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.

  • $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.