For systems exhibiting signs of intermittency, one could examine the average
times of regular motion for a few values of l. The critical value of l c could then be
estimated by assuming (6.16) to be a valid scaling law for the system.
Similarly, when transient chaos is observed, an estimate for l c could be obtained
by examining the variation of transient lengths, assuming (6.16) to be valid. As for
the period-doubling criterion the observations could be obtained analytically,
numerically or experimentally.
Predictions Based on Observed Quasiperiodic Motion The quasiperiodic route
to chaos follows a path of subsequent Hopf bifurcations, as explained in Sect. 6.4.2.
When a third Hopf bifurcation is just to appear, the system is likely to turn chaotic.
If, for a given system, one can set up analytical conditions for the occurrence of
the first two bifurcations, then a predictive tool for quasiperiodic motion is at hand.
For some systems the distance (in parameter space) from quasiperiodic motion to
chaos is so short that a quasiperiodic criterion is effectively also a chaos criterion.
6.5.2 Searching for Homoclinic Tangles and Smale
Horseshoes
Of the many efforts offered by mathematicians aiming to understand chaos, those
relating to homoclinic tangling and horseshoe maps seem especially promising. It
seems that most chaotic processes, if not all, are somehow related to the presence of
horseshoe maps in the dynamics of the system. By the point of view to be discussed
in this section, the presence of homoclinic tangling provides a fundamental
mechanism behind the creation of horseshoe maps. Thus, we consider the following
path of events:
Homoclinic intersections ) Homoclinic tangling ) Horseshoe maps
) Extreme sensitivity to initial conditions ) Possibility of chaos
The first three terms will be defined below. For the moment we note that if the
hypothesis regarding the path to chaos holds true, then a predictive, sufficient
criterion may take as its starting point a search for homoclinic intersections.
In a section to follow we describe a particular method for locating homoclinic
intersections for a certain class of systems, the so-called Melnikov method. To
understand this method, and to grasp a probable mechanism behind chaos, we need
to explain the concepts involved and how these relate to each other. Rigorous
definitions and proofs can be found elsewhere (e.g., Guckenheimer and Holmes
1983).
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6 Chaotic Vibrations
times of regular motion for a few values of l. The critical value of l c could then be
estimated by assuming (6.16) to be a valid scaling law for the system.
Similarly, when transient chaos is observed, an estimate for l c could be obtained
by examining the variation of transient lengths, assuming (6.16) to be valid. As for
the period-doubling criterion the observations could be obtained analytically,
numerically or experimentally.
Predictions Based on Observed Quasiperiodic Motion The quasiperiodic route
to chaos follows a path of subsequent Hopf bifurcations, as explained in Sect. 6.4.2.
When a third Hopf bifurcation is just to appear, the system is likely to turn chaotic.
If, for a given system, one can set up analytical conditions for the occurrence of
the first two bifurcations, then a predictive tool for quasiperiodic motion is at hand.
For some systems the distance (in parameter space) from quasiperiodic motion to
chaos is so short that a quasiperiodic criterion is effectively also a chaos criterion.
6.5.2 Searching for Homoclinic Tangles and Smale
Horseshoes
Of the many efforts offered by mathematicians aiming to understand chaos, those
relating to homoclinic tangling and horseshoe maps seem especially promising. It
seems that most chaotic processes, if not all, are somehow related to the presence of
horseshoe maps in the dynamics of the system. By the point of view to be discussed
in this section, the presence of homoclinic tangling provides a fundamental
mechanism behind the creation of horseshoe maps. Thus, we consider the following
path of events:
Homoclinic intersections ) Homoclinic tangling ) Horseshoe maps
) Extreme sensitivity to initial conditions ) Possibility of chaos
The first three terms will be defined below. For the moment we note that if the
hypothesis regarding the path to chaos holds true, then a predictive, sufficient
criterion may take as its starting point a search for homoclinic intersections.
In a section to follow we describe a particular method for locating homoclinic
intersections for a certain class of systems, the so-called Melnikov method. To
understand this method, and to grasp a probable mechanism behind chaos, we need
to explain the concepts involved and how these relate to each other. Rigorous
definitions and proofs can be found elsewhere (e.g., Guckenheimer and Holmes
1983).
350
6 Chaotic Vibrations