Abstract
A Koopman decomposition is a powerful method of analysis for fluid flows leading to an apparently linear description of nonlinear dynamics in which the flow is expressed as a superposition of fixed spatial structures with exponential time dependence. Attempting a Koopman decomposition is simple in practice due to a connection with Dynamic Mode Decomposition (DMD). However, there are nontrivial requirements for the Koopman decomposition and DMD to overlap which mean it is often difficult to establish whether the latter is truly approximating the former. Here, we focus on nonlinear systems containing multiple simple invariant solutions where it is unclear how to construct a consistent Koopman decomposition, or how DMD might be applied to locate these solutions. First, we derive a Koopman decomposition for a heteroclinic connection in a StuartLandau equation revealing two possible expansions. The expansions are centred about the two fixed points of the equation and extend beyond their linear subspaces before breaking down at a crossover point in state space. Welldesigned DMD can extract the two expansions provided that the time window does not contain this crossover point. We then apply DMD to the NavierStokes equations near to a heteroclinic connection in lowReynolds number (Re=O(100)) plane Couette flow where there are multiple simple invariant solutions beyond the constant shear basic state. This reveals as many different Koopman decompositions as simple invariant solutions present and again indicates the existence of crossover points between the expansions in state space. Again, DMD can extract these expansions only if it does not include a crossover point.
Original language  English 

Pages (fromto)  127 
Number of pages  27 
Journal  Journal of Fluid Mechanics 
Volume  879 
Early online date  19 Sep 2019 
DOIs  
Publication status  Published  25 Nov 2019 
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Jacob Page
 School of Mathematics  Lectureship/Readership in Applied and Computational Mathemat
Person: Academic: Research Active (Teaching)