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

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