Quantitative Finance for Physicists

7.7 REFERENCES FOR FURTHER READING

Two popular books, the journalistic report by Gleick [8] and the

‘‘first-hand’’ account by Ruelle [9], offer insight into the science of

chaos and the people behind it. The textbook by Hilborn [5] provides

a comprehensive description of the subject. The interrelations be-

tween the chaos theory and fractals are discussed in detail in [10].

7.8 EXERCISES

1. Consider the quadratic map X

k

¼ X

k1

2

þ C, where C > 0.

(a) Prove that C ¼ 0:25 is a bifurcation point.

(b) Find fixed points for C ¼ 0:125. Define what point is an

attractor and what is its attraction basin for X > 0.

2. 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: X

k

¼ X

k1

þ tF(X

k1

,Y

k1

,Z

k1

)

where the time step t can be assigned 0.01.

86 Nonlinear Dynamical Systems