### Abstract

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.

Original language | English |
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Pages (from-to) | 789-805 |

Number of pages | 17 |

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 15 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2016 Jan 1 |

### Keywords

- Absolute and convective instability and stability
- Bifurcation
- Binary fluid mixture
- Collision dynamics
- Hierarchical network structure
- Localized structure

### ASJC Scopus subject areas

- Analysis
- Modelling and Simulation

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## Cite this

*SIAM Journal on Applied Dynamical Systems*,

*15*(2), 789-805. https://doi.org/10.1137/15M1010968