Spectral solution of the two dimensional Rossler chaotic system

[patric44]

TL;DR Summary

The validity of spectral solution in chaotic systems.

Hello everyone,
This is a question related to a research article involving a chaotic system that required to be solved numerically. I was trying to employ one of the spectral methods to solve the two dimensional version of the so called Rossler chaotic system:
$$\frac{dx(t)}{dt}=-y(t),\\\frac{dy(t)}{dt}=x(t)+ay(t).$$
Unfortunately the solution is valid only of the small values of $t$, i.e., $t<5$. My question is whether the spectral methods are suitable for this problem in the first place, and if there is another polynomial based technique available.

 

[S.G. Janssens]

Is $a$ a parameter? If so, then the system you provide is planar, autonomous, and linear. You could easily solve it by hand. In what sense do you think it is chaotic?

If you insist on solving it numerically, my hint would be to look at the pair of eigenvalues and their relative magnitude, as a function of $a$. (From your graph it looks like you are in the range $|a| < 2$.)

 

[patric44]

Is $a$ a parameter? If so, then the system you provide is planar, autonomous, and linear. You could easily solve it by hand. In what sense do you think it is chaotic?

If you insist on solving it numerically, my hint would be to look at the pair of eigenvalues and their relative magnitude, as a function of $a$. (From your graph it looks like you are in the range $|a| < 2$.)

Thanks for your response. In fact, the mentioned system is the trivial case of the system under consideration. Although, the spectral methods fails to approximate its solution (we consider $a=0.1$ in the given graph). I'm wondering if the choice of spectral methods, tau method for example, is a bad idea. Also, I hope there is an alternative method based on polynomial expansions that may be used for more accurate solution---not only for small subintervals of the domain of definition.

 

[S.G. Janssens]

For $a = 0.1$ you have a complex conjugate pair of eigenvalues, so they have the same magnitude, and the system is not stiff. I would think that pretty much any reasonable numerical method would be able to approximate the solution on a large time interval.

Also, I assume you are aware that you could solve the system exactly and explicitly, for all parameter values $a$? If your only purpose of all of this is to find an approximation to the solution, I would recommend just writing down the explicit solution and plotting that.

 

[pasmith]

Depending on the value of $a$, the eigenvalues will have positive

Summary: The validity of spectral solution in chaotic systems.

Hello everyone,
This is a question related to a research article involving a chaotic system that required to be solved numerically. I was trying to employ one of the spectral methods to solve the two dimensional version of the so called Rossler chaotic system:
$$\frac{dx(t)}{dt}=-y(t),\\\frac{dy(t)}{dt}=x(t)+ay(t).$$
Unfortunately the solution is valid only of the small values of $t$, i.e., $t<5$. My question is whether the spectral methods are suitable for this problem in the first place, and if there is another polynomial based technique available.

In this case we know that for $|a| \leq 2$ the solutions both grow (or decay) in amplitude at a rate $e^{at/2}$, so on $[0,\infty)$ you could use basis functions $\phi_n(t) = e^{at/2}P_n(t/(1+t))$ where $P_n$ are orthogonal on $[0,1]$ with respect to the inner product $\int_0^1 w(s)f(s)g(s)\,ds$. This means that the $\phi_n$ are orthogonal with respect to $$\int_0^\infty \frac{e^{-at}}{(1 + t)^2} w\left(\frac{t}{1+t}\right) f(t) g(t)\,dt.$$