Experimental factors also contributed to the observed irregular motion. First, with such a large number of cells, there is a competition between the natural hexagonal structure associated with the unbounded system and the ring of cells at the circular boundary. Second, most other studies were conducted at atmospheric pressure in an open room. The presence of a diffusion flame and the convection of the room air strongly perturb the cellular pattern. Third, we cannot overstate the importance of the design and construction of the Pagni burner for the observation of the ordered states. In other experiments (by us) using a porous plug with a square geometry, the ordered states were never found. The cellular flames had a motion that was reminiscent of pictures of the irregular structures reported in most other experiments.
The disordered states, the intermittently ordered states and the pulsating cellular states described here all bifurcate from the ordered states. Their dynamics can be found in the solutions to the equations that describe the flame front. The nonperiodic dynamics observed in previous experiments occurs because of boundary conditions and external noise which cause the system to constantly make transitions among ordered states with slightly different (large) numbers of cells. This motion overwhelms the intrinsic chaotic motion and makes it unobservable.
The chaotic motion of the ordered states in which ordered cells subtly change their shape and size is the most unusual and unexpected aspect of this study. This motion causes subtle, but important effects in the other states of cellular flames. The experimental data conclusively establishes that low amplitude, nonperiodic oscillations are characteristic of ordered cellular states. The difficult question is whether these oscillations are chaotic and intrinsic to the structure of cellular flames, or whether they are extrinsic, due to the amplification of external noise (from nonaxial velocity components of the premixed gases) by the cellular structure. The answer to this question involves highly technical issues in dynamics. The preponderance of our experimental evidence to date suggests that the chaotic motion is intrinsic, and that it is associated with the unique topological nature of the cusps and folds of cellular flames.
The current tools of analysis of chaotic systems grew out of studies of the properties of strange attractors that describe the dynamics arising from ordinary differential equations. The pictures in Figures 1, 4, and 5 demonstrate the considerable differences among the spatial characteristics of these states; yet none of the current measures of chaotic dynamics distinguishes among these four states. Calculations of the dimension of the chaotic attractor yield inconclusive results in each case. The power spectra have similar high frequency fall-offs. These techniques work for "low-dimensional" dynamical systems, such as chaotic pulsating flames which are described by coupled sets of ordinary differential equations, but fail badly when applied to more complicated systems, such as cellular flames, which are described by partial dlifferential equations.