The discovery had a significant impact on the scientific community, calling for a
shift in the way we look at computer models and experiments. For some systems
(e.g., the weather system) even the most refined mathematical modeling and the
most powerful computers will not help in producing accurate long-term predictions.
On the other hand, the discovery also implied that complicated behavior does not
necessarily call for a complicated mathematical model.
Following the discovery of Lorenz, a huge number of systems displaying
extreme sensitivity to initial conditions and chaotic behavior have been observed.
They were always there, of course, but no one had looked at them with the right
eyes
1 . You may observe them in nature, in mathematical models, in laboratory
experiments, in human behavior, chemical reactions, fluid flow, history, economy,
biology, in a dripping tap, the beat of a chicken’s heart, the changing weather, a
boxing match, a buckled beam, the firing of nerve cells, waterfalls, mechanical
valves, etc. Opening your eyes it is hard to avoid the impression that non-chaotic
regular behavior is associated only with a minor subset of systems in the real world.
… And now everything is chaos? No, it is not. It takes some specific conditions to
produce chaos. Linear systems cannot behave chaotically. Neither can
less-than-three autonomous first-order nonlinear differential equations. On the other
hand, to put it a little sharply most real-world systems are chaotic in the sense that it
is impossible to predict their state in a very distant future. For many of these
systems we are able to predict their state within an accuracy and scale of time that
serve our needs; they are weakly chaotic. Other systems are strongly chaotic, that is,
we cannot predict their state with any reasonable accuracy within the time-scale of
interest. Let us have a look at a simple example.
6.2 First Example of a Chaotic System
Consider a harmonically forced Duffing equation with negative linear stiffness:
€ x þ b_ x À
1
2
x þ
1
2
x
3
¼ p cosðXtÞ:
ð6:1Þ
This equation represents a single-mode approximation to the magnetically
buckled beam in Fig. 6.1, the now classical Moon’s beam (e.g., Moon 1980). The
equation governs the deflection x(t) of the beam-tip in response to the harmonic
loading of amplitude p and frequency X, in the presence of linear damping with
coefficient b (0 < b ( 1). All variables and parameters of the equation are
nondimensional.
1
Poincaré seems to have been aware of what we now call chaos, while in the 1880s studying the
three-body problem. But he was probably too far ahead of his contemporaries for anyone to
appreciate the importance of this, and there were no computers to perform long-range integration
of nonlinear differential equations.
6.1 Introduction
325
shift in the way we look at computer models and experiments. For some systems
(e.g., the weather system) even the most refined mathematical modeling and the
most powerful computers will not help in producing accurate long-term predictions.
On the other hand, the discovery also implied that complicated behavior does not
necessarily call for a complicated mathematical model.
Following the discovery of Lorenz, a huge number of systems displaying
extreme sensitivity to initial conditions and chaotic behavior have been observed.
They were always there, of course, but no one had looked at them with the right
eyes
1 . You may observe them in nature, in mathematical models, in laboratory
experiments, in human behavior, chemical reactions, fluid flow, history, economy,
biology, in a dripping tap, the beat of a chicken’s heart, the changing weather, a
boxing match, a buckled beam, the firing of nerve cells, waterfalls, mechanical
valves, etc. Opening your eyes it is hard to avoid the impression that non-chaotic
regular behavior is associated only with a minor subset of systems in the real world.
… And now everything is chaos? No, it is not. It takes some specific conditions to
produce chaos. Linear systems cannot behave chaotically. Neither can
less-than-three autonomous first-order nonlinear differential equations. On the other
hand, to put it a little sharply most real-world systems are chaotic in the sense that it
is impossible to predict their state in a very distant future. For many of these
systems we are able to predict their state within an accuracy and scale of time that
serve our needs; they are weakly chaotic. Other systems are strongly chaotic, that is,
we cannot predict their state with any reasonable accuracy within the time-scale of
interest. Let us have a look at a simple example.
6.2 First Example of a Chaotic System
Consider a harmonically forced Duffing equation with negative linear stiffness:
€ x þ b_ x À
1
2
x þ
1
2
x
3
¼ p cosðXtÞ:
ð6:1Þ
This equation represents a single-mode approximation to the magnetically
buckled beam in Fig. 6.1, the now classical Moon’s beam (e.g., Moon 1980). The
equation governs the deflection x(t) of the beam-tip in response to the harmonic
loading of amplitude p and frequency X, in the presence of linear damping with
coefficient b (0 < b ( 1). All variables and parameters of the equation are
nondimensional.
1
Poincaré seems to have been aware of what we now call chaos, while in the 1880s studying the
three-body problem. But he was probably too far ahead of his contemporaries for anyone to
appreciate the importance of this, and there were no computers to perform long-range integration
of nonlinear differential equations.
6.1 Introduction
325