Three-Parameter Continuation.

We now consider curves in three parameters of each of the codimension-two points encountered in this model, by freeing the parameter $\epsilon=$ PAR(3). First we continue the first inclination flip stored at label 7 in s.3

make tenth

Note that ITWIST=1 in h.kpr.10, so that the adjoint is also continued, and there is one fixed condition IFIXED(1)=13 so that test function $\psi_{13}$ has been frozen. Among the output there is a codimension-three point (zero of $\psi_9$) where the neutrally twisted homoclinic orbit collides with the saddle-node curve

 BR  PT  TY LAB    PAR(1)     ...   PAR(2)        PAR(3)        PAR(29)    ...     
  1  28  UZ  14  1.282702E-01 ... -2.519325E+00 5.744770E-01 -4.347113E-09 ...

The other detected inclination flip (at label 8 in s.3) is continued similarly

make eleventh

giving among its output another codim 3 saddle-node inclination-flip point

 
 BR  PT  TY LAB    PAR(1)     ...   PAR(2)        PAR(3)        PAR(29)    ...  
  1  27  UZ  14  1.535420E-01 ... -2.458100E+00 1.171705E+00 -1.933188E-07 ...

Output beyond both of these codim 3 points is spurious and both computations end in an MX point (no convergence).

To continue the non-central saddle-node homoclinic orbits it is necessary to work on the data without the solution $\phi (t)$. We restart from the data saved at LAB=8 and LAB=13 in s.7 and s.8 respectively. We could continue these codim 2 points in two ways, either by appending the defining condition $\psi_{16} =0$ to the continuation of saddle-node homoclinic orbits (with IEQUIB=2, etc.), or by appending $\psi_{9} =0$ to the continuation of a saddle homoclinic orbit (with IEQUIB=1. The first approach is used in the example mtn, for contrast we shall adopt the second approach here.

Figure 23.8: Projection onto the (PAR(3),PAR(2))-plane of the non-central saddle-node homoclinic orbit curves (labeled 1 and 2) and the inclination-flip curves (labeled 3 and 4)

make twelfth
make thirteenth

The projection onto the $(\epsilon ,k)$-plane of all four of these codimension-two curves is given in Figure 23.8. The intersection of the inclination-flip lines with one of the non-central saddle-node homoclinic lines is apparent. Note that the two non-central saddle-node homoclinic orbit curves are almost overlaid, but that as in Figure 23.6 the orbits look quite distinct in phase space.