Quantitative Finance for Physicists

that this periodicity is not described with a single frequency, and the

maximums of its peaks vary. Finally, at very high values of

r(r> 313), the system acquires a single stable limit cycle. This fascin-

ating manifold of solutions is not an exclusive feature of the Lorenz

model. Many nonlinear dissipative systems exhibit a wide spectrum of

solutions including chaotic regimes.

7.5 PATHWAYS TO CHAOS

A number of general pathways to chaos in nonlinear dissipative

systems have been described in the literature (see, e.g., [5] and refer-

ences therein). All transitions to chaos can be divided into two major

groups: local bifurcations and global bifurcations. Local bifurcations

occur in some parameter range, but the trajectories become chaotic

when the system control parameter reaches the critical value. Three

types of local bifurcations are discerned: period-doubling, quasi-peri-

odicity, and intermittency. Period-doubling starts with a limit cycle at

some value of the system control parameter. With further change of

40

20

0

−20

−40

02468101214t

Y(t)

Y(0) = 1.00

Y(0) = 1.01

Figure 7.9 Sensitivity of the Lorenz model to the initial conditions for p ¼

10, b ¼ 8/3 and r ¼ 28.

82 Nonlinear Dynamical Systems