# Growing patterns

## TL;DR

This paper explores how expanding pattern-forming regions select specific crystalline states, using mathematical and numerical tools to analyze the complex bifurcation phenomena involved.

## Contribution

It provides new insights into pattern selection during growth, highlighting the role of bifurcation diagrams and analytical tools in understanding this process.

## Key findings

- Pattern selection depends on growth rates and bifurcation structures.
- Mathematical tools can elucidate the complexity of pattern formation.
- Numerical simulations support theoretical predictions.

## Abstract

Pattern forming systems allow for a wealth of states, where wavelengths and orientation of patterns varies and defects disrupt patches of monocrystalline regions. Growth of patterns has long been recognized as a strong selection mechanism. We present here recent and new results on the selection of patterns in situations where the pattern-forming region expands in time. The wealth of phenomena is roughly organized in bifurcation diagrams that depict wavenumbers of selected crystalline states as functions of growth rates. We show how a broad set of mathematical and numerical tools can help shed light into the complexity of this selection process.