Four Types of Chaotic Dynamics
in Cellular Flames

Theoretical Studies of Intermittently Ordered States

Intermittently ordered states are examples of heteroclinic connections. These dynamical structures are thought to play an important role in turbulence. Stone and Holmes (1991) reviewed the recent literature on this subject. Armbruster, Guckenheimer and Holmes (1988) proved the existence of structurally stable heteroclinic cycles in certain differential equations invariant under continuous symmetry groups, such as 0(2), the group of rotations and reflections in the plane, which is the relevant symmetry group for our experiments.

In the language of dynamics the evolution of the system in time is described by the motion of a point in state space. The signature of heteroclinic connections is that the system visits the neighborhood of ordered states for varying lengths of time.

Each ordered state corresponds to a fixed point which becomes unstable as the total flow rate is increased. For isobutane-air flames there is always an ordered state (with different numbers of inner and outer cells, corresponding to a different point) that becomes stable and no intermittently ordered states are observed. For propane-air flames there are ranges of flow rates in which ALL the fixed points are unstable. The system is attracted to the neighborhoods of these fixed points where it is repelled along the unstable direction. An orbit which connects different fixed points is called a heteroclinic orbit, hence the name, heteroclinic connection.

In an experiment an intermittently ordered state is observed when the system passes near one of the unstable fixed points. The system can remain in the vicinity of the fixed point for various lengths of time corresponding to a range of residence times. When the system moves outside the neighborhood of the fixed point, it is moving along the unstable manifold connecting the fixed points and the ordered state is no longer discernible. Such motion is depicted in frames 2-13 of Figure 4. Notice the qualitatively different spatial character of this state compared with that of the disordered state.

Although heteroclinic connections are believed to occur widely in a variety of fluid flows (Moffatt, 1986), the spatial and temporal characteristics of a fluid exhibiting this dynamics has not been studied in detail. None of the numerical studies of cellular flames has identified similar dynamics although heteroclinic connections are well known in studies of the one-dimensional Kuramoto-Sivashinsky equation (Kevrekidis, Nicolenko, and Scovel, 1990). The studies using the hydrodynamical model of cellular flames have not specifically discussed heteroclinic connections.

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