Lorentz Chaos - The 'Butterfly Effect'

[unscientific]

Homework Statement

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 $\dot E$.[/B]

Homework Equations

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..

 

[unscientific]

any insight on last part?

 

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any help on the chaos bit?

 

[unscientific]

bumpp on chaos

 

[unscientific]

butterfly bumpping

 

[unscientific]

bump on lorentz chaos

 

[unscientific]

Solved.