The orbit is split into parts and AUTO sees it as an NDIM-dimensional object. The first part goes from the equilibrium to the point that is furthest from the equilibrium. Then follow shifted copies of the orbit, which travel from the point back to the point . The last part goes from the point back to the equilibrium. The derivatives with respect to time of the point that is furthest from the equilibrium are stored at the parameters PAR(NPARX-NDIM+1...NPARX).
If ITWIST=1, and this was also the case in the preceding run, then a copy of the adjoint vector at is stored at the parameters PAR(NPARX-NDIM*2+1...NPARX-NDIM) and Lin's method can be used to do homoclinic branch switching. To be more precise, the individual parts and are at distances away from each other, along the Lin vector , at the left- and right-hand end points. These gaps are at parameters PAR(20+2*i). Moreover, each part (except ) ends at at a Poincaré section which goes through and is perpendicular to .
The times that each part takes are stored as follows: PAR(10), PAR(11) and PAR(19+2*i) for . Through a continuation in problem parameters, gaps , and times it is possible to switch from a -homoclinic to an -homoclinic orbit.
If ITWIST=0, the adjoint vector is not computed and Lin's method is not used. Instead, AUTO produces a gap =PAR(22) at the right-hand end point of , measuring the distance between the stable manifold of the equilibrium and . This technique can also be used to find 2-homoclinic orbits, by varying in and , similar to the method described before, but only if the unstable manifold in one-dimensional. Because this method is more limited than the method using Lin vectors, we do not recommend it for normal usage.
To switch back to a normal homoclinic orbit, set ISTART back to a positive value such as 1. Now HomCont has lost all the information about the adjoint, so if ITWIST is set to 0, HomCont does a normal continuation without the adjoint, and if ITWIST is set to 1, one needs to do a Newton dummy step first to recalculate the adhoint.