Tutorials

• Tutorials
• • ODE examples
  • DAE examples
  • DDE examples
  • Examples based on ModelingToolkit
  • PDEs: bifurcations of equilibria
  • PDEs: automatic bifurcation diagram
  • PDEs: bifurcations of periodic orbits
  • PDEs based on FEM with Gridap.jl
  • Symmetries, freezing, waves, fronts

The tutorials are rated by the following scale of difficulty

1. 🟢 basic knowledge of (numerical) bifurcation theory (following equilibria / periodic orbits)
2. 🟡 advanced knowledge of (numerical) bifurcation theory (codim 2 bifurcations of equilibria)
3. 🟠 high level of knowledge of (numerical) bifurcation theory (codim 2 bifurcations of periodic orbits, tweaking the methods)
4. 🟤 very advanced tutorial, research level

There are three levels of automatization of the computation in these tutorials:

1. fully automatic bifurcation diagram (aBD) computation (only for equilibria): one uses 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

These examples are specific to ODEs.

Computation of equilibria

• 🟢 Neural mass equation
• 🟢 Temperature model
• 🟢 pp2 example from AUTO07p (aBD + Hopf aBS)

Codimension 2 bifurcations of equilibria

• 🟡 CO oxidation (codim 2)
• 🟡 Extended Lorenz-84 model (codim 2 + BT/ZH aBS)

Periodic orbits

We provide some examples focused on the computation of periodic orbits. Here is one where we present the different ways to compute periodic orbits.

• 🟡 Neural mass equation (Hopf aBS)

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

DDE examples

See the tutorials of DDEBifurcationKit.jl.

Examples based on ModelingToolkit

• 🟢 Neural mass equation - MTK

PDEs: bifurcations of equilibria

• 🟡 Temperature model (codim 2)
• 🟡 Temperature model with ApproxFun, no AbstractArray
• 🟡 2d Swift-Hohenberg equation: snaking, Finite Differences
• 🟤 2d generalized Bratu–Gelfand problem
• 🟠 2d Swift-Hohenberg equation (non-local) on the GPU
• 🟠 3d Swift-Hohenberg equation, Finite differences

PDEs: 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

PDEs: bifurcations of periodic orbits

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

PDEs based on FEM with Gridap.jl

• 🟡 2d Bratu–Gelfand problem with Gridap.jl

Symmetries, freezing, waves, fronts

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