APPLICATION OF DYNAMICAL-SYSTEMS THEORY TO NONLINEAR COMBUSTION INSTABILITIES

Two important approximations have been incorporated in much of the wor k with approximate analysis of unsteady motions in combustion chambers : 1) truncation of the series expansion to a finite number of modes, a nd 2) time-averaging. A major purpose of the present analysis is to in vestigate the limitations of those approximations. A continuation meth od is used to determine the limit cycle behavior of the time-dependent amplitudes of the longitudinal acoustic modes in a combustion chamber . The results show that time-averaging works well only when the system is slightly unstable. tn addition, the stability boundaries predicted by the two-mode approximation are shown to be artifacts of the trunca tion of the system. Systems of two, four, and six modes are analyzed a nd show that more modes are needed to analyze more unstable systems. F or the six-mode approximation with an unstable second-mode, two bifurc ations are found to exist: 1) a pitchfork bifurcation leading to a new branch of limit cycles, and 2) a torus bifurcation leading to quasipe riodic motions.