I have a problem. I am using two different numerical methods to try to “solve” the Lorentz attractor. Those are the Euler method (RK(s=1)) and the trapezoidal method with fixed-point iterations.
The problem is that providing both methods the same initial parameters and the same iteration parameters (step-size, time-interval) I get two different “solutions”:
Euler method
Trapezoid method:
I have revised my script several times and I cannot find any error. So I am starting to think that the solution will depend on the method. If so, please tell me, so I can stop looking for the mistake in the script.
You have stumbled across one of the key features of the Lorenz attractor: sensitive dependence on initial conditions (also known as the butterfly effect). The phenomenon you observe is a natural outcome of applying approximate solution methods to a system like the Lorenz attractor that exhibits sensitive dependence on initial conditions.
Sensitive dependence on initial conditions means that if you take two nearby points $p_1,q_1$ and regard them as initial conditions for two solutions at time $t_1$, and then you flow along those solutions curves to points $p_2,q_2$ at some time $t_2$, the points $p_2,q_2$ will almost certainly be farther apart than the initial points $p_1,q_1$ were.
Approximation methods such as two you are using do not provide exact solutions. Suppose you start from an initial condition $r_0$ at some time $t_0$, and then you flow along a solution curve to a point $r_1$ at time $t_1$. An approximation method (such as Euler's method, or the trapezoidal rule) will not produce the point $r_1$ exactly; it will instead produce only an approximation to $r_1$. If you use two different approximations, it is almost certion that your two approximations $p_1,q_1$ will not only be different from $r_1$ but will also be different from each other. If they are good approximations, at least $p_1,q_1$ should both be close to $r_1$ and therefore close to each other.
But now suppose you flow along the solutions curves with initial conditions $p_1,q_1$ to a later time $t_2$, arriving at $p_2,q_2$. As said above, the distance between $p_2,q_2$ is quite likely to be larger than the distance between $p_1,q_1$. On top of that, your approximations still are not giving you exact values of $p_2,q_2$, they are piling up an additional further error. And so, as you flow further and further, you begin to lose all control over exact positions.