# Detection of bifurcation points of Equilibria

The bifurcations are detected during a call to `br = continuation(prob, alg, contParams::ContinuationPar;kwargs...)`

by turning on the following flags:

`contParams.detect_bifurcation = 2`

The bifurcation points are located by looking at the spectrum **e.g.** by monitoring the unstable eigenvalues. The eigenvalue λ is declared unstable if `real(λ) > contParams.tol_stability`

. The located bifurcation points are then returned in `br.specialpoint`

.

The rightmost eigenvalues are computed by default to detect bifurcations. Hence, the number of eigenvalues with positive real parts must be finite (*e.g.* small). This might require to consider $-F(x,p)=0$ instead of (E).

## Precise detection of bifurcation points using Bisection

Note that the bifurcation points detected when `detect_bifurcation = 2`

can be rather *crude* localization of the true bifurcation points. Indeed, we only signal that, in between two continuation steps *which can be large*, a (several) bifurcation has been detected. Hence, we only have a rough idea of where the bifurcation is located, unless your `dsmax`

is very small... This can be improved as follows.

If you choose `detect_bifurcation = 3`

, a **bisection algorithm** is used to locate the bifurcation points more precisely. It means that we recursively track down the change in stability. Some options in `ContinuationPar`

control this behavior:

`n_inversion`

: number of sign inversions in the bisection algorithm`max_bisection_steps`

maximum number of bisection steps`tol_bisection_eigenvalue`

tolerance on real part of eigenvalue to detect bifurcation points in the bisection steps

If this is still not enough, you can use a Newton solver to locate them very precisely. See Fold / Hopf Continuation.

During the bisection, the eigensolvers are called like `eil(J, nev; bisection = true)`

in order to be able to adapt the solver precision.

## Large scale computations

The user must specify the number of eigenvalues to be computed (like `nev = 10`

) in the parameters `::ContinuationPar`

passed to `continuation`

. Note that `nev`

is automatically incremented whenever a bifurcation point is detected ^{[1]}. Also, there is an option in `::ContinuationPar`

to save (or not) the eigenvectors. This can be useful in memory limited environments (like on GPUs).

## List of detected bifurcation points

Bifurcation | index used |
---|---|

Fold | fold |

Hopf | hopf |

Bifurcation point (single eigenvalue stability change, Fold or branch point) | bp |

Not documented | nd |

## Eigensolver

The user must provide an eigensolver by setting `NewtonOptions.eigsolver`

where `newton_options`

is located in the parameter `::ContinuationPar`

passed to continuation. See `NewtonPar`

and `ContinuationPar`

for more information on the composite type of the options passed to `newton`

and `continuation`

.

The eigensolver is highly problem dependent and this is why the user should implement / parametrize its own eigensolver through the abstract type `AbstractEigenSolver`

or select one among List of implemented eigen solvers.

## Generic bifurcation

By this we mean a change in the dimension of the Jacobian kernel. The detection of Branch point is done by analysis of the spectrum of the Jacobian.

The detection is triggered by setting `detect_bifurcation > 1`

in the parameter `::ContinuationPar`

passed to `continuation`

.

## Fold bifurcation

The detection of **Fold** point is done by monitoring the monotonicity of the parameter.

The detection is triggered by setting `detect_fold = true`

in the parameter `::ContinuationPar`

passed to `continuation`

. When a **Fold** is detected on a branch `br`

, a point is added to `br.foldpoint`

allowing for later refinement using the function `newton_fold`

.

## Hopf bifurcation

The detection of Hopf point is done by analysis of the spectrum of the Jacobian.

The detection is triggered by setting `detect_bifurcation > 1`

in the parameter `::ContinuationPar`

passed to `continuation`

. When a **Hopf point** is detected, a point is added to `br.specialpoint`

allowing for later refinement using the function `newton_hopf`

.

- 1In this case, the Krylov dimension is not increased because the eigensolver could be a direct solver. You might want to increase this dimension using the callbacks in
`continuation`

.