EMBER (Emergent and Macroscopic Behaviour ExtRaction)
Stochastic continuation toolbox written in Java:

https://dsweb.siam.org/Software/ember-emergent-and-macroscopic-behaviour-extraction

Runner-up - DSWeb 2018 Software Contest with Dr. Spencer Thomas (NPL) and Prof. Anne Skeldon (Surrey)
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AUTO Tutorial for localised patterns
 

With Bjorn Sandstede, this tutorial is part of the workshop The stability of coherent structures and patterns, (11-12th June 2012). The course materials can be downloaded from:
  • course notes, pdf,
  • source codes and documentation, zip.
Help on installing AUTO under various platforms can be found here.
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2D Localised Pattern Codes for the Swift-Hohenberg equation

These codes (tgz) were created to produce all the figures in the paper:
Localized hexagon patterns in the planar Swift-Hohenberg equation,
DJB Lloyd, B Sandstede, D Avitabile and AR Champneys,  SIAM J. Appl. Dyn. Sys. 7(3) 1049-1100, 2008. pdf, hexagon movie, ladder movie
and may be downloaded from the SIAM J. Appl. Dyn. Sys. webpage. The list of programs in localised_pattern_codes.tgz are given below:
Requirements: Matlab with optimization toolbox (tested on version 2007b) and AUTO07p.
(FSOLVE in the optimization toolbox is used to solve the BVPS. However, BVPS have been set-up
so that any globalised Newton solver will work.)

To untar files use: tar xvzf localised_pattern_codes.tgz

Note: All sub-directories have README files to allow immediate running of all codes.

Matlab codes:

Matlab codes: 1D BVP solvers:

/1D_SH/solve_SH1D.m
- solves 1D quadratic/cubic Swift-Hohenberg equation BVP on Half line. Finds a localised pulse and computes its stability with respect to perturbations on the full line. Uses Fourier differentiation matrices.

/1D_SH/solve_SH1Dfinite.m
- solves 1D quadratic/cubic Swift-Hohenberg equation BVP on Half line. Finds a localised pulse and computes its stability with respect to perturbations on the full line. Uses finite differences and sparse matrices to speed up computations.

Matlab codes: 2D BVP solvers:

/BVPS/SH2DBVPFOUR_hex_10.m
- Solves 2D quadratic/cubic Swift-Hohenberg equation BVP on the positive quadrant with Neumann BCS using Fourier differentiation matrices. Code finds a localised planar <10> hexagon pulse and plots the solution.

/BVPS/SH2DBVPFOUR_hex_11.m
- Solves 2D quadratic/cubic Swift-Hohenberg equation BVP on the positive quadrant with Neumann BCS using Fourier differentiation matrices. Code finds a localised planar <11> hexagon pulse and plots the solution.

/BVPS/SH2DBVPFOUR_hexagon.m
- Solves 2D quadratic/cubic Swift-Hohenberg equation BVP on the positive quadrant with Neumann BCS using Fourier differentiation matrices. Code finds a localised hexagon patch and plots the solution.

/BVPS/SH2DBVPFOUR_rhomboid.m
- Solves 2D quadratic/cubic Swift-Hohenberg equation BVP on the positive quadrant with Neumann BCS using Fourier differentiation matrices. Code finds a localised rhomboid patch and plots the solution.

/Hexagon_Maxwell/Continue_Maxwell.m
- Gets initial data and continues Hexagon Maxwell curve in two parameters of the quadratic/cubic Swift-Hohenberg equation. Calls compute_Maxwell.m and SH2DBVPFOUR.m

/radial_SH/solve_radial_SH.m
- solves quadratic/cubic radial Swift-Hohenberg equation BVP on [0,L] with Neumann bcs at r=L. Uses L'Hopitals rule for r=0 boundary conditions. Finds a localised pulse and computes its stability with respect to perturbations on the half line. Uses finite differences and sparse matrices to speed up computations.

Matlab codes: 2D IVP solvers

/IVPS/swifthohen2DETD_hex.m
- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code computes figure 1(a) of "Localised Hexagon patterns in the planar Swift-Hohenberg equation" by Lloyd, Sandstede, Avitabile and Champneys, SIADS 2008.

/IVPS/swifthohen2DETD_hexpatch.m
- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code finds a hexagon patch.

/IVPS/swifthohen2DETD_front10.m
- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code finds <10> hexagon pulse in the Swift-Hohenberg equation.

/IVPS/swifthohen2DETD_front11.m
- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code finds <11> hexagon pulse in the Swift-Hohenberg equation.

/IVPS/swifthohen2DETD_radial.m
- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code finds a localised ring in the Swift-Hohenberg equation.

/IVPS/swifthohen2DETD_randompatch.m
- solves quadratic/cubic Swift-Hohenberg equation IVP with periodic BCs on [-L,L]^2 computation is based on v = fft2(u) and first-order exponential time stepping of Cox and Matthews (2002). Code starts from a localised random patch.

AUTO codes:

Note: All codes are tested on AUTO07p. Initial data is supplied for immediate running. Conversion scripts and Matlab codes for data handling (procurement of initial data and post-processing of AUTO output), are supplied. README files in each tar file for instruction on immediate running and data handling.

/Fourier_cont.tgz
- Code computes pulses on a finite cylinder of the quadratic/cubic Swift-Hohenberg equation using a Fourier-cosine projection in the circumference direction. Code computes <10> and <11> hexagon pulses on the half line/cylinder.

/periodicSH.tgz
- Continues periodic solutions, Maxwell curves and localised pulses of the 1D Swift-Hohenberg equation with periodic boundary conditions.

/polarftSH.tgz
- Continues hexagon patches in the 2D quadratic/cubic Swift-Hohenberg equation using a Fourier-cosine projections in the angular direction as described in section 4.4 of "Localised Hexagon patterns in the planar Swift-Hohenberg equation" by Lloyd, Sandstede, Avitabile and Champneys, SIADS 2008.

/radialodeSHC.tgz
- Continues radial localised pulses in the radial quadratic/cubic Swift-Hohenberg equation.

/SH1Dstability.tgz
- Continues pulses in the 1D quadratic/cubic Swift-Hohenberg equation and the leading eigenfunction.

/radialodeSH2hexeig.zip
- Continues radial pulses and hexagon eigenfunction in the quadratic/cubic Swift-Hohenberg equation. Code traces out the hexagon pitchfork locus in the linear and quadratic bifurcation parameters.

/to_matlab_autox
- Converts AUTO output files b.foo and s.foo to matlab readable files. To use type $autox to_matlab.autox foo convertedfoo