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We find a stable equilibrium (EP) for and
at
by time integration. We continue this equilibrium with free parameter
for decreasing values of
by running testEquilMLfast.m:
p=[0.11047;0.1];ap1=[1]; [x0,v0]=init_EP_EP(@MLfast,[0.047222;0.32564],p,ap1); opt=contset; opt=contset(opt,'Singularities',1); opt=contset(opt,'MaxNumPoints',65); opt=contset(opt,'MinStepsize',0.00001); opt=contset(opt,'MaxStepsize',0.01); opt=contset(opt,'Backward',1); [x,v,s,h,f]=cont(@equilibrium,x0,[],opt); cpl(x,v,s,[3 1]);
The output is as follows:
>> testEquilMLfast first point found tangent vector to first point found label = H , x = ( 0.036756 0.294770 0.075658 ) First Lyapunov coefficient = 8.238955e+00 label = LP, x = ( -0.033738 0.136501 -0.020727 ) a=-1.036700e+01 label = NE , x = ( -0.119894 0.045956 0.033207 ) label = LP, x = ( -0.244914 0.008514 0.083257 ) a=-2.697425e+00 elapsed time = 0.1 secs npoints curve = 65
We find a Hopf (H) bifurcation point at
The detected Hopf point is used to start a limit cycle continuation. We choose test intervals and
collocation points for the discretization.
testEquilMLfast x1=x(1:2,s(2).index);p=[x(end,s(2).index);0.1]; [x0,v0]=init_H_LC(@MLfast,x1,p,ap1,0.0001,30,4); opt=contset; opt=contset(opt,'IgnoreSingularity',1); opt=contset(opt,'Singularities',1); opt=contset(opt,'MaxNumPoints',50); [x2,v2,s2,h2,f2]=cont(@limitcycle,x0,v0,opt); plotcycle(x2,v2,s2,[1 2]);The output is as follows:
>> testLCMLfast first point found tangent vector to first point found label = H , x = ( 0.036756 0.294770 0.075658 ) First Lyapunov coefficient = 8.238955e+00 label = LP, x = ( -0.033738 0.136501 -0.020727 ) a=-1.036700e+01 label = NE , x = ( -0.119894 0.045956 0.033207 ) label = LP, x = ( -0.244914 0.008514 0.083257 ) a=-2.697425e+00 elapsed time = 0.1 secs npoints curve = 65 first point found tangent vector to first point found Limit point cycle (period = 4.222012e+00, parameter = 8.456948e-02) Normal form coefficient = -2.334578e-01 elapsed time = 4.0 secs npoints curve = 50
The periodic orbit is initially unstable. We detect a limit point of cycles LPC at . At this point the stability is gained. Afterwards the stability is preserved but the period tends to infinity and the periodic orbits end in a homoclinic orbit. The results can be seen in Figure 21.
We now compute a curve of fold bifurcations of limit cycles.
The starting vector x0 is calculated from the LPC on the previously computed curve of limit cycles, using init_LPC_LPC. Continuation is done using a call to the standard continuer with limitpointcycle as curve definition file.
We free both and
to continue the LPC curve through this LPC point. The computations are done by executing the scripttestLPCMLfast.m:
testEquilMLfast x1=x(1:2,s(2).index);p=[x(end,s(2).index);0.1]; [x0,v0]=init_H_LC(@MLfast,x1,p,ap1,0.0001,30,4); opt=contset; opt=contset(opt,'IgnoreSingularity',1); opt=contset(opt,'Singularities',1); opt=contset(opt,'MaxNumPoints',50); opt=contset(opt,'FunTolerance',0.0000001); opt=contset(opt,'VarTolerance',0.0000001); [x2,v2,s2,h2,f2]=cont(@limitcycle,x0,v0,opt); [x0,v0]=init_LPC_LPC(@MLfast,x2,s2(2),[1 2],30,4); opt=contset; opt=contset(opt,'FunTolerance',0.0001); opt=contset(opt,'VarTolerance',0.0001); opt=contset(opt,'MaxNumPoints',30); %opt=contset(opt,'Backward',1); opt=contset(opt,'Singularities',1); [x3,v3,s3,h3,f3]=cont(@limitpointcycle,x0,v0,opt); plotcycle(x3,v3,s3,[1 2]);
The output is as follows:
>> testLPCMLfast first point found tangent vector to first point found label = H , x = ( 0.036756 0.294770 0.075658 ) First Lyapunov coefficient = 8.238955e+00 label = LP, x = ( -0.033738 0.136501 -0.020727 ) a=-1.036700e+01 label = NE , x = ( -0.119894 0.045956 0.033207 ) label = LP, x = ( -0.244914 0.008514 0.083257 ) a=-2.697425e+00 elapsed time = 0.4 secs npoints curve = 65 first point found tangent vector to first point found Limit point cycle (period = 4.222012e+00, parameter = 8.456948e-02) Normal form coefficient = -2.334578e-01 elapsed time = 3.9 secs npoints curve = 50 first point found tangent vector to first point found elapsed time = 7.1 secs npoints curve = 30
The results are plotted using the standard plot function plotcycle where the fourth argument is used to select the coordinates. The results can be seen in Figure 22. We note that it shrinks to a single point. The labels of the plot are added manually .