Tutorials

There are three levels of tutorials:

1. fully automatic bifurcation diagram (aBD) computation (only for equilibria): one uses the function bifurcationdiagram and let it compute the diagram fully automatically. Another possibility is to use deflated continuation.
2. semi-automatic bifurcation diagram computation: one uses automatic branch switching (aBS) to compute branches at specified bifurcation points
3. manual bifurcation diagram computation: one does not use automatic branch switching. This has only educational purposes or for complex problems where aBS fails.

ODE examples

We present examples in the case of ODEs. Although BifurcationKit.jl is not geared towards them, we provide some specific methods which allow to study the bifurcations of ODE in a relatively efficient way.

• Neural mass equation (Hopf aBS)
• CO-oxydation (codim 2)
• Extended Lorenz-84 model (codim 2 + BT/ZH aBS)
• pp2 example from AUTO07p (aBD + Hopf aBS)

Here are some examples more oriented towards the computation of periodic orbits. Here is one for aBS from period-doubling bifurcations of periodic orbits

• Period doubling in Lur'e problem (PD aBS)

In the next tutorial, we show how to refine a periodic orbit guess obtained from numerical simulation. We also show how to perform continuation of PD/NS points using Shooting or Collocation.

• Periodic predator-prey model

In the next tutorial, we showcase the detection of Chenciner bifurcations. This is a relatively advanced tutorial, so we don't give much explanations. The reader should get first familiar with the above simpler examples.

• Steinmetz-Larter model

In the next tutorial, we showcase aBS from Bautin/HH to curve of Fold/NS of periodic orbits.

• Lorenz-84 model, take 2.

Homoclinic orbits

Based on the package HclinicBifurcationKit.jl and its docs.

• Autonomous electronic circuit (aBS from BT)
• Nonlinear laser model

DAE examples

• Colpitts–type Oscillator (Intermediate)

DDE examples

See the tutorials of DDEBifurcationKit.jl.

Examples based on ModelingToolkit

• Neural mass equation - MTK

Bifurcation of Equilibria

• Temperature model (Simplest example)
• Temperature model with ApproxFun, no AbstractArray (intermediate)
• 2d Swift-Hohenberg equation: snaking, Finite Differences
• 2d generalized Bratu–Gelfand problem
• 2d Swift-Hohenberg equation (non-local) on the GPU, periodic BC (Advanced)
• 3d Swift-Hohenberg equation, Finite differences

Automatic bifurcation diagram

• 1d Swift-Hohenberg equation (Automatic)
• Deflated Continuation in the Carrier Problem
• 1d Kuramoto–Sivashinsky Equation
• Automatic diagram of 2d Bratu–Gelfand problem (Intermediate)

Solving PDEs using Finite elements with Gridap.jl

• 2d Bratu–Gelfand problem with Gridap.jl (Intermediate)

Bifurcation diagrams with periodic orbits

• 1d Brusselator (automatic)
• 1d Brusselator (advanced user)
• Brusselator 1d with periodic BC using FourierFlows.jl (experienced user)
• Period doubling in the Barrio-Varea-Aragon-Maini model
• 2d Ginzburg-Landau equation (finite differences, codim 2, Hopf aBS)
• 2d Ginzburg-Landau equation (shooting)
• 1d Langmuir–Blodgett transfer model (advanced)

Symmetries, freezing, waves, fronts

• Fronts in 1d autocatalytic model (Automatic)
• Modulated fronts in 1d autocatalytic model (Manual)
• 1d Ginzburg-Landau equation (TW)
• Detonation engine