Fig. 6.11(b) shows the boxed region in Fig. 6.11(a), corresponding to
quasiperiodic motion, enlarged 200 times. Narrow periodic and chaotic windows
are seen to intervene in the quasiperiodic regime.
6.4.3 The Transient Route
This route to chaos involves chaotic transients of gradually increasing length.
It is quite common with nonlinear systems to observe chaotic motion at the
initial parts of a time-series. During this time there is a positive largest Lyapunov
exponent, and long-term predictions seem impossible. Often, however, there is a
globally attracting equilibrium point or limit cycle to which the solution quickly
converges. The motion starts chaotic but ends up regular.
Sometimes such chaotic transients increase in length as a system parameter is
varied. They may last for so long that, by any reasonable means, the chaos observed
must be considered a stationary feature. This is the transient route to chaos. It does
not involve a sharp transition from regular to chaotic motion. Or rather, the transition value that appears in a bifurcation map depends on the time chosen for
transient cut-off. If the cut-off time is raised, then some solutions that were formerly
doomed chaotic now have time to settle down into regular motion.
It has been conjectured that the transient route is related to the collision of a
chaotic attractor with a coexisting unstable limit cycle (Grebogi et al. 1985).
Returning to Fig. 6.11(a), showing a bifurcation diagram for the non-shallow
arch, the transient route is observed upon entering the chaotic regime at X = X 4
from the right. For this figure the first 4,000 periods of the driving force were
considered as transients. Raising the cut-off limit for transients even further, the
chaotic transition zone at X 4 will move to the left.
It follows from the above discussion that transient cut-off times should be chosen
with care (see e.g. Table 1.4 in Sect. 1.7.5). Some use a general limit beyond which
chaotic motion is considered stationary, e.g., Moon (1987) suggests 5,000 periods
of a fundamental system frequency. However, when the system under study models
a particular system of the real world, it is recommended to choose the transient limit
relative to the total life or operating time of the system. For example, 5,000 fundamental periods may be a short time when studying dynamics of turbine
machinery, but a very long time for the study of ship motions due to sea waves.
6.4.4 The Intermittency Route
Chaotic intermittency reveals itself in a time-series plot of the motion. Here periods
of regular motion alternate with periods of chaos.
The route starts with regular periodic motion. Then, as a system parameter is
varied, short bursts of chaos pop up in between long periods of regular motion.
6.4 Universal Routes to Chaos
345
quasiperiodic motion, enlarged 200 times. Narrow periodic and chaotic windows
are seen to intervene in the quasiperiodic regime.
6.4.3 The Transient Route
This route to chaos involves chaotic transients of gradually increasing length.
It is quite common with nonlinear systems to observe chaotic motion at the
initial parts of a time-series. During this time there is a positive largest Lyapunov
exponent, and long-term predictions seem impossible. Often, however, there is a
globally attracting equilibrium point or limit cycle to which the solution quickly
converges. The motion starts chaotic but ends up regular.
Sometimes such chaotic transients increase in length as a system parameter is
varied. They may last for so long that, by any reasonable means, the chaos observed
must be considered a stationary feature. This is the transient route to chaos. It does
not involve a sharp transition from regular to chaotic motion. Or rather, the transition value that appears in a bifurcation map depends on the time chosen for
transient cut-off. If the cut-off time is raised, then some solutions that were formerly
doomed chaotic now have time to settle down into regular motion.
It has been conjectured that the transient route is related to the collision of a
chaotic attractor with a coexisting unstable limit cycle (Grebogi et al. 1985).
Returning to Fig. 6.11(a), showing a bifurcation diagram for the non-shallow
arch, the transient route is observed upon entering the chaotic regime at X = X 4
from the right. For this figure the first 4,000 periods of the driving force were
considered as transients. Raising the cut-off limit for transients even further, the
chaotic transition zone at X 4 will move to the left.
It follows from the above discussion that transient cut-off times should be chosen
with care (see e.g. Table 1.4 in Sect. 1.7.5). Some use a general limit beyond which
chaotic motion is considered stationary, e.g., Moon (1987) suggests 5,000 periods
of a fundamental system frequency. However, when the system under study models
a particular system of the real world, it is recommended to choose the transient limit
relative to the total life or operating time of the system. For example, 5,000 fundamental periods may be a short time when studying dynamics of turbine
machinery, but a very long time for the study of ship motions due to sea waves.
6.4.4 The Intermittency Route
Chaotic intermittency reveals itself in a time-series plot of the motion. Here periods
of regular motion alternate with periods of chaos.
The route starts with regular periodic motion. Then, as a system parameter is
varied, short bursts of chaos pop up in between long periods of regular motion.
6.4 Universal Routes to Chaos
345