In this talk, I will give a panorama of BifurcationKit.jl, a Julia package to perform numerical bifurcation analysis of large dimensional equations (PDE, nonlocal equations, etc) using Matrix-Free / Sparse Matrix formulations of the problem. Notably, numerical bifurcation analysis can be done entirely on GPU.

BifurcationKit incorporates continuation algorithms (PALC, deflated continuation, ...) which can be used to perform fully automatic bifurcation diagram computation of stationary states. I will showcase this with the 2d Bratu problem. I will also show an example of neural network that runs entirely on GPU.

Additionally, by leveraging on the above methods, the package can also seek for periodic orbits of Cauchy problems by casting them into an equation of high dimension. It is by now, one of the only softwares which provides parallel (Standard / Poincaré) shooting methods and finite differences based methods to compute periodic orbits in high dimensions. I will present an application highlighting the ability to fine tune BifurcationKit to get performance.