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
k1
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
k1
þ tF(X
k1
,Y
k1
,Z
k1
)
where the time step t can be assigned 0.01.
86 Nonlinear Dynamical Systems

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