Homework Help: Lorentz Chaos - The 'Butterfly Effect'

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1. Apr 28, 2015

unscientific

1. The problem statement, all variables and given/known data

Given the lorentz system for $\sigma=10, b = \frac{8}{3}, r = 28$, and $x(t)$ from the first lorentz system, show that we can solve for y(t) and z(t) for the modified lorentz system by finding $\dot E$.

2. Relevant equations

3. The attempt at a solution

I have found the 3 fixed points. They are at the origin $(0,0,0)$, and $C^{+} = \left( \sqrt{b(r-1)}, \sqrt{b(r-1)}, r-1 \right)$ and $C^{-} = \left(-\sqrt{b(r-1)}, -\sqrt{b(r-1)} , r-1 \right)$. For $r = 28$, all three points are unstable.

It turns out that the points $C^{+}, C^{-}$ are only stable for $1 < r < 25$.

For a dynamical system, the Lyapunov exponent $\lambda$ is related to the trajectory in phase space by
$$|\delta V(t) | = |\delta V_0| e^{\lambda t}$$
So does this mean that for $\lambda > 0$ these trajectories are replled from one unstable point to another unstable point? I think this is the 'butterfly effect' described somewhere.

Also, I have re-expressed the equations:
$$\dot e_x + \dot x = \sigma \left[ (e_y - e_x) + (y-x) \right]$$
$$\dot e_y + \dot y = rx - (e_y + y) - x(e_z + z)$$
$$\dot e_z + \dot z = x(e_y + y) - b(e_z + z)$$
$$\dot E = \frac{2}{\sigma} e_x \dot e_x + 2 e_y \dot e_y + 2 e_z \dot e_z$$

I'm not sure what the question wants..

Last edited: Apr 28, 2015
2. May 3, 2015

Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. May 3, 2015

unscientific

My problem with this question stems from the confusion of what they want. I tried re-reading this question but it still didn't make sense to me. Would appreciate some insight from others.

4. May 4, 2015

unscientific

any insight on last part?

5. May 7, 2015

unscientific

any help on the chaos bit?

6. May 10, 2015

unscientific

bumpp on chaos

7. May 14, 2015

unscientific

butterfly bumpping

8. May 16, 2015

unscientific

bump on lorentz chaos

9. May 17, 2015

Solved.