20
Several mathematicians considered similar
problems (János Neumann among others) and
achieved partial results but the most comprehensive research was carried out by Edward Lorenz.
Lorenz wanted to forecast weather for a longer time period with the help of computer
modelling. He noticed that in the course of modelling on the basis of deterministic equation systems when the same weather data are used as
input but with six-digit accuracy in one case and
with three-digit accuracy in the other case the
results of the two cases were incomparably different. When entering the data he was thinking
rightfully that it does not matter whether the
accuracy of data is 1000th or 10,000th °C. The
two curves printed by the computer—as graphs
describing the behaviour of weather systems—
ran together initially and then the difference
between them was increasing. One of the conclusions of Lorenz was that if a physical system’s
behaviour is not periodic then—no matter what
the system is like otherwise—its movement cannot be predicted (in Gleick 1988).
Accurate weather forecast interested the
renowned János Neumann earlier and faced with
the difficulties (Neumann 1949). Among others
he recognised that there could be so-called instability points in a complex dynamic system like
the weather system that are especially sensitive
regarding the operation of the system as a whole,
i.e. at these places a slight physical effect could
significantly modify the operation of the system.
Neumann did not recognise, however, that instability could occur at any point of chaotic systems. After a lecture of Edward Lorenz in 1979
this unpredictable behaviour of chaotic systems
was called initially butterfly effect. The term
comes from the visualization of Lorenz: in systems like, for example, the weather in theory it is
possible that the flaps of a butterfly’s wings in
Brazil could trigger a process that in the end
causes a tornado in Texas. Today in science the
term sensitivity to initial conditions is used
instead of the butterfly effect.
Lorenz realised that the behaviour of chaotic
systems is always aperiodic. At first glance the
aperiodic behaviour of a weather system could be
doubted as in moderate climate winter, spring,
summer and autumn follow each other in every
year (three seasons are repeated in monsoon
areas, etc.) showing periodicity. Lorenz was
thinking, however, of periodicity with mathematical accuracy, i.e. the accurate repetition of
weather events and that is not met by weather due
to its variability. The same weather should appear
in every hour of every day in a year as in the same
hour of the same day in the previous year with the
same amount of precipitation and the same temperature. This is far from the weather we experience. Aperiodic behaviour also means that
changes in the system in the long term cannot be
predicted, maybe roughly estimated at best.
Aperiodic systems are very frequent in nature:
apart from weather the “related” climatic system,
ecological systems and its subsystems (e.g. various animal populations) operate in the same way.
Chaos related research of Lorenz turned better
known in the scientific public when James Yorke,
the renowned mathematician read the paper published in 1963 and gave to the also famous mathematician, Steve Smale. In the end Yorke named
Chaos theory and he developed the field significantly (Li and Yorke 1975). The biologist, Robert
May and his co-author revealed further conditions of chaotic systems. They studied the reproduction of populations using mathematical
methods (May and Oster 1976) proving that if the
reproduction rate (r) exceeds a critical value
(found to be r = 3) the curve of the graph constructed from the data is broken into two, bifurcation occurs. (Discussed in detail in Sect. 2.5.)
Bifurcations initially cause periods of 2, 4, 8, 16
before chaos without regular periods is started.
The paper of May discussing chaos in population
biology was published in Nature, one of the most
distinguished scientific journals (May 1976).
Scientists started to understand that in nature
non-linear changes are fundamental, irregularities are very frequent. There is still, however,
some kind of regularity behind chaos as it will be
seen in Sect. 2.4.
Complex adaptive systems can be interpreted
as a special type of chaotic systems. Such systems are composed of system elements that are
2 Structure and Operation of Systems, Models of the Global Earth System
Several mathematicians considered similar
problems (János Neumann among others) and
achieved partial results but the most comprehensive research was carried out by Edward Lorenz.
Lorenz wanted to forecast weather for a longer time period with the help of computer
modelling. He noticed that in the course of modelling on the basis of deterministic equation systems when the same weather data are used as
input but with six-digit accuracy in one case and
with three-digit accuracy in the other case the
results of the two cases were incomparably different. When entering the data he was thinking
rightfully that it does not matter whether the
accuracy of data is 1000th or 10,000th °C. The
two curves printed by the computer—as graphs
describing the behaviour of weather systems—
ran together initially and then the difference
between them was increasing. One of the conclusions of Lorenz was that if a physical system’s
behaviour is not periodic then—no matter what
the system is like otherwise—its movement cannot be predicted (in Gleick 1988).
Accurate weather forecast interested the
renowned János Neumann earlier and faced with
the difficulties (Neumann 1949). Among others
he recognised that there could be so-called instability points in a complex dynamic system like
the weather system that are especially sensitive
regarding the operation of the system as a whole,
i.e. at these places a slight physical effect could
significantly modify the operation of the system.
Neumann did not recognise, however, that instability could occur at any point of chaotic systems. After a lecture of Edward Lorenz in 1979
this unpredictable behaviour of chaotic systems
was called initially butterfly effect. The term
comes from the visualization of Lorenz: in systems like, for example, the weather in theory it is
possible that the flaps of a butterfly’s wings in
Brazil could trigger a process that in the end
causes a tornado in Texas. Today in science the
term sensitivity to initial conditions is used
instead of the butterfly effect.
Lorenz realised that the behaviour of chaotic
systems is always aperiodic. At first glance the
aperiodic behaviour of a weather system could be
doubted as in moderate climate winter, spring,
summer and autumn follow each other in every
year (three seasons are repeated in monsoon
areas, etc.) showing periodicity. Lorenz was
thinking, however, of periodicity with mathematical accuracy, i.e. the accurate repetition of
weather events and that is not met by weather due
to its variability. The same weather should appear
in every hour of every day in a year as in the same
hour of the same day in the previous year with the
same amount of precipitation and the same temperature. This is far from the weather we experience. Aperiodic behaviour also means that
changes in the system in the long term cannot be
predicted, maybe roughly estimated at best.
Aperiodic systems are very frequent in nature:
apart from weather the “related” climatic system,
ecological systems and its subsystems (e.g. various animal populations) operate in the same way.
Chaos related research of Lorenz turned better
known in the scientific public when James Yorke,
the renowned mathematician read the paper published in 1963 and gave to the also famous mathematician, Steve Smale. In the end Yorke named
Chaos theory and he developed the field significantly (Li and Yorke 1975). The biologist, Robert
May and his co-author revealed further conditions of chaotic systems. They studied the reproduction of populations using mathematical
methods (May and Oster 1976) proving that if the
reproduction rate (r) exceeds a critical value
(found to be r = 3) the curve of the graph constructed from the data is broken into two, bifurcation occurs. (Discussed in detail in Sect. 2.5.)
Bifurcations initially cause periods of 2, 4, 8, 16
before chaos without regular periods is started.
The paper of May discussing chaos in population
biology was published in Nature, one of the most
distinguished scientific journals (May 1976).
Scientists started to understand that in nature
non-linear changes are fundamental, irregularities are very frequent. There is still, however,
some kind of regularity behind chaos as it will be
seen in Sect. 2.4.
Complex adaptive systems can be interpreted
as a special type of chaotic systems. Such systems are composed of system elements that are
2 Structure and Operation of Systems, Models of the Global Earth System