Singularity induced bifurcation and fold points in inviscid transonic flow

Abstract

Transonic inviscid flow equation of elliptichyperbolic type when written in terms of the velocity components and similarity variable results in a second order nonlinear ODE having several features typical of differential algebraic equations rather than ODEs. These features include the fold singularities (e.g. folded nodes and saddles, forward and backward impasse points), singularity induced bifurcation behavior and singularity crossing phenomenon. We investigate the above properties and conclude that the quasilinear DAEs of transonic flow have interesting properties that do not occur in other known quasilinear DAEs, for example, in MHD. Several numerical examples are included.

Inflection, Canards and Folded Singularities in Excitable Systems: Application to a 3D FitzHugh–Nagumo Model

“…Therefore, after some rearrangement and substitutions for dy/dx and dz/dx, we obtain an implicit equation which we rewrite, after twice completing a square, as (34) with: K = ε 2 (x + εα) 2 + y 2 . One can show that the solution to equation (34) corresponds to a bubble (i.e.…”

“…Such dynamical systems have been successful in modelling many complex phenomena (e.g. chemical reactions [11], neuronal activity [43], semiconductor lasers [33], material science [42], astrophysics [34], etc.) where the timing and ordering of various processes ensures correct functions.…”