subharmonics and amplitude jumps. To deal competently with resonances in
engineering structures, you do not need to know about the philosophical discussions that were triggered by their discovery several hundred years ago (which was
about the possible presence of heavenly harmonic phenomena). However, the
discovery of chaotic phenomena has changed the way we look at nature and on
mathematical models and computer output, as did the discovery of relativistic and
quantum phenomena early in the twentieth century. So, before we embark on the
more practical aspects of chaos, we briefly touch upon its general role in science.
In the beginning, there was Newton… Why do we set up mathematical models?
Usually we do so to understand, explain, predict or control particular aspects of
physical environment. Sometimes we are completely satisfied with a description on
the average, in which case we rely on stochastic models. For example, temperature
is a quite useful measure for describing the average kinetic energy of a huge number
of molecules in a gas; Few will care about the individual molecules. At other times
an averaged description is inadequate for the purpose. Then deterministic models
are needed, devoid of stochastic components. For example, when constructing a
grandfather clock we rely on the deterministic pendulum equation.
With deterministic systems all future and past states are uniquely defined by
equations of motions and sets of initial conditions. That is, if you know the state of
a deterministic system at a given instance of time, then you are able to predict its
behavior at all times. This is at the very core of Newtonian mechanics, laid down in
the Principia some 300 years ago (Newton 1686). During sequels of successful
modeling and experiments the belief did almost take root in our minds that we
could predict the future of any (non-atomic) system, though this might require an
extremely detailed mathematical model. Complicated phenomena simply called for
more complicated models, and bigger computers to simulate them. Ironically, it
took only a most primitive computer to definitely demolish the belief.
…Then came a meteorologist, named Lorenz … For many mathematical models
the behavior is insensitive to the specific choice of initial conditions. For example,
simulating on a computer the nonlinear pendulum equation € h þ ðg=lÞ sin h ¼ 0 with
initial conditions hðt 0 Þ ¼ _
h þ ðt 0 Þ ¼ k, you will get virtually identical results
whether k = 1.00 or k = 1.01. Checking the predictions in the laboratory you may
even succeed by starting the pendulum with k = 0.9, and still conclude that you are
able to predict the states of the system.
However, in 1961 Edward Lorenz, research meteorologist at MIT simulating
weather-models on a computer, discovered that a simple third-order model displayed extreme sensitivity to initial conditions. Starting with initial conditions that
were arbitrarily close in phase space, Lorenz observed the computer simulations
quickly running out of step. To predict the future of the system its initial conditions
were to be specified with infinite accuracy. This is impossible, with computers and
in laboratory experiments, and so the true behavior of a simple and purely deterministic system was seemingly unpredictable.
324
6 Chaotic Vibrations
engineering structures, you do not need to know about the philosophical discussions that were triggered by their discovery several hundred years ago (which was
about the possible presence of heavenly harmonic phenomena). However, the
discovery of chaotic phenomena has changed the way we look at nature and on
mathematical models and computer output, as did the discovery of relativistic and
quantum phenomena early in the twentieth century. So, before we embark on the
more practical aspects of chaos, we briefly touch upon its general role in science.
In the beginning, there was Newton… Why do we set up mathematical models?
Usually we do so to understand, explain, predict or control particular aspects of
physical environment. Sometimes we are completely satisfied with a description on
the average, in which case we rely on stochastic models. For example, temperature
is a quite useful measure for describing the average kinetic energy of a huge number
of molecules in a gas; Few will care about the individual molecules. At other times
an averaged description is inadequate for the purpose. Then deterministic models
are needed, devoid of stochastic components. For example, when constructing a
grandfather clock we rely on the deterministic pendulum equation.
With deterministic systems all future and past states are uniquely defined by
equations of motions and sets of initial conditions. That is, if you know the state of
a deterministic system at a given instance of time, then you are able to predict its
behavior at all times. This is at the very core of Newtonian mechanics, laid down in
the Principia some 300 years ago (Newton 1686). During sequels of successful
modeling and experiments the belief did almost take root in our minds that we
could predict the future of any (non-atomic) system, though this might require an
extremely detailed mathematical model. Complicated phenomena simply called for
more complicated models, and bigger computers to simulate them. Ironically, it
took only a most primitive computer to definitely demolish the belief.
…Then came a meteorologist, named Lorenz … For many mathematical models
the behavior is insensitive to the specific choice of initial conditions. For example,
simulating on a computer the nonlinear pendulum equation € h þ ðg=lÞ sin h ¼ 0 with
initial conditions hðt 0 Þ ¼ _
h þ ðt 0 Þ ¼ k, you will get virtually identical results
whether k = 1.00 or k = 1.01. Checking the predictions in the laboratory you may
even succeed by starting the pendulum with k = 0.9, and still conclude that you are
able to predict the states of the system.
However, in 1961 Edward Lorenz, research meteorologist at MIT simulating
weather-models on a computer, discovered that a simple third-order model displayed extreme sensitivity to initial conditions. Starting with initial conditions that
were arbitrarily close in phase space, Lorenz observed the computer simulations
quickly running out of step. To predict the future of the system its initial conditions
were to be specified with infinite accuracy. This is impossible, with computers and
in laboratory experiments, and so the true behavior of a simple and purely deterministic system was seemingly unpredictable.
324
6 Chaotic Vibrations