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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

    Result for Euler Method

  • Trapezoid method:

    Result for 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.

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  • $\begingroup$ You should always first compare the same method using different step sizes. Even then you will find fast divergence of the approximative paths in chaotic (=highly non-linear) systems. $\endgroup$ – Lutz Lehmann Aug 23 '16 at 16:38
  • $\begingroup$ @LutzL: chaotic (=highly non-linear) systems – chaotic and highly non-linear are not equivalent. Highly non-linear systems can exhibit a regular dynamics. Chaotic systems are always considerably non-linear, though – whether highly non-linear depends on how you exactly define that word. $\endgroup$ – Wrzlprmft Aug 24 '16 at 8:12
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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.

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