One basic idea for utilizing chaos explores the fact that many chaotic attractors
have imbedded within them an infinite number of unstable periodic orbits. Though
this can be proved mathematically, you may grasp it intuitively by reconsidering the
period-doubling route to chaos (cf. Fig. 6.9): At each period-doubling a stable
periodic orbit turns unstable in favor of a new stable periodic orbit, having twice the
period of oscillation. The unstable periodic orbit remains a possible type of motion,
though usually we do not observe it in numerical simulations or laboratory
experiments—just like pencils are rarely observed standing in their upright position
of unstable equilibrium. Right beyond the k’th period-doubling there are k unstable
periodic orbits with periods 1, 2, 4, …, 2
k−1 , along with the newly emerged stable
period-2
k orbit. Recall from Sect. 6.4.1 that the sequence of period-doublings
accumulates at a critical value of the bifurcation parameter. Beyond that point the
number of unstable periodic orbits has grown to become infinite, and chaos is the
stable type of motion.
Now, assume we observe a system in a chaotic state of motion. The system is
capable of displaying an unlimited number of different periodic motions, but we do
not observe them because they are all unstable. If any particular unstable orbit could
be stabilized by choice, then one would be able to produce and control a variety of
observable periodic motions. Physically, the task of stabilizing an unstable orbit
corresponds to making a pencil stand upright on the palm of your hand, that is, to
stabilize its unstable equilibrium. What would you do? A workable approach is to
move the hand quickly from side to side and back and forth. Mathematically this
trick corresponds to moving a specified fixed point of a Poincaré map. So, to
stabilize a particular orbit of a chaotically vibrating system one needs to: 1) obtain
an experimental Poincaré map, 2) locate a particular fixed point of this map, and 3)
move the fixed point systematically in time by applying small perturbations to a
control parameter of the system.
Curiously, without even requiring a mathematical model of the system to be
controlled, in a number of cases this technique has proved to be rather easy to
apply. For example, several of the unstable periodic orbits for a chaotic
magneto-elastic ribbon, clamped in a magnetic field, has been stabilized by using
very small adjustments of the magnetic field strength (Ditto et al. 1990; Ditto and
Pecora 1993).
6.10 Closing Comments
Summary Let us state the most troublesome feature of any nonlinear system: For a
nonlinear system you may have more than one outcome, and you may not be able to
predict which one it will be. Even if you can, you may not be able to predict details
of it, because it may turn out to be chaotic.
However, we have tools at our disposal for detecting if the response is really
chaotic and hence unpredictable, or just complicated. There are tools for
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6 Chaotic Vibrations
have imbedded within them an infinite number of unstable periodic orbits. Though
this can be proved mathematically, you may grasp it intuitively by reconsidering the
period-doubling route to chaos (cf. Fig. 6.9): At each period-doubling a stable
periodic orbit turns unstable in favor of a new stable periodic orbit, having twice the
period of oscillation. The unstable periodic orbit remains a possible type of motion,
though usually we do not observe it in numerical simulations or laboratory
experiments—just like pencils are rarely observed standing in their upright position
of unstable equilibrium. Right beyond the k’th period-doubling there are k unstable
periodic orbits with periods 1, 2, 4, …, 2
k−1 , along with the newly emerged stable
period-2
k orbit. Recall from Sect. 6.4.1 that the sequence of period-doublings
accumulates at a critical value of the bifurcation parameter. Beyond that point the
number of unstable periodic orbits has grown to become infinite, and chaos is the
stable type of motion.
Now, assume we observe a system in a chaotic state of motion. The system is
capable of displaying an unlimited number of different periodic motions, but we do
not observe them because they are all unstable. If any particular unstable orbit could
be stabilized by choice, then one would be able to produce and control a variety of
observable periodic motions. Physically, the task of stabilizing an unstable orbit
corresponds to making a pencil stand upright on the palm of your hand, that is, to
stabilize its unstable equilibrium. What would you do? A workable approach is to
move the hand quickly from side to side and back and forth. Mathematically this
trick corresponds to moving a specified fixed point of a Poincaré map. So, to
stabilize a particular orbit of a chaotically vibrating system one needs to: 1) obtain
an experimental Poincaré map, 2) locate a particular fixed point of this map, and 3)
move the fixed point systematically in time by applying small perturbations to a
control parameter of the system.
Curiously, without even requiring a mathematical model of the system to be
controlled, in a number of cases this technique has proved to be rather easy to
apply. For example, several of the unstable periodic orbits for a chaotic
magneto-elastic ribbon, clamped in a magnetic field, has been stabilized by using
very small adjustments of the magnetic field strength (Ditto et al. 1990; Ditto and
Pecora 1993).
6.10 Closing Comments
Summary Let us state the most troublesome feature of any nonlinear system: For a
nonlinear system you may have more than one outcome, and you may not be able to
predict which one it will be. Even if you can, you may not be able to predict details
of it, because it may turn out to be chaotic.
However, we have tools at our disposal for detecting if the response is really
chaotic and hence unpredictable, or just complicated. There are tools for
382
6 Chaotic Vibrations