Demo 3: Subcritical pitchfork in a 2D crystal

A 2D hexagonal Lennard-Jones crystal is compressed uniaxially. At a critical strain (~14.5 %) the symmetric (uniformly compressed) equilibrium becomes unstable and two buckled configurations emerge — a pitchfork bifurcation. The buckling mode is a vortex-like in-plane rotation of atoms about the centre of a lattice triangle. The two buckled branches correspond to opposite rotation senses. The bifurcation is subcritical: the buckled branches first extend backward in strain (with a fold/turning point) before continuing forward — creating a hysteresis region. In this notebook we will use LACT to capture this phenomenon by tracing three solution curves:

Curve	Description
QS	Quasi-static loading starting from undeformed state
Continuation A	Seeded just before the end of QS, continued backward — traces the buckled branch
Continuation B	Seeded well before bifurcation, continued forward past it on the symmetric (unstable) branch