Starting Strategies.

There are four possible starting procedures for continuation.

(i)

Data can be read from a previously-obtained output point from AUTO (e.g., from continuation of a periodic orbit up to large period; note that if the end-point of the data stored is not close to the equilibrium, a phase shift must be performed by setting ISTART=4). These data can be read from fort.8 (saved to s.xxx) by making IRS correspond to the label of the data point in question.

(ii)

Data from numerical integration (e.g., computation of a stable periodic orbit, or an approximate homoclinic obtained by shooting) can be read in from a data file using the general AUTO utility @fc or us (see earlier in the manual). The numerical data should be stored in a file xxx.dat, in multi-column format according to the read statement

       READ(...,*) T(J),(U(I,J),I=1,NDIM)

where $T$ runs in the interval [0,1]. After running @fc or us the restart data is stored in the format of a previously computed solution in s.dat. When starting from this solution IRS should be set to 1 and the value of ISTART is irrelevant.

(iii)

By setting ISTART=2, an explicit homoclinic solution can be specified in the routine STPNT in the usual AUTO format, that is $U=...(T)$ where $T$ is scaled to lie in the interval $[0,1]$.

(iv)

The choice ISTART=3, allows for a homotopy method to be used to approach a homoclinic orbit starting from a small approximation to a solution to the linear problem in the unstable manifold [#!DoFrMo:93!#]. For details of implementation, the reader is referred to Section 5.1.2. of ChKu:94, under the simplification that we do not solve for the adjoint $u(t)$ here. The basic idea is to start with a small solution in the unstable manifold, and perform continuation in PAR(11)=$2T$ and dummy initial-condition parameters $\xi_i$ in order to satisfy the correct right-hand boundary conditions, which are defined by zeros of other dummy parameters $\omega_i$. More precisely, the left-hand end point is placed in the tangent space to the unstable manifold of the saddle and is characterized by NUNSTAB coordinates $\xi_i$ satisfying the condition

$$\xi_1^2 + \xi_2^2 + \ldots +\xi_{\tt NUNSTAB}^2 = \epsilon _0^2,$$

where $\epsilon _0$ is a user-defined small number. At the right-hand end point, NUNSTUB values $\omega_i$ measure the deviation of this point from the tangent space to the stable manifold of the saddle.

Suppose that IEQUIB=0,1 and set IP=12+IEQUIB*NDIM. Then

PAR(IP)	: $\epsilon _0$
PAR(IP+i)	: $\ \ \xi_{\tt i}$, i=1,2,...,NUNSTAB
PAR(IP+NUNSTAB+i)	: $\ \ \omega_{\tt i}$, i=1,2,...,NUNSTAB

Note that to avoid interference with the test functions (i.e. PAR(21)-PAR(36)), one must have IP+2*NUNSTAB < 21.

If an $\omega_i$ is vanished, it can be frozen while another dummy or system parameter is allowed to vary in order to make consequently all $\omega_i=0$. The resulting final solution gives the initial homoclinic orbit provided the right-hand end point is sufficiently close to the saddle. See Chapter 23 for an example, however, we recommend the homotopy method only for ``expert users''.

To compute the orientation of a homoclinic orbit (in order to detect inclination-flip bifurcations) it is necessary to compute, in tandem, a solution to the modified adjoint variational equation, by setting ITWIST=1. In order to obtain starting data for such a computation when restarting from a point where just the homoclinic is computed, upon increasing ITWIST to 1, AUTO generates trivial data for the adjoint. Because the adjoint equations are linear, only a single step of Newton's method is required to enable these trivial data to converge to the correct unique bounded solution. This can be achieved by making a single continuation step in a trivial parameter (i.e. a parameter that does not appear in the problem).

Decreasing ITWIST to 0 automatically deletes the data for the adjoint from the continuation problem.