R. Veltz
** BifurcationKit.jl**
*
Inria Sophia-Antipolis,
2020
*
pseudo-arclength-continuation
periodic-orbits
floquet
gpu
bifurcation-diagram
deflation
newton-krylov
URL,
RIS,
BibTex

This Julia package aims at performing automatic bifurcation analysis of possibly large dimensional equations F(u,λ)=0 where λ∈ℝ.
It incorporates continuation algorithms (PALC, deflated continuation, ...) which provide a predictor $(u_1,p_1)$ from a known solution $(u_0,p_0)$. A Newton-Krylov method is then used to correct this predictor and a Matrix-Free eigensolver is used to compute stability and bifurcation points.
Despite initial focus on large scale problems, the package can easily handle low dimensional problems.
By leveraging on the above method, the package can also seek for periodic orbits of Cauchy problems by casting them into an equation $F(u,p)=0$ of high dimension. It is by now, one of the only softwares which provides shooting methods AND methods based on finite differences or collocation to compute periodic orbits.
The current package focuses on large scale nonlinear problems and multiple hardwares. Hence, the goal is to use Matrix Free methods on GPU (see PDE example and Periodic orbit example) or on a cluster to solve non linear PDE, nonlocal problems, compute sub-manifolds...
One design choice is that we try not to require u to be a subtype of an AbstractArray as this would forbid the use of spectral methods like the one from ApproxFun.jl. For now, our implementation does not allow this for Hopf continuation and computation of periodic orbits.
```
@{veltz:hal-02902346,
author = "R. Veltz",
title = "BifurcationKit.jl",
year = 2020,
institution = "Inria Sophia-Antipolis",
month = "Jul",
keywords = "pseudo-arclength-continuation, periodic-orbits, floquet, gpu, bifurcation-diagram, deflation, newton-krylov",
url = "https://hal.archives-ouvertes.fr/hal-02902346"
}
```