# BifurcationKit.jl

This Julia package aims at performing **bifurcation analysis** of large dimensional equations F(u,λ)=0 where λ∈ℝ.

It incorporates a pseudo arclength continuation algorithm which provides a *predictor* (u1,λ1) from a known solution (u0,λ0). A Newton-Krylov method is then used to correct this predictor and a Matrix-Free eigensolver is used to compute stability and bifurcation points.

By leveraging on the above method, it can also seek for periodic orbits of Cauchy problems by casting them into an equation F(u,λ)=0 of high dimension. **It is by now, one of the only softwares which provides shooting methods AND methods based on finite differences 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...

**If you use this package for your work, please cite it!! Open source development strongly depends on this. It is referenced on HAL-Inria as follows:**

```
@misc{veltz:hal-02071874,
TITLE = {{PseudoArcLengthContinuation.jl}},
AUTHOR = {Veltz, Romain},
URL = {https://hal.inria.fr/hal-02071874},
YEAR = {2019},
MONTH = Mar,
KEYWORDS = {Pseudo Arclength Continuation},
PDF = {https://hal.inria.fr/hal-02071874/file/PseudoArcLengthContinuation.jl-master.zip},
HAL_ID = {hal-02071874},
HAL_VERSION = {v1},
}
```

## Installation

This package requires Julia >= v1.3.0

To install it, please run

`] add https://github.com/rveltz/BifurcationKit.jl`

## Website

The package is located here.

## Examples of bifurcation diagrams

Snaking in 2D Swift Hohenberg | Periodic orbits in Brusselator |

Period doubling BVAM Model | Ginzburg-Landau 2d |

## Main features

- Newton-Krylov solver with generic linear / eigen
*preconditioned*solver. Idem for the arc-length continuation. - Newton-Krylov solver with nonlinear deflation and preconditioner. It can be used for branch switching for example.
- Bifurcation points are located using a bisection algorithm
- Branch, Fold, Hopf bifurcation point detection of stationary solutions.
- Automatic branch switching at simple branch points
- Automatic branch switching at simple Hopf points to periodic orbits
- Fold / Hopf continuation based on Minimally Augmented formulation, with Matrix Free / Sparse Jacobian.
- Periodic orbit computation and continuation using Shooting or Finite Differences.
- Branch, Fold, Neimark-Sacker, Period Doubling bifurcation point detection of periodic orbits.
- Computation and Continuation of Fold of periodic orbits

Custom state means, we can use something else than `AbstractArray`

, for example your own `struct`

.

**Note that you can combine most of the solvers, like use Deflation for Periodic orbit computation or Fold of periodic orbits family.**

Features | Matrix Free | Custom state | Tutorial | GPU |
---|---|---|---|---|

Newton | Y | Y | All | |

Newton + Deflation | Y | Y | 4, 5 | |

Continuation (Natural, Secant, Tangent) | Y | Y | All | |

Branching point detection | Y | Y | All | |

Fold point detection | Y | Y | All | |

Hopf detection | Y | Y | 6 - 9 | |

Fold Point continuation | Y | Y | 1, 8 | |

Hopf continuation | Y | `AbstractArray` |
5 | |

Branch switching at Branch / Hopf points | Y | `AbstractArray` |
3 | |

Periodic Orbit (FD) Newton / continuation | Y | `AbstractVector` |
6, 8 | |

Periodic Orbit with Parallel Poincaré / Standard Shooting Newton / continuation | Y | `AbstractArray` |
6, 7, 9 | |

Fold, Neimark-Sacker, Period doubling detection | Y | `AbstractVector` |
6 - 9 | |

Continuation of Fold of periodic orbits | Y | `AbstractVector` |
8 |

## To do or grab

Without a priority order:

- improve compatibility with
`DifferentialEquations.jl`

- Add interface to other iterative linear solvers (cg, minres,...) from IterativeSolvers.jl