On Solving ODE equations and specifying number of samples

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I am trying to understand the following set of equations given here: https://matlabgeeks.com/tips-tutorials/modeling-with-odes-in-matlab-part-5b/

The equations are those of a chaotic Lorenz system. The tutorial is quite easy to understand but what I do not follow is how to set the number of data points to generate i.e., the length of the time series? Which parameter helps to decide to generate how many data points will be generated. Can somebody please help? I have looked into other resources as well but I could not understand. For instance, by trial and error I found that if I specify eps = 0.000001; T = [0 45] then the number of data points are about 7000. If I want the number of data points to 10,000 I don't know what the values of these parameters should be.

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matlabode

1 Answers

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As described in the article (and the previous parts 1 and 2 of the series), the sequence of sample points is generated dynamically so that each segment contributes about the same amount of truncation error towards the global error, weighted by the absolute and relative tolerances. Additionally, it uses interpolation inside the segment to produce 3 inner points so that a plot will appear curved also for large tolerances. That is, the internal segmentation is given by T(1:4:end), the other points are interpolated.

You can also prescribe your own sample times, the values there get likewise interpolated from the "dense output", the interpolations over the internally produced segmentation.

T = linspace(t0, tend, 7000);
Y = ode45('lorenz', T, Y0, options);

You could also extract the dense output via

sol = ode45('lorenz', [t0 tend], Y0, options);

and then use the provided interpolation to compute samples at arbitrary times

Y = deval(sol,T);

In Empirical error proof Runge-Kutta algorithm ... I also computed the error for the Lorenz system for a fixed-step RK method, which shows the same divergence of the solutions after a relatively short time.