Question

I’m using ode45() in matlab for some work in dynamics, using this procedure to calculate the largest Lyaponov exponent of the Lorenz system.

The process involves solving a system of differential equations starting from x0, and comparing this with a trajectory starting very close to x0.

At each time step, the second trajectory needs to be re-adjusted before advancing the time step, so I would like to be able to call ode45() just once – is this possible?

A starting attempt for an alternative solution is presented here, but it doesn’t work; as far as I can see, the matrices r and R obtained below should be similar:

% Time 
ti = 0; tf = 1; res = 10;
T = linspace(ti, tf, res);

% Solve the system first time
[~,R] = ode45('tbmLorenz',T,x0);

% Alternate trajectory
r = zeros(size(R));

% Temporary cordinates
temp = zeros(3,3);

% Solve for second trajectory
for i = 2:(res-1)
    % Time step
    ts = T((i-1):(i+1));

    % Solve three steps
    [~,temp] = ode45('tbmLorenz',ts,r(i-1,:));

    r_i = temp(2,:)

    % Code to alter r_i goes here

    % Save this
    r(i,:) = r_i;
end

… but they aren’t:

r =
    1.0000    3.0000    4.0000
    9.7011   20.6113    7.4741
   29.9265   16.4290   79.0449
   -5.7096  -15.2075   49.2946
  -12.4917  -13.6448   44.7003
  -13.6131  -13.8826   45.0346
  -13.5061  -13.1897   45.4782
  -13.0538  -13.0119   44.5473
  -13.4463  -13.8155   44.4783
         0         0         0
>> R
R =
    1.0000    3.0000    4.0000
    9.7011   20.6139    7.4701
   29.9663   16.5049   79.1628
   -5.7596  -15.2745   49.3982
  -12.4738  -13.5598   44.7800
  -13.5440  -13.8432   44.9084
  -13.5564  -13.3049   45.4568
  -13.1016  -12.9980   44.6882
  -13.3746  -13.7095   44.4364
  -13.7486  -13.6991   45.4092

That the last line of r is zero is not a problem.

Any ideas? Cheers!
\T

Answer 1

Ode45 is an adaptive algorithm with varying step size by definition. You can fix the step size if you want, but then you are not using ode45, you’re using something else.

If you want to solve this type of problem, I suggest you use a fixed time step. This link has code for what you need.

If you insist on using ode45, then you must adjust the simulation file itself with time-interpolated inputs.