correct tuto sh2d FD

docs/src/index.md

@@ -64,7 +64,7 @@ Needless to say, if you use regular arrays, you don't need to worry about what f
 
 We make the same requirements as `KrylovKit.jl`. Hence, we refer to its [docs](https://jutho.github.io/KrylovKit.jl/stable/#Package-features-and-alternatives-1) for more information. We additionally require the following methods to be available:
 
-- `Base.length(x)`: it is used in the constraint equation of the pseudo arclength continuation method (see [`continuation`](@ref) for more details). If `length` is not available for your "vector", define it `length(x) = 1` and adjust tuning the parameter `θ` in `PALC`.
+- `Base.length(x)`: it is used in the constraint equation of the pseudo arclength continuation method (see [`continuation`](@ref) for more details). If `length` is not available for your "vector", define `length(x) = 1` and adjust the parameter `θ` in `PALC`.
 - `Base.copyto!(dest, in)` this is required to reduce the allocations by avoiding too many copies
 - `Base.eltype` must be extended to your vector type. It is mainly used for branching.
 

docs/src/intro-abs.md

@@ -48,63 +48,6 @@ We refer to [Branch switching](@ref abs-nonsimple-eq) for more details.
 We provide a graph of bifurcations of equilbria and periodic orbits that can be detected in `BifurcationKit`. An arrow from say `Equilibrium` to `Hopf` means that Hopf bifurcations can be detected while continuing equilibria. Each object of codim 0 (resp. 1) can be continued in one (resp. 2) parameters.
 
 
-```@diagram mermaid
-graph LR
-    S[  ]
-    C[ Equilibrium ]
-    PO[ Periodic orbit ]
-    BP[ Fold/simple branch point ]
-    H[ Hopf \n :hopf]
-    CP[Cusp]
-    BT[ Bogdanov-Takens ]
-    ZH[ZH]
-    HH[HH]
-    GH[Bautin]
-    FPO[ Fold Periodic orbit ]
-    NS[ Neimark-Sacker ]
-    PD[ Period Doubling ]
-    BPC[BPC]
-
-    S --> C
-    S --> PO
-    C --> nBP[ non simple\n branch point ]
-    C --> BP
-    C --> H
-
-    BP --> CP
-    BP --> BT
-
-    PO --> H
-    PO --> FPO
-    PO --> NS
-    PO --> PD
-
-    FPO --> GH
-    FPO --> BPC
-    FPO --> R1
-
-    NS --> R1
-    NS --> R3
-    NS --> R4
-    NS --> CH
-    NS --> LPNS
-    NS --> PDNS
-    NS --> R2
-    NS --> NSNS
-
-    PD --> PDNS
-    PD --> R2
-    PD --> LPPD
-    PD --> GPD
-
-    H --> BT
-    H --> ZH
-    BP --> ZH
-    H --> HH
-    H --> GH
-
-    Codim --> Codim0 --> Codim1 --> Codim2
-```
 
 ## Summary of branching procedures
 

docs/src/tutorials/tutorials2.md

@@ -11,7 +11,7 @@ $$-(I+\Delta)^2 u+l\cdot u +\nu u^2-u^3 = 0$$
 
 with Neumann boundary conditions. This full example is in the file `example/SH2d-fronts.jl`. This example is also treated in the MATLAB package [pde2path](http://www.staff.uni-oldenburg.de/hannes.uecker/pde2path/). We use a Sparse Matrix to express the operator $L_1=(I+\Delta)^2$.
 
-```@example sh2d_fd
+```@example sh2dFD
 using DiffEqOperators, Parameters
 using BifurcationKit, Plots, SparseArrays
 import LinearAlgebra: I, norm
@@ -39,9 +39,10 @@ function Laplacian2D(Nx, Ny, lx, ly)
 	return A, D2x
 end
 ```
+
 We also write the functional and its Jacobian which is a Sparse Matrix
 
-```julia
+```@example sh2dFD
 function F_sh(u, p)
 	@unpack l, ν, L1 = p
 	return -L1 * u .+ (l .* u .+ ν .* u.^2 .- u.^3)
@@ -59,7 +60,7 @@ d3F_sh(u, p, dx1, dx2, dx3) = (-6 .* dx2 .* dx3) .* dx1
 
 We first look for hexagonal patterns. This is done with
 
-```julia
+```@example sh2dFD
 X = -lx .+ 2lx/(Nx) * collect(0:Nx-1)
 Y = -ly .+ 2ly/(Ny) * collect(0:Ny-1)
 
@@ -114,7 +115,7 @@ which produces the results
 
 with `sol_hexa` being
 
-```@example sh2d_fd
+```@example sh2dFD
 println("--> norm(sol) = ",norminf(sol_hexa.u))
 heatmapsol(sol_hexa.u)
 ```

docs/src/tutorials/tutorials3.md

@@ -8,7 +8,7 @@ Depth = 3
 !!! unknown "References"
     This example is taken from **Numerical Bifurcation Analysis of Periodic Solutions of Partial Differential Equations,** Lust, 1997.
 
-We look at the Brusselator in 1d. The equations are as follows
+We look at the Brusselator in 1d (see [^Lust]). The equations are as follows
 
 $$\begin{aligned} \frac { \partial X } { \partial t } & = \frac { D _ { 1 } } { l ^ { 2 } } \frac { \partial ^ { 2 } X } { \partial z ^ { 2 } } + X ^ { 2 } Y - ( β + 1 ) X + α \\ \frac { \partial Y } { \partial t } & = \frac { D _ { 2 } } { l ^ { 2 } } \frac { \partial ^ { 2 } Y } { \partial z ^ { 2 } } + β X - X ^ { 2 } Y \end{aligned}$$
 
@@ -366,7 +366,7 @@ br_po = continuation(
 	br, 1,
 	# arguments for continuation
 	opts_po_cont,
-	PoincareShootingProblem(Mt, prob_ode, Rodas4P(); jacobian = BK.FiniteDifferencesMF());
+	PoincareShootingProblem(Mt, prob_ode, Rodas4P(); jacobian = BK.FiniteDifferencesMF(), update_section_every_step = 0);
 	# the next option is not necessary
 	# it speeds up the newton iterations
 	# by combining the linear solves of the bordered linear system
@@ -379,3 +379,8 @@ br_po = continuation(
 and you should see:
 
 ![](brus-psh-cont.png)
+
+
+## References
+
+[^Lust]:> **Numerical Bifurcation Analysis of Periodic Solutions of Partial Differential Equations,** Lust, 1997.

docs/src/tutorials/tutorials3b.md

@@ -561,7 +561,7 @@ Bifurcation points:
 - #  4,    ns at p ≈ 1.87667870 ∈ (1.86813520, 1.87667870), |δp|=9e-03, [converged], δ = ( 2,  2), step =  32, eigenelements in eig[ 33], ind_ev =   6
 ```
 
-# References
+## References
 
 [^Lust]:> **Numerical Bifurcation Analysis of Periodic Solutions of Partial Differential Equations,** Lust, 1997.