Continuation of Central Saddle-Node Homoclinics.

Local bifurcation analysis shows that at $K=6.0,\ Z=0.06729762\ldots$, the system has a saddle-node equilibrium

$$(X^0,Y^0) = (5.738626\ldots,0.5108401\ldots),$$

with one zero and one negative eigenvalue. Direct simulations reveal a homoclinic orbit to this saddle-node, departing and returning along its central direction (i.e., tangent to the null-vector).

Starting from this solution, stored in the file mtn.dat, we continue the saddle-node central homoclinic orbit with respect to the parameters $K$ and $Z$ by copying the demo and running it

@dm mtn
make first

The file mtn.f contains approximate parameter values

$$K={\tt PAR(1)}=6.0,\ Z={\tt PAR(2)}=0.06729762,$$

as well as the coordinates of the saddle-node

$$X^0={\tt PAR(12)}=5.738626,\ Y^0={\tt PAR(13)}=0.5108401,$$

and the length of the truncated time-interval

$$T_0={\tt PAR(11)} = 1046.178 \: .$$

Since a homoclinic orbit to a saddle-node is being followed, we have also made the choices

$${\tt IEQUIB = 2 \quad NUNSTAB =0 \quad NSTAB = 1 }$$

in h.mtn.1. The two test-functions, $\psi_{15}$ and $\psi_{16}$, to detect non-central saddle-node homoclinic orbits are also activated, which must be specified in three ways. Firstly, in h.mtn.1, NPSI is set to 2 and the active test functions IPSI(I),I=1,2 are chosen as 15 and 16. This sets up the monitoring of these test functions. Secondly, in c.mtn.1 user-defined functions (NUZR=2) are set up to look for zeros of the parameters corresponding to these test functions. Recall that the parameters to be zeroed are always the test functions plus 20. Finally, these parameters are included in the list of continuation parameters (NICP,(ICP(I),I=1 NICP)).

Among the output there is a line

  BR    PT  TY LAB    PAR(1)    ...     PAR(2)        PAR(35)       PAR(36)    
   1    27  UZ   5  6.10437E+00 ...   6.932475E-02 -6.782898E-07  8.203437E-02

indicating that a zero of the test function IPSI(1)=15 This means that at

$$D_1=(K^1,Z^1)=(6.6104\ldots, 0.069325\ldots)$$

the homoclinic orbit to the saddle-node becomes non-central, namely, it returns to the equilibrium along the stable eigenvector, forming a non-smooth loop. The output is saved in b.1, s.1 and d.1. Repeating computations in the opposite direction along the curve, IRS=1, DS=-0.01 in c.mtn.2,

make second

one obtains

  BR    PT  TY LAB    PAR(1)     ...    PAR(2)        PAR(35)       PAR(36)  
   1    34  UZ   9  5.180323E+00 ...  6.385506E-02  3.349720E-09  9.361957E-02

which means another non-central saddle-node homoclinic bifurcation occurs at

$$D_2=(K^2,Z^2)=(5.1803\ldots,0.063855\ldots).$$

Note that these data were obtained using a smaller value of NTST than the original computation (compare c.mtn.1 with c.mtn.2). The high original value of NTST was only necessary for the first few steps because the original solution is specified on a uniform mesh.