Synchronization subspace

Since the synchronous CW solution is stable within the synchronization subspace, its stability in the whole phase space is determined by its transverse stability, i.e. the stability with respect to perturbations of the initial data transverse to the synchronization subspace. The analysis of the transverse stability of synchronous CW solutions can be carried out by inspecting the transverse characteristic equation [29]... 

Fig. 6.21. The regions in the f — r] parameter space, for which an orbit started from randomly chosen initial condition is attracted to the synchronization subspace Ms This corresponds to the asymptotic synchronization of the outer lasers, (a) 5 coupled lasers, (b) 10 coupled lasers.

Note, that Fig. 6.15(a) reveals the local stability properties of the synchronous stationary states. In order to have an insight into the transverse stability of the whole subspace Mg, we perform also a numerical analysis and show the results in Fig. 6.15(b). A grid 100x100 was introduced to discretize a square region of the parameters rj and initial condition near the subspace, but not exactly symmetric. The gray area shows the regions, for which the orbit was attracted to the subspace, i.e. for which it was synchronized. 

In fig. 2 we show the potential energy diagram in the r/a space illustrated in fig. 1. In this subspace the synchronous supra-supra reaction path passes over a local maximum and there is a synchronous channel involving a coplanar syn diradicaloid structure. When rotation about the CC bond is considered, the syn diradicaloid structure turns out to be a local maximum. True transition states exist only for two reaction paths associated with... 

BFGS = Broyden - Fletcher - Goldfarb - Shanno DFP = Davidson-Fletcher-Powell EF = eigenvector following GDIIS = geometry optimization by direct inversion of the iterative subspace LST = linear synchronous transit QST = quadratic synchronous transit RFO = rational function optimization. 

CPR = conjugate peak refinement GDIIS = geometry direct inversion in the iterative subspace GE = gradient extremal LST = linear synchronous transit LTP = line then plane LUP = locally updated planes NR = Newton-Raph-son P-RFO = partitioned rational function optimization QA = quadratic approximation QST = quadratic synchronous transit SPW = self-penalty walk STQN = synchronous transit-guided quasi-Newton TRIM = trust radius image minimization TS = transition structure. 

---