Transitions between streamline topologies of structurally stable Hamiltonian flows in multiply connected domains

Abstract

h i g h l i g h t s• A combinatorial procedure providing a list of possible transitions between streamline topologies. • It is applicable to any physical phenomena described by 2D Hamiltonian vector fields.• It provides many global and generic transitions between streamline topologies. • It is applicable to snapshots of streamline patterns observed experimentally. • A new data compression algorithm for a large amount of long-time flow evolution data. a b s t r a c tWe consider Hamiltonian vector fields with a dipole singularity satisfying the slip boundary condition in two-dimensional multiply connected domains. One example of such Hamiltonian vector fields is an incompressible and inviscid flow in exterior multiply connected domains with a uniform flow, whose Hamiltonian is called the stream function. Here, we are concerned with topological structures of the level sets of the Hamiltonian, which we call streamlines by analogy from incompressible fluid flows. Classification of structurally stable streamline patterns has been considered in Yokoyama and Sakajo (2013), where a procedure to assign a unique sequence of words, called the maximal word, to these patterns is proposed. Thanks to this procedure, we can identify every streamline pattern with its representing sequence of words up to topological equivalence. In the present paper, based on the theory of word representations, we propose a combinatorial method to provide a list of possible transient structurally unstable streamline patterns between two different structurally stable patterns by simply comparing their maximal word representations without specifying any Hamiltonian. Although this method cannot deal with topological streamline changes induced by bifurcations, it reveals the existence of many non-trivial global transitions in a generic sense. We also demonstrate how the present theory is applied to fluid flow problems with vortex structures.

Discrete representations of orbit structures of flows for topological data analysis

“…The theory has been extended to the structurally stable Hamiltonian flows in the presence of a uniform flow and solid boundaries [22], in which it is additionally shown that every streamline topology of structurally stable Hamiltonian vector fields is represented by a sequence of letters, called a maximal word. Furthermore, it is found that streamline topology of orbit structures are in one-to-one correspondence with labeled directed rooted trees [20].…”

Combinatorial structures of the space of Hamiltonian vector fields on compact surfaces

“…On the other hand, the topologies of such fluids also can be changed by switching combinatorial structures of separatrices. In the codimension zero and one cases, such graph structures are characterized [5,6,7,9,10]. In particular, vertices correspond to structurally stable Hamiltonian vector fields and edges correspond to "generic" intermediate vector fields.…”

Relations among Hamiltonian, area-preserving, and non-wandering flows on surfaces

“…The difference between Hamiltonian and area-preserving flows on closed surfaces can be represented by harmonic flows, which are generated by the dual vector fields of harmonic one-forms. The topological invariants of Hamiltonian flows with finitely many singular points on compact surfaces are constructed from integrable systems points and dynamical systems points of views, and the structural stability are characterized [5,20,25,27,28,34]. On the other hand, any area-preserving flows are non-wandering flows.…”