Modal decomposition and normal form for hydrodynamic flows: Examples from cellular flame patterns

Abstract

Spatio-temporal complexity of hydrodynamic flows may be reduced through modal decomposition, especially in systems with symmetries. The symmetries of the most significant modes can then be used to deduce normal form equations associated with the observed state. In turn, the normal form equations can be used to deduce bifurcations to and from the given state. We illustrate this process using two spatio-temporal cellular states on a circular flame front. The first example contains a pair of uniformly rotating cells. Principle component analysis shows that two coherent structures capture most of the dynamics and suggests that the state is a broken-parity traveling mode. Other experimentally observed states, such as modulated rotating states and a heteroclinic cycle between two spatially orthonormal states result from secondary bifurcations from the rotating state. The second example, referred to as the hopping mode, visually appears to have significantly more complicated dynamics. However, modal decomposition shows that it consists of two parity broken states moving at different angular velocities. The corresponding normal form contains a codimension-three steady-state bifurcation leading to a homoclinic cycle whose spatio-temporal characteristics are similar to those of hopping states. We use these examples to propose a methodology to combine coherent structures that form a single, possibly time-dependent entity which we refer to as a generalized coherent structure. The process can reduce the number of entities needed to expand complex spatio-temporal states. The paper is dedicated to the memory of Michael Gorman, whose experiments on cellular flame fronts and relentless demands for better theoretical understanding of the patterns motivated the study.