Dm. Wolf et Sr. Sanders, MULTIPARAMETER HOMOTOPY METHODS FOR FINDING DC OPERATING POINTS OF NONLINEAR CIRCUITS, IEEE transactions on circuits and systems. 1, Fundamental theory andapplications, 43(10), 1996, pp. 824-838
This paper introduces multiparameter homotopy methods for finding de o
perating points. The question of whether adding extra real or complex
parameters to a single-parameter homotopy function can lead to improve
d solution paths is investigated, It is shown that no number of added
real parameters can lead to local fold avoidance, but that generic fol
ds may be efficiently avoided by complexifying the homotopy parameter
and tracing a closed curve in complex parameter space around the criti
cal fold value. A combination of real 2-parameter homotopy and complex
parameter homotopy is shown to be sufficient for avoiding real fork b
ifurcations and enumerating all real, locally connected branches. Addi
tionally, the potential of complex parameter homotopy methods for find
ing all circuit solutions Is explored. Results from algebraic geometry
indicate that if an analytic homotopy function with a single complex
parameter is irreducible, then there exist regular paths through the c
omplex parameter plane connecting any solution of H(x, lambda') = 0 to
any other solution of H(x, lambda') = 0. Thus, in principle at least,
complex parameter homotopy can be used to find all circuit solutions.