Tutorials

• Tutorials
• • 1. ODE examples
  • • Computation of equilibria
    • Codimension 2 bifurcations of equilibria
    • Periodic orbits
    • Homoclinic orbits
  • 2. Multi-parameters continuation
  • 3. DAE examples
  • 4. DDE examples
  • 5. SciML examples based on ModelingToolkit
  • 6. SciML examples based on Catalyst
  • 7. PDEs: bifurcations of equilibria
  • 8. PDEs: automatic bifurcation diagram
  • 9. PDEs: bifurcations of periodic orbits
  • 10. PDEs based on FEM with Gridap.jl
  • 11. 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.

1. 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

2. Multi-parameters continuation

Based on the package MultiParamContinuation.jl and its docs. This functionality allows to continue solutions in two parameters.

• 🟢 sphere example
• 🟢 sphere example based in BifurcationKit

In the next tutorial, we continue solutions (equilibria, Hopf points, periodic orbits) as function of two free parameters:

• 🟢 ABC problem

3. DAE examples

• 🟡 Colpitts–type Oscillator

4. DDE examples

See the tutorials of DDEBifurcationKit.jl.

5. SciML examples based on ModelingToolkit

ModelingToolkit provides a tailored interface to BifurcationKit and the user is encouraged to have a look at its specific documentation.

6. SciML examples based on Catalyst

Catalyst provides a tailored interface to BifurcationKit and the user is encouraged to have a look at its specific documentation

7. 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

8. 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

9. 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

10. PDEs based on FEM with Gridap.jl

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

11. Symmetries, freezing, waves, fronts

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