Contents

Generate system files for the extended Lorenz-84 model

This Jupyter Notebook generates the system files for the extended Lorenz-84 model given by

\[\begin{split} \begin{cases} \begin{aligned} \dot{X} &=-Y^{2}-Z^{2}-\alpha X+\alpha F-\xi U^{2} \\ \dot{Y} &=X Y-\beta X Z-Y+G \\ \dot{Z} &=\beta X Y+X Z-Z \\ \dot{U} &=-\delta U+\xi U X+S \end{aligned} \end{cases} \end{split}\]

In this system, \(X\) models the intensity of a baroclinic wave, \(Y\) and \(Z\) the sin and cos coefficients of the wave respectively, the variable \(U\) is added to study the influence of external parameters such as temperature. We fix the parameters as follows \(\alpha=0.25, \beta=1, G=0.25, \delta=1.04\) and \(\xi=0.987\). The continuation parameters are \(F\) and \(S\).

These are used in the extended Lorenz-84 model demo.

Add MatCont path and load sym package if GNU Octave is used

matcontpath = '../';
addpath(matcontpath);
addpath([matcontpath, 'Utilities'])
if isOctave
  pkg load symbolic % for GNU Octave
end

Set the system name

system_name = 'extendedLorenz84';

Create coordinates and parameter names as strings

coordsnames = {'X', 'Y', 'Z', 'U'};
parnames = {'F', 'S'};

Create symbols for coordinates and parameters

The array par is the array of symbols in the same order as parnames. Due to the following two lines we may, for example, use either alpha or par(1). There should no changes be need of this code.

syms(parnames{:});       % create symbol for alpha and delta
par=cell2sym(parnames);  % now alpha1 is par(1) etc
syms(coordsnames{:});    % create symbol for alpha and delta
coords=cell2sym(coordsnames); % create 1 x n vector for coordinates

Define fixed parameters

alpha = 0.25;
beta = 1;
G = 0.25;
delta = 1.04;
xi = 0.987;

Define the system

dX_dt = -Y^2-Z^2-alpha*X+alpha*F-xi*U^2;
dY_dt = X*Y-beta*X*Z-Y+G;
dZ_dt = beta*X*Y+X*Z-Z;
dU_dt = -delta*U+xi*U*X+S;
system = [dX_dt; dY_dt; dZ_dt; dU_dt];

In general there are no modifications needed after this line.

Differentiate and generate code (directional derivatives)

Exporting it to <system_name>.m. This method uses directional derivatives. Then using polarization identities derivatives can be calculated in arbitrary direction.

suc = generate_directional_derivatives(...
  system,...   % n x 1 array of derivative symbolic expressions
  coords,... % 1 x n array of symbols for states
  par,...      % 1 x np array of symbols used for parameters
  system_name,... % argument specifying the system name
  [matcontpath, 'Systems/']... % directory to save to file
);

Higher-order parameter-dependent multi-linear form.

Exporting it to <system_name>_multilinearforms.m. These multi-linear forms are currently only used in the computation of the parameter-dependent center manifold for the codimension two Bogdanov-Takens bifurcation.

order = 3;
suc = generate_multilinear_forms(system_name, system, coords, par, order, ...
        [matcontpath, 'Systems/']);