 0.02;initset = 882, 0 <, 82, -1.5<, 82, 1.5 <, 80, 0<, 8-2, 0<,81.5, 2<, 8-0.5, 2<, 81.5, -2<, 8-0.5, -2<, 8-0.5, 2 <,8f@mD, ∂∂∂∂<, 8f@mD, -∂∂∂∂<, 8f@mD + ∂∂∂∂,0<, 8f@mD - ∂∂∂∂,0<<;plrange = 88-2, 2<, 8-2, 2<<; asprat = 1; labshift = 18; imsize = 360;arrowflag = True; arrowvec = 81 ê 2<;t0 = 0.0; h = 0.02; nsteps = 800; bothdirflag = True;2 bifurc3.nbrangeflag = True; ranger = 88-2.1, 2.1<, 8-2.1, 2.1<<;Now we choose a small number of m-values for a bifurcation sequence.parmlist = 88-0.5<, 8-0.25<, 80.0<, 80.25<, 80.5<<;bifurc@initset , t0, h, nsteps, 1, 2, parmlistD;Bifurcation sequence for parmlist = 88-0.5<, 8-0.25<, 80.<, 80.25<, 80.5<<-2 -1.5 -1 -0.5 0.5 1 1.5 2x-2-1.5-1-0.50.511.52yTranscritical Bifurcation 8m<=8 -0.50<bifurc3.nb 3-2 -1.5 -1 -0.5 0.5 1 1.5 2x-2-1.5-1-0.50.511.52yTranscritical Bifurcation 8m<=8 -0.25<4 bifurc3.nb-2 -1.5 -1 -0.5 0.5 1 1.5 2x-2-1.5-1-0.50.511.52yTranscritical Bifurcation 8m<=8 0.00<bifurc3.nb 5-2 -1.5 -1 -0.5 0.5 1 1.5 2x-2-1.5-1-0.50.511.52yTranscritical Bifurcation 8m<=8 0.25<6 bifurc3.nb-2 -1.5 -1 -0.5 0.5 1 1.5 2x-2-1.5-1-0.50.511.52yTranscritical Bifurcation 8m<=8 0.50<The equilibrium for m = 0 is an interesting composite. It looks like a stable node on the right, and like asaddle on the left. As our final task in this notebook, we will make a movie of the saddle-node bifurcation. We will concen-trate the frames of the movie around m = 0. We make 41 frames at m-intervals of 0.015, with m running from -0.3to -0.3. We remove the axes for a clearer view, and we place a red dot on the saddle, a blue dot on the stable node.The function refgraph below defines the dot graphs.refgraph := Module@8temp1, temp2<, display = False; ptsize = 0.025;setcolor@8Red<D; temp1 = dots @88f@mD,0<<D;setcolor@8Blue<D; temp2 = dots@88g@mD,0<<D;display = True; setcolor@8Black<D; 8temp2, temp1<Dbifurc3.nb 7parmlist = Module@8ans, i<, ans = 8<;Do@ans = Append@ans, 80.015 * i<D, 8i, -20, 20<D; ansD88-0.3<, 8-0.285<, 8-0.27<, 8-0.255<, 8-0.24<, 8-0.225<, 8-0.21<,8-0.195<, 8-0.18<, 8-0.165<, 8-0.15<, 8-0.135<, 8-0.12<,8-0.105<, 8-0.09<, 8-0.075<, 8-0.06<, 8-0.045<, 8-0.03<,8-0.015<, 80<, 80.015<, 80.03<, 80.045<, 80.06<, 80.075<, 80.09<,80.105<, 80.12<, 80.135<, 80.15<, 80.165<, 80.18<, 80.195<,80.21<, 80.225<, 80.24<, 80.255<, 80.27<, 80.285<, 80.3<<initset = 882, 0 <, 82, -1.5<, 82, 1.5 <, 80, 0<, 8-2, 0<,81.5, 2<, 8-0.5, 2<, 81.5, -2<, 8-0.5, -2<, 8-0.5, 2 <,8f@mD, ∂∂∂∂<, 8f@mD, -∂∂∂∂<, 8f@mD + ∂∂∂∂,0<, 8f@mD - ∂∂∂∂,0<<;plrange = 88-2, 2<, 8-2, 2<<; asprat = 1;labshift = 18; imsize = 360; axon = False;arrowflag = True; arrowvec = 81 ê 2<;totdig = 6; decdig = 3;t0 = 0.0; h = 0.02; nsteps = 800; bothdirflag = True;rangeflag = True; ranger = 88-2.1, 2.1<, 8-2.1, 2.1<<;bifurc@initset , t0, h, nsteps, 1, 2, parmlist, refgraphD;Bifurcation sequence for parmlist =88-0.3<, 8-0.285<, 8-0.27<, 8-0.255<, 8-0.24<, 8-0.225<, 8-0.21<, 8-0.195<,8-0.18<, 8-0.165<, 8-0.15<, 8-0.135<, 8-0.12<, 8-0.105<, 8-0.09<, 8-0.075<,8-0.06<, 8-0.045<, 8-0.03<, 8-0.015<, 80<, 80.015<, 80.03<, 80.045<,80.06<, 80.075<, 80.09<, 80.105<, 80.12<, 80.135<, 80.15<, 80.165<,80.18<, 80.195<, 80.21<, 80.225<, 80.24<, 80.255<, 80.27<, 80.285<, 80.3<<8 bifurc3.nbTranscritical Bifurcation 8m<=8 -0.300<bifurc3.nb