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This paper presents a detailed analysis of the stability and network structure of thermal convection patterns of mixtures containing two miscible fluids. The stability of steady spatially localized solutions consisting of an even number of convection cells (even-SP) is investigated in detail because even-SPs emerge as transient states resulting from collisions between counterpropagating periodic traveling pulses, while odd-SPs do not. The even-SP branch is examined to analyze the distribution of the eigenvalues and the corresponding eigenfunctions to understand the stable/unstable direction in the vicinity of each solution and to investigate how they are connected to each other. Further, we construct a hierarchical network diagram consisting of even-SP solutions as nodes and stable and unstable manifolds connecting between them as edges, which shows a skeleton of transition processes of an arbitrary initial condition and asymptotic states.

A Skeleton of Collision Dynamics: Hierarchical Network Structure among Even-Symmetric Steady Pulses in Binary Fluid Convection