Application of dynamical systems theory to nonlinear combustion instabilities

Two important approximations have been incorporated much of the work with approximate of unsteady motions in combustion chambers: of the series expansion to a finite of modes, and time averaging. A major of the analysis reported in this paper has to investigate the limitations of those approximations. particular two fundamental problems nonlinear behavior are discussed: the conditions which stable limit cycles of a linearly unstable may exist; and conditions under which bifurcations the limit cycle may occur. A continuation is used to determine the limit cycle behavior the equations representing the time dependent of the longitudinal acoustic modes a cylindrical combustion chamber. The system all linear processes and second-order nonlinear dynamics. The results presented show time averaging works well only when the system , in some sense, only slightly unstable. In , the stability boundaries predicted by the -mode approximation are shown to be artifacts the truncation of the system. Systems of two, , and six modes are analyzed and show that modes are needed to analyze more unstable . For the six-mode approximation with an second mode two bifurcations are found to . A pitchfork bifurcation causes a new branch limit cycles to exist in which the odd acoustic are excited. This new branch of limit cycles undergoes a torus bifurcation that causes the to exhibit stable quasi-periodic motions.