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Switching between Saddle-Node and Saddle Homoclinic Orbits.

Now we can switch to continuation of saddle homoclinic orbits at the located codim 2 points $ D_1$ and $ D_2$.
make third
starts from $ D_1$. Note that now
NUNSTAB = 1     IEQUIB = 1
has been specified in h.mtn.3. Also, test functions $ \psi_9$ and $ \psi_{10}$ have been activated in order to monitor for non-hyperbolic equilibria along the homoclinic locus. We get the following output
  BR    PT  TY LAB    PAR(1)     ...    PAR(2)        PAR(29)       PAR(30)    
   1    10      11  7.114523E+00 ...  7.081751E-02 -4.649861E-01  3.183429E-03
   1    20      12  9.176810E+00 ...  7.678731E-02 -4.684912E-01  1.609294E-02
   1    30      13  1.210834E+01 ...  8.543468E-02 -4.718871E-01  3.069638E-02
   1    40  EP  14  1.503788E+01 ...  9.428036E-02 -4.743794E-01  4.144558E-02
The fact that PAR(29) and PAR(30) do not change sign indicates that there are no further non-hyperbolic equilibria along this family. Note that restarting in the opposite direction with IRS=11, DS=-0.02
make fourth
will detect the same codim 2 point $ D_1$ but now as a zero of the test-function $ \psi_{10}$
  BR    PT  TY LAB    PAR(1)     ...    PAR(2)        PAR(29)       PAR(30)
  1    10  UZ  15  6.610459E+00  ...  6.932482E-02 -4.636603E-01  1.725013E-09
Note that the values of PAR(1) and PAR(2) differ from that at label 4 only in the sixth significant figure.

Actually, the program runs further and eventually computes the point $ D_2$ and the whole lower family of $ P$ emanating from it, however, the solutions between $ D_1$ and $ D_2$ should be considered as spurious22.1, therefore we do not save these data. The reliable way to compute the lower family of $ P$ is to restart computation of saddle homoclinic orbits in the other direction from the point $ D_2$

make fifth
This gives the lower family of $ P$ approaching the BT point (see Figure 22.1)
  BR    PT  TY LAB    PAR(1)     ...   PAR(2)        PAR(29)       PAR(30)    
   1    10      15  4.966429E+00 ... 6.298418E-02 -4.382426E-01  4.946824E-03
   1    20      16  4.925379E+00 ... 7.961214E-02 -3.399102E-01  3.288447E-02
   1    30      17  7.092267E+00 ... 1.587114E-01 -1.692842E-01  3.876291E-02
   1    40  EP  18  1.101819E+01 ... 2.809825E-01 -3.482651E-02  2.104384E-02
The data are appended to the stored results in b.1, s.1 and d.1. One could now display all data using the AUTO  command @p 1 to reproduce the curve $ P$ shown in Figure 22.1.

Figure 22.2: Approximation by a large-period cycle
\begin{figure}
\epsfysize 10.0cm
\centerline{\epsffile{include/mtn2.ps}}\end{figure}
Figure 22.3: Projection onto the ($ K,D_0$)-plane of the three-parameter curve of non-central saddle-node homoclinic orbit
\begin{figure}
\epsfysize 9.0cm
\centerline{\epsffile{include/mtn3.ps}}\end{figure}
It is worthwhile to compare the homoclinic curves computed above with a curve $ T_0=const$ along which the system has a limit cycle of constant large period $ T_0=1046.178$, which can easily be computed using AUTO or locbif. Such a curve is plotted in Figure 22.2. It obviously approximates well the saddle homoclinic loci of $ P$, but demonstrates much bigger deviation from the saddle-node homoclinic segment $ D_1D_2$. This happens because the period of the limit cycle grows to infinity while approaching both types of homoclinic orbit, but with different asymptotics: as $ -\ln\Vert\alpha-\alpha^*\Vert$, in the saddle homoclinic case, and as $ \Vert\alpha-\alpha^*\Vert^{-1}$ in the saddle-node case.


next up previous contents
Next: Three-Parameter Continuation. Up: HomCont Demo : mtn. Previous: Continuation of Central Saddle-Node   Contents
Gabriel Lord 2007-11-19