The Primary Branch of Homoclinics.

First, we locate a homoclinic orbit using the homotopy method. The file kpr.f already contains approximate parameter values for a homoclinic orbit, namely $\lambda=$PAR(1)=-1.851185, $k=$PAR(2)=-0.15. The files c.kpr.1 and h.kpr.1 specify the appropriate constants for continuation in $2T$=PAR(11) (also referred to as PERIOD) and the dummy parameter $\omega_1$=PAR(17) starting from a small solution in the local unstable manifold;

make first

Among the output there is the line

     BR    PT  TY LAB    PERIOD        L2-NORM     ...    PAR(17)    ...
      1    29  UZ   2  1.900184E+01  1.693817E+00  ...  4.433433E-09 ...

which indicates that a zero of the artificial parameter $\omega_1$ has been located. This means that the right-hand end point of the solution belongs to the plane that is tangent to the stable manifold at the saddle. The output is stored in files b.1, s.1, d.1. Upon plotting the data at label 2 (see Figure 23.1) it can be noted that although the right-hand projection boundary condition is satisfied, the solution is still quite away from the equilibrium.

The right-hand endpoint can be made to approach the equilibrium by performing a further continuation in $T$ with the right-hand projection condition satisfied (PAR(17) fixed) but with $\lambda$ allowed to vary.

Figure 23.1: Projection on the $(x,y)$-plane of solutions of the boundary value problem with $2T=19.08778$.

Figure 23.2: Projection on the $(x,y)$-plane of solutions of the boundary value problem with $2T = 60.0$.

make second

the output at label 4, stored in kpr.2,

   BR   PT TY   LAB    PERIOD       L2-NORM     ...    PAR(1)     ...
   1    35  UZ   4  6.000000E+01  1.672806E+00  ... -1.851185E+00 ...

provides a good approximation to a homoclinic solution (see Figure 23.2).

The second stage to obtain a starting solution is to add a solution to the modified adjoint variational equation. This is achieved by setting both ITWIST and ISTART to 1 (in h.kpr.3), which generates a trivial guess for the adjoint equations. Because the adjoint equations are linear, only a single Newton step (by continuation in a trivial parameter) is required to provide a solution. Rather than choose a parameter that might be used internally by AUTO , in c.kpr.3 we take the continuation parameter to be PAR(11), which is not quite a trivial parameter but whose affect upon the solution is mild.

make third

The output at the second point (label 6) contains the converged homoclinic solution (variables (U(1), U(2), U(3)) and the adjoint (U(4), U(5), U(6))). We now have a starting solution and are ready to perform two-parameter continuation.

The fourth run

make fourth

continues the homoclinic orbit in PAR(1) and PAR(2).

Figure 23.3: Projection on the $(x,y)$-plane of solutions $\phi (t)$ at 1 ( $\lambda =-1.825470, k=-0.1760749$) and 2 ( $\lambda =-1.686154, k=-0.3183548$).

Figure 23.4: Three-dimensional blow-up of the solution curves $\phi (t)$ at labels 1 (dotted) and 2 (solid line) from Figure 3.8.

Note that several other parameters appear in the output. PAR(10) is a dummy parameter that should be zero when the adjoint is being computed correctly; PAR(29), PAR(30), PAR(33) correspond to the test functions $\psi_9$,$\psi_{10}$ and $\psi_{13}$. That these test functions were activated is specified in three places in c.kpr.4 and h.kpr.4 as described in Section 20.6.

Note that at the end-point of the family (reached when after NMX=50 steps) PAR(29) is approximately zero which corresponds to a zero of $\psi_9$, a non-central saddle-node homoclinic orbit. We shall return to the computation of this codimension-two point later. Before reaching this point, among the output we find two zeroes of PAR(33) (test function $\psi_{13}$) which gives the accurate location of two inclination-flip bifurcations,

 BR  PT  TY LAB    PAR(1)     ...     PAR(2)        PAR(10)   ...    PAR(33)  
  1   6  UZ  10 -1.801662E+00 ... -2.002660E-01 -7.255434E-07 ... -1.425714E-04
  1  12  UZ  11 -1.568756E+00 ... -4.395468E-01 -2.156353E-07 ...  4.514073E-07

That the test function really does have a regular zero at this point can be checked from the data saved in b.3, plotting PAR(33) as a function of PAR(1) or PAR(2). Figure 23.3 presents solutions $\phi (t)$ of the modified adjoint variational equation (for details see ChKuSa:95) at parameter values on the homoclinic family before and after the first detected inclination flip. Note that these solutions were obtained by choosing a smaller step DS and more output (smaller NPR) in c.kpr.4. A blow-up of the region close to the origin of this figure is shown in Figure 23.4. It illustrates the flip of the solutions of the adjoint equation while moving through the bifurcation point. Note that the data in this figure were plotted after first performing an additional continuation of the solutions with respect to PAR(11).

Continuing in the other direction

make fifth

we approach a Bogdanov-Takens point

  
 BR    PT  TY LAB    PAR(1)     ...    PAR(10)    ...    PAR(33)    
  1    50  EP  13 -1.938276E+00 ... -7.523344E+00 ...  6.310810E+01

Figure 23.5: Computed homoclinic orbits approaching the BT point

Note that the numerical approximation has ceased to become reliable, since PAR(10) has now become large. Phase portraits of homoclinic orbits between the BT point and the first inclination flip are depicted in Figure 23.5. Note how the computed homoclinic orbits approaching the BT point have their endpoints well away from the equilibrium. To follow the homoclinic orbit to the BT point with more precision, we would need to first perform continuation in $T$ (PAR(11)) to obtain a more accurate homoclinic solution.