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ISTART

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ISTART=1 : This option is obsolete in the current version. It may be used as a flag that a solution is to be restarted from a previously computed point or from numerical data converted into AUTO format using us or @fc. In this case IRS>0.
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ISTART=2 : If IRS=0, an explicit solution must be specified in the subroutine STPNT in the usual format.
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ISTART=3 : The ``homotopy'' approach is used for starting, see Section 20.7 for more details. Note that this is not available with the choice IEQUIB=2.
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ISTART=4 : A phase-shift is performed for homoclinic orbits to let the equilibrium (either fixed or non-fixed, depending on IEQUIB) correspond to $ t=0$ and $ t=1$. This is necessary if a periodic orbit that is close to a homoclinic orbit is continued into a homoclinic orbit.
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ISTART=-N, $ N=1,2,3,\ldots$ : Homoclinic branch switching: this description is for reference only. We refer to the demo in Chapter 27 to see how this can be used in actual practice and to OlChKr:03 for theory and background.

The orbit is split into $ N+1$ parts and AUTO sees it as an $ (N+1)\times$NDIM-dimensional object. The first part $ u_0$ goes from the equilibrium to the point $ x_0$ that is furthest from the equilibrium. Then follow $ N-1$ shifted copies of the orbit, which travel from the point $ x_0$ back to the point $ x_0$. The last part $ U_N$ goes from the point $ x_0$ back to the equilibrium. The derivatives $ \dot{x}_0$ 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 $ \Psi$ at $ x_0$ 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 $ u_i$ and $ u_{i+1}$ are at distances $ \varepsilon_i$ away from each other, along the Lin vector $ \Psi$, at the left- and right-hand end points. These gaps $ \varepsilon_i$ are at parameters PAR(20+2*i). Moreover, each part (except $ u_{N+1}$) ends at at a Poincaré section which goes through $ x_0$ and is perpendicular to $ \dot{x}_0$.

The times $ T_i$ that each part $ u_i$ takes are stored as follows: $ T_0=$PAR(10), $ T_N=$PAR(11) and $ T_i=$PAR(19+2*i) for $ i=1\ldots N-1$. Through a continuation in problem parameters, gaps $ \varepsilon_i$, and times $ T_i$ it is possible to switch from a $ 1$-homoclinic to an $ N$-homoclinic orbit.

If ITWIST=0, the adjoint vector is not computed and Lin's method is not used. Instead, AUTO produces a gap $ \varepsilon$=PAR(22) at the right-hand end point $ p$ of $ u_{N+1}$, measuring the distance between the stable manifold of the equilibrium and $ p$. This technique can also be used to find 2-homoclinic orbits, by varying in $ \varepsilon$ and $ T_1$, 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.


next up previous contents
Next: NREV, IREV Up: HomCont-Constants. Previous: ITWIST   Contents
Gabriel Lord 2007-11-19