EMBER (Emergent and Macroscopic Behaviour ExtRaction)

Stochastic continuation toolbox written in Java:

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

AUTO Tutorial for localised patterns

With

Help on installing AUTO under various platforms can be found

2D Localised Pattern Codes for the Swift-Hohenberg equation

These codes (

DJB Lloyd, B Sandstede, D Avitabile and AR Champneys, SIAM J. Appl. Dyn. Sys. 7(3) 1049-1100, 2008.__pdf__,

and may be downloaded from the SIAM J. Appl. Dyn. Sys. webpage. The list of programs in

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

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)

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:Help on installing AUTO under various platforms can be found

__here__.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.

__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