Failing parametrizations: julia and matlab code

Abstract: This code contains all implementations to produce the results for the following manuscript: A. K. Stoychev, U. J. Römer, "Failing Parametrizations: What can go Wrong when Approximating Spectral Submanifolds", Nonlinear Dynamics 111, 5963-6000 (2023), doi: 10.1007/s11071-022-08154-3

Refer to the file "Code_Failing_Parametrizations_final.zip" for all files and a readme. A preliminary version of the code that was prepared for the preprint with the same title is also available. TechnicalRemarks: Seven systems are treated in the manuscript sections 3.1 - 3.8. The simulations for sections 3.1 - 3.7 are implemented in Julia (available at https://julialang.org/), the simulation for section 3.8 uses a modification of the Matlab toolbox SSMTool-2.1 (available at https://github.com/haller-group/SSMTool-2.1).

To execute the Julia code, setup Julia and a suitable editor (e.g. VSCode, https://www.julia-vscode.org/); the implementation was performed in Julia version 1.7.2.

Before any of the .jl-files can be executed for the first time, the necessary packages must be downloaded.
For this, press "]" (without quotation marks) in the Julia REPL to enter the package manager, then execute the following commands:

add LinearAlgebra
add Combinatorics
add SparseArrays
add DifferentialEquations
add Optim
add JuMP
add Ipopt
add NLsolve
add Plots
add PyPlot
add PlotlyJS
add TaylorSeries
add LaTeXStrings

To execute the Matlab code, install Matlab (commercial license required); the implementation was performed in Matlab version R2021b.

At the beginning of every Matlab session (after you open Matlab), navigate to the subfolder "External\SSMTool-2.1-master" and run the script "install.m".

To run the original system by Jain and Haller (2021), navigate to the subfolder "External\SSMTool-2.1-master\examples\vonKarmanBeam" and run the Workbook "vonKarmanBeamWorkbook.mlx".
(Note that none of the other examples contained in SSMTool-2.1 (available at https://github.com/haller-group/SSMTool-2.1) is included in this reference of the code.)

To run the modified system as discussed in our manuscript (increased forcing amplitude, modified coefficients), navigate to the subfolder "External\SSMTool-2.1-master\examples\vonKarmanBeam_Modified" and run the Workbook "vonKarmanBeamWorkbook_Modified.mlx".
(Note that the runtime may be several hours due to the reference result calculated via continuation with coco.)
The modified coefficients are stored in the file \External\SSMTool-2.1-master\examples\vonKarmanBeam_Modified\Alternative_Parametrization\Coefficients\SSM_Coeffs_von_Karman_10EL_Modified.mat which is loaded by the simulation script "vonKarmanBeamWorkbook_Modified.mlx".