Verify the equilibrium points of the Lorenz model (7.4.3).

*3. Calculate the Lyapunov exponent of the logistic map as a function of the parameter A.

*4. Implement the algorithm for simulating the Lorenz model.

(a) Reproduce the ‘‘butterfly’’ trajectories depicted in Figure 7.8.

(b) Verify existence of the periodicity window at r ¼ 150.

(c) Verify existence of the limit cycle at r ¼ 350.

Hint: Use a simple algorithm: Xk ¼ Xk
1 þ tF(Xk
1, Yk
1, Zk
1)

where the time step t can be assigned 0.01.

**1. Verify how a sum of Gaussians can reproduce a distribution with the power-law tails in the spirit of [18].
**2. Discuss the recent polemics on the power-law tails of stock prices [27–29].
**3. Discuss the scaling properties of financial time series reported in [30].