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 algorithmmax_bisection_stepsmaximum number of bisection stepstol_bisection_eigenvaluetolerance 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.