Separatrix construction

Thus, if both the original and biased systems obey TST so that the above-mentioned derivation holds, hyperdynamics can provide considerable acceleration compared to direct-MD simulations. However, in practice, the applicability of hyperdynamics is limited by the availability of low-overhead bias potentials. Indeed, while some different forms have been proposed in the last few years, often they are computationally expensive, tailored to a limited class of systems or built on sets of restrictive assumptions about the nature of the separatrix. The main challenge, which is the subject of active research in different groups, thus remains the construction of bias potentials that are simple, efficient, generic, and transferable. We present below one recent advance in this area. 

Figure 21. Construction of the exact separatrix on the surface of section for a symmetric double-well model potential, (a) The unstable manifold (b) Superposition of the stable (dashed) and unstable (solid) manifolds, (c) The exact separatrix, which is a union of portions of the above manifolds, (d) Turnstiles superimposed on the separatrix. [From S. K. Gray and S. A. Rice, J. Chem. Phys. 86, 2020 (1987).]...

Denote by Ki and K2 both rings, into the union of which the circles 71 and 72 split the torus Construct a new surface Pi by adding to the separatrix... 

In Chap. 13 we will consider the bifurcations of a homoclinic loop to a saddle equilibrium state. We start with the two-dimensional case. First of all, we investigate the question of the stability of the separatrix loop in the generic case (non-zero saddle value), as well as in the case of a zero saddle value. Next, we elaborate on the cases of arbitrarily finite codimensions where the so-called Dulac sequence is constructed, which allows one to determine the stability of the loop via the sign of the first non-zero term in this sequence. 

In the same section we give the bifurcation diagrams for the codimension two case with a first zero saddle value and a non-zero first separatrix value (the second term of the Dulac sequence) at the bifurcation point. Leontovich s method is based on the construction of a Poincare map, which allows one to consider homoclinic loops on non-orientable two-dimensional surfaces as well, where a small-neighborhood of the separatrix loop may be a Mobius band. Here, we discuss the bifurcation diagrams for both cases. 

---