Three-Parameter Continuation.

Finally, we can follow the curve of non-central saddle-node homoclinic orbits in three parameters. The extra continuation parameter is =PAR(3). To achieve this we restart at label 4, corresponding to the codim 2 point . We return to continuation of saddle-node homoclinics, NUNSTAB=0,IEQUIB=2, but append the defining equation to the continuation problem (via NFIXED=1, IFIXED(1)=15). The new continuation problem is specified in r.mtn.6 and s.mtn.6.

make sixth

Notice that we set ILP=1 and choose PAR(3) as the first continuation parameter so that AUTO can detect limit points with respect to this parameter. We also make a user-defined function (NUZR=1) to detect intersections with the plane . We get among other output

  BR    PT  TY LAB    PAR(3)        L2-NORM    ...    PAR(1)        PAR(2)
   1    22  LP  19  1.081212E-02  5.325894E+00 ...  5.673631E+00  6.608184E-02
   1    31  UZ  20  1.000000E-02  4.819681E+00 ...  5.180317E+00  6.385503E-02

the first line of which represents the value at which the homoclinic curve P has a tangency with the branch of fold bifurcations. Beyond this value of , P consists entirely of saddle homoclinic orbits. The data at label 20 reproduce the coordinates of the point . The results of this computation and a similar one starting from in the opposite direction (with DS=-0.01) are displayed in Figure 18.3.