CL_MATCONT: the MLFast example

In §8.4.4 we studied a continuation of limit cycles in the MLfast Morris-Lecar model and noted that the limit cycles were approaching a homoclinic orbit. We will now approach this homoclinic even closer, and then start its continuation from the large limit cycle. The result is shown in Figure 31.

>> init;
>> 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);
first point found
tangent vector to first point found
label = H , x = ( 0.036756 0.294770 0.075659 )
First Lyapunov coefficient = 8.234573e+000
label = LP, x = ( -0.033738 0.136501 -0.020727 )
a=-1.036706e+001
label = H , x = ( -0.119894 0.045956 0.033207 )
Neutral saddle
label = LP, x = ( -0.244915 0.008514 0.083257 )
a=2.697414e+000

elapsed time  = 0.4 secs
npoints curve = 65
>> 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,'MaxStepSize',1);
>> opt=contset(opt,'IgnoreSingularity',1);
>> opt=contset(opt,'Singularities',1);
>> opt=contset(opt,'MaxNumPoints',200);
>> [x2,v2,s2,h2,f2]=cont(@limitcycle,x0,v0,opt);
first point found
tangent vector to first point found
Limit point cycle (period = 4.222011e+000, parameter = 8.456948e-002)
Normal form coefficient = -2.334576e-001
Limit point cycle (period = 5.653399e+001, parameter = 7.293070e-002)
Normal form coefficient = 1.132235e+000
Limit point cycle (period = 5.739877e+001, parameter = 7.293070e-002)
Normal form coefficient = 3.266287e+000
Limit point cycle (period = 8.938964e+001, parameter = 7.293071e-002)
Normal form coefficient = -1.537206e-001

elapsed time  = 86.6 secs
npoints curve = 200
>> p(ap1) = x2(end,end);
>> T = x2(end-1,end)/2;
>> [x0,v0]=init_LC_Hom(@MLfast, x2(:,end), s2(:,end), p, [1 2], 40, 4,...
>>          [0 1 1], T, 0.01, 0.01);
>> opt=contset(opt,'MaxNumPoints',15);
>> [xh,vh,sh,hh,fh] = cont(@homoclinic,x0,v0,opt);
first point found
tangent vector to first point found
elapsed time  = 4.4 secs
npoints curve = 15
>> plotcycle(xh,vh,sh,[1 2]);

The above computations can be done by running the script homoc1. The picture is presented in Figure 31. Similar tests can be done by using the testrun testmyml.

Figure 31: Computed curve of homoclinic-to-saddle orbits started from a limit cycle with large period.
\begin{figure}
\centering
\centerline{\psfig{figure=ex/MLFasthom.eps,height=8cm}}
\end{figure}