Fold / Hopf Continuation

• Fold / Hopf Continuation
• • Fold continuation
  • Hopf continuation
  • Setting the jacobian
  • Newton refinement
  • Codim 2 continuation
  • Advanced use
  • Algorithmic details (Fold)
  • Algorithmic details (Hopf)
  • References

In this page, we explain how to perform continuation of Fold / Hopf points and detect the associated bifurcations.

For this to work best, it is advised to have an analytical expression for the jacobian. See the tutorial Temperature model (Simplest example) for more details.

A quite complete example for detection of codim 2 bifurcations of equilibria is Extended Lorenz-84 model (codim 2 + BT/ZH aBS) although it is for ODEs.

List of detected codim 2 bifurcation points

In a nutshell, all you have to do (see below) is to call continuation(br, ind_bif, lens2) to continue the bifurcation point stored in br.specialpoint[ind_bif] and set proper options.

Fold continuation

The continuation of Fold bifurcation points is based on a Minimally Augmented^[Govaerts] formulation which is an efficient way to detect singularities. The continuation of Fold points is based on the formulation

\[G(u,p) = (F(u,p), \sigma(u,p))\in\mathbb R^{n+1}\quad\quad (F_f)\]

where the test function $g$ is solution of

\[\left[\begin{array}{cc} dF(u,p) & w \\ v^{\top} & 0 \end{array}\right]\left[\begin{array}{c} r \\ \sigma(u,p) \end{array}\right]=\left[\begin{array}{c}0_{n} \\1\end{array}\right]\quad\quad (M_f)\]

where $w,v$ are chosen in order to have a non-singular matrix $(M_f)$. More precisely, $v$ (resp. $w$) should be close to a null vector of dF(u,p) (resp. dF(u,p)'). During continuation, the vectors $w,v$ are updated so that the matrix $(M_f)$ remains non-singular ; this is controlled with the argument update_minaug_every_step (see below).

note that there are very simplified calls for this, see Newton refinement below. In particular, you don't need to set up the Fold Minimally Augmented problem yourself. This is done in the background.

Detection of codim 2 bifurcation points

You can detect the following codim 2 bifurcation points by using the option detect_codim2_bifurcation in the method continuation. Under the hood, the detection of these bifurcations is done by using Event detection as explained in Event Handling.

• the detection of Cusp (Cusp) is done by the detection of Fold bifurcation points along the curve of Folds by monitoring the parameter component of the tangent.
• the detection of Bogdanov-Takens (BT) is performed using the test function^[Bindel] $\psi_{BT}(p) = \langle w(p),v(p)\rangle$
• the detection of Zero-Hopf (ZH) is performed by monitoring the number of eigenvalues $\lambda$ such that $\Re\lambda > \min\limits_{\nu\in\Sigma(dF)}|\Re\nu|$ and $\Im\lambda > \epsilon$ where $\epsilon$ is the Newton tolerance.

Hopf continuation

The continuation of Fold bifurcation points is based on a Minimally Augmented (see ^[Govaerts] p. 87) formulation which is an efficient way to detect singularities. The continuation of Hopf points is based on the formulation

\[G(u,\omega,p) = (F(u,\omega,p), \Re\sigma(u,\omega,p), \Im\sigma(u,\omega,p))\in\mathbb R^{n+2}\quad\quad (F_h)\]

where the test function $g$ is solution of

\[\left[\begin{array}{cc} dF(u,p)-i\omega I_n & w \\ v^{\top} & 0 \end{array}\right]\left[\begin{array}{c} r \\ \sigma(u,\omega,p) \end{array}\right]=\left[\begin{array}{c} 0_{n} \\ 1 \end{array}\right]\quad\quad (M_h)\]

where $w,v$ are chosen in order to have a non-singular matrix $(M_h)$. More precisely, $w$ (resp. $v$) should be a left (resp. right) approximate null vector of $dF(u,p)-i\omega I_n$. During continuation, the vectors $w,v$ are updated so that the matrix $(M_h)$ remains non-singular ; this is controlled with the argument update_minaug_every_step (see below).

note that there are very simplified calls to this, see Newton refinement below. In particular, you don't need to set up the Hopf Minimally Augmented problem yourself. This is done in the background.

Detection of codim 2 bifurcation points

You can detect the following codim 2 bifurcation points by using the option detect_codim2_bifurcation in the method continuation. Under the hood, the detection of these bifurcations is done by using Event detection as explained in Event Handling.

• the detection of Bogdanov-Takens (BT) is performed using the test function^[Bindel],^[Blank] $\psi_{BT}(p) = \langle w(p),v(p)\rangle$
• the detection of Bautin (GH) is based on the test function $\psi_{GH}(p) = \Re(l_1(p))$ where $l_1$ is the Lyapunov coefficient defined in Simple Hopf point.
• the detection of Zero-Hopf (ZH) is performed by monitoring the eigenvalues.
• the detection of Hopf-Hopf (HH) is performed by monitoring the eigenvalues.

The continuation of Hopf points is stopped at BT and when $\omega<100\epsilon$ where $\epsilon$ is the newton tolerance.

Setting the jacobian

In order to apply the newton algorithm to $F_f$ or $F_h$, one needs to invert the jacobian. This is not completely trivial as one must compute this jacobian and then invert it. You can select the following jacobians for your computations (see below):

• [Default] for jacobian_ma = :autodiff, automatic differentiation is applied to $F_f$ (or $F_h$) and the matrix is then inverted using the provided linear solver. In particular, the jacobian is formed. This is very well suited for small dimensions (say < 100)
• for jacobian_ma = :minaug, a specific procedure for evaluating the jacobian $F_f$ (or $F_h$) and inverting it (without forming the jacobian!) is used. This is well suited for large dimensions.

Newton refinement

Once a Fold / Hopf point has been detected after a call to br = continuation(...), it can be refined using newton iterations. Let us say that ind_bif is the index in br.specialpoint of a Fold / Hopf point. This guess can be refined as follows:

outfold = newton(br::AbstractBranchResult, ind_bif::Int;  
	normN = norm, options = br.contparams.newton_options,
	bdlinsolver = BorderingBLS(options.linsolver),
	start_with_eigen = false, kwargs...)

For the options parameters, we refer to Newton.

It is important to note that for improved performances, a function implementing the expression of the hessian should be provided. This is by far the fastest. Reader interested in this advanced usage should look at the code example/chan.jl of the tutorial Temperature model (Simplest example).

Codim 2 continuation

To compute the codim 2 curve of Fold / Hopf points, one can call continuation with the following options

where the options are as above except with have an additional parameter axis lens2 which is used to locate the bifurcation points.

See Temperature model (Simplest example) for an example of use.

Advanced use

Here, we expose the solvers that are used to perform newton refinement or codim 2 continuation in case the above methods fails. This is useful in case it is too involved to expose the linear solver options. An example of advanced use is the continuation of Folds of periodic orbits, see Continuation of Fold of periodic orbits.

Algorithmic details (Fold)

If we write $(s,\sigma)$ the solution of the adjoint problem associated to $(M_f)$, one can show^[Govaerts] that the differential of $\sigma$ satisfies: $ \partial \sigma + \langle s,\partial dF \cdot r\rangle = 0 $ This allows to compute the jacobian of the Fold functional to use for the Newton algorithm

\[\left[\begin{array}{cc} \partial_{u}F(u,p) & \partial_pF(u,p) \\ \partial_x\sigma(u,p) & \partial_p\sigma(u,p) \end{array}\right].\]

Algorithmic details (Hopf)

We recall that the unknowns are $(x,p,\omega)$. The jacobian of the Hopf functional to use for the Newton algorithm

\[\left[\begin{array}{ccc} \partial_{u}F & \partial_pF & 0 \\ \partial_x\sigma_r & \partial_p\sigma_r & \partial_\omega\sigma_r\\ \partial_x\sigma_i & \partial_p\sigma_i & \partial_\omega\sigma_i \end{array}\right].\]

References