quantifying and visualizing the chaotic response, so that it can be described in
useful terms. It is sometimes possible to predict conditions for chaos, so that one
can design around or perhaps utilize it. Time-horizons of predictability can be
estimated, wherein at least some prediction is meaningful. We know a basic
mechanism behind chaos: the repeated stretching and folding of phase space, in turn
creating extreme sensitivity to initial conditions. A number of systems have been
mentioned, mathematical and physical, for which chaos has been observed and at
least partially explored. And we are aware that chaos is not a ‘singular’ phenomenon, i.e. something produced artificially by computers simulating degenerate
systems with pathological parameters. Rather, chaos is such a common phenomenon within all areas of science, nature and human daily life, that one wonders
why computers were ever required to open our eyes to it.
Bringing your Knowledge to the Kitchen Though chaos may cause a plenitude of
problems one should not overlook the more joyful aspects. Hopefully you will find
excitement in observing chaotic phenomena wherever you go.
A reasonably equipped kitchen is a good place to start. For example, in a typical
refrigerator one may observe bottles or (worse) cooling elements that rattle periodically or sometimes chaotically in response to the purely deterministic 50 or
60 Hz AC-input. Also, depending on the volume flow, the kitchen tap may drip
periodically, chaotically or not at all.
To study flow-induced vibrations you can pour a little milk into a bowl, which
has a heavy glass lid, and cook it for a while in a microwave oven. Spectacular
Hopf bifurcations may arise when vapor begins escaping at the lid in a periodic
fashion. With full power, you may even witness bifurcations into chaos (then milk
will spill all over the place and you realize that curiosity has its price).
There are even chaotic toys and art on the commercial market, e.g., a device
termed ‘space ball’ which is basically a chaotic pendulum.
Life on the Edge of Chaos Our genes seem to appreciate chaos. Ears, eyes, brains,
etc. are highly responsive to changes in stimuli. Watching a pendulum that starts
swinging we lose alertness and become bored as soon as the motion settles down
into orderly and predictable oscillations. The chaotic ‘space ball’ sells because it
mixes up the orderly motion of an ordinary pendulum with a chaotic process. Order
is appreciated because it allows for predicting events of the future – an important
factor for evolutionary survival. Our nervous system is alert to disturbances of
predictable order, because these force us to revise predictions and possibly take
actions to prevent or achieve something.
Complete randomness is just as boring as complete order. For example, pure
tones and random noise are interesting only to the moment when we find out they
arise from the radio (and thus present no danger or chance of benefit). When this
radio plays a piece of music we are more alert, since music represents a more proper
mix of repeated and non-repeated stimuli. You may find that many of the interesting, exciting and delightful things in life are characterized by ‘structured disorder’, that is: By something in-between perfect order and pure randomness.
6.10 Closing Comments
383
useful terms. It is sometimes possible to predict conditions for chaos, so that one
can design around or perhaps utilize it. Time-horizons of predictability can be
estimated, wherein at least some prediction is meaningful. We know a basic
mechanism behind chaos: the repeated stretching and folding of phase space, in turn
creating extreme sensitivity to initial conditions. A number of systems have been
mentioned, mathematical and physical, for which chaos has been observed and at
least partially explored. And we are aware that chaos is not a ‘singular’ phenomenon, i.e. something produced artificially by computers simulating degenerate
systems with pathological parameters. Rather, chaos is such a common phenomenon within all areas of science, nature and human daily life, that one wonders
why computers were ever required to open our eyes to it.
Bringing your Knowledge to the Kitchen Though chaos may cause a plenitude of
problems one should not overlook the more joyful aspects. Hopefully you will find
excitement in observing chaotic phenomena wherever you go.
A reasonably equipped kitchen is a good place to start. For example, in a typical
refrigerator one may observe bottles or (worse) cooling elements that rattle periodically or sometimes chaotically in response to the purely deterministic 50 or
60 Hz AC-input. Also, depending on the volume flow, the kitchen tap may drip
periodically, chaotically or not at all.
To study flow-induced vibrations you can pour a little milk into a bowl, which
has a heavy glass lid, and cook it for a while in a microwave oven. Spectacular
Hopf bifurcations may arise when vapor begins escaping at the lid in a periodic
fashion. With full power, you may even witness bifurcations into chaos (then milk
will spill all over the place and you realize that curiosity has its price).
There are even chaotic toys and art on the commercial market, e.g., a device
termed ‘space ball’ which is basically a chaotic pendulum.
Life on the Edge of Chaos Our genes seem to appreciate chaos. Ears, eyes, brains,
etc. are highly responsive to changes in stimuli. Watching a pendulum that starts
swinging we lose alertness and become bored as soon as the motion settles down
into orderly and predictable oscillations. The chaotic ‘space ball’ sells because it
mixes up the orderly motion of an ordinary pendulum with a chaotic process. Order
is appreciated because it allows for predicting events of the future – an important
factor for evolutionary survival. Our nervous system is alert to disturbances of
predictable order, because these force us to revise predictions and possibly take
actions to prevent or achieve something.
Complete randomness is just as boring as complete order. For example, pure
tones and random noise are interesting only to the moment when we find out they
arise from the radio (and thus present no danger or chance of benefit). When this
radio plays a piece of music we are more alert, since music represents a more proper
mix of repeated and non-repeated stimuli. You may find that many of the interesting, exciting and delightful things in life are characterized by ‘structured disorder’, that is: By something in-between perfect order and pure randomness.
6.10 Closing Comments
383