| Topic: |
Science > Philosophy |
| User: |
"Frederick" |
| Date: |
14 Oct 2003 04:14:01 PM |
| Object: |
On Chaos |
MATHEMATICS: ON CHAOS
ScienceWeek http://www.scienceweek.com
The following points are made by J. Stark and K. Hardy (Science
2003 301:1192):
1) Even the most practically minded scientist, let alone a
mathematician, rarely expects to see a connection between their
research interests and their kitchen. For someone who has worked
in nonlinear dynamics for many years, it thus came as quite a
shock to buy a new microwave oven and find that it had a "chaos
defrost" setting.
2) This setting is indeed based on "chaos theory", the popular
name for nonlinear dynamics. A simple chaotic system is used to
generate an irregular heating sequence, which can reduce the time
required to defrost food by up to 60%. The use of nonlinear
dynamics may not be essential in this particular application:
Probabilistic or control theory techniques might generate an
equally efficient sequence. Nonetheless, it shows that nonlinear
dynamics is becoming useful in practical applications.
3) It is surprising that this has taken so long, given that most
real systems are nonlinear. As a branch of mathematics, modern
nonlinear dynamics has been around for more than a century. In
the 1960s, a spurt of creative activity led to exciting
mathematical breakthroughs. In a pioneering paper, Lorenz (1963)
described a simple nonlinear model that exhibited unexpected
complex dynamical behavior. The implication was that some
phenomena previously dismissed as random fluctuations or
experimental artifacts might in fact have explicable causes that
are amenable to control. But few scientists in other subjects
took much notice.
4) Other disciplines began to pay attention to nonlinear dynamics
in the early 1980s, when two key papers made it possible to go
beyond model systems and analyze observational and experimental
data. Takens (1980) showed how the data generated by a system are
related to the underlying dynamics, and Grassberger and Procaccia
(1983) reported a method for characterizing the system's
complexity. Initially, these ideas were primarily used to
determine whether particular real systems were "chaotic". These
early results were scientifically important but had perhaps
limited relevance for practical applications. In recent years,
there has been a noticeable shift in emphasis toward solving
practical problems in engineering, medicine, and other areas.
5) Some applications are in areas that have historically played a
key role in the development of nonlinear dynamics, such as
celestial mechanics. For example, techniques used to design
spacecraft trajectories no longer rely on writing down the
relevant equations and solving them on a computer, but on a
sophisticated understanding of the global structure of all
possible solutions to these equations. The geometry of these
solutions can be used to guide a spacecraft to its destination
with minimum fuel, enabling missions that were previously deemed
impossible. Similarly, ideas from nonlinear dynamics are finding
their way into weather prediction. Again, one of the key ideas is
to look at many solutions at once.
Science http://www.sciencemag.org
--------------------------------
THEORETICAL PHYSICS: CHAOS AND NONLINEAR DYNAMICS
In general, a nonlinear dynamical system is a system described by
time-dependent differential equations such that the rates of
change of one or more dependent variables of the system depend in
a nonlinear fashion on the variables themselves. Certain
nonlinear dynamical systems, some of which are of great
scientific interest, exhibit "chaotic dynamics". In this context,
the term "chaos" refers to unpredictable behavior arising in a
system that obeys deterministic laws but exhibits
unpredictability. The essential idea is that in certain systems
small perturbations may produce a cascade of larger
perturbations, so that eventually the behavior of such systems
cannot be predicted from prior states no matter if the systems
appear simple and obey deterministic laws. Examples of chaotic
nonlinear dynamical systems are the weather and populations of
organisms, and instances of chaotic dynamics have now been
documented in most scientific disciplines.
Because the differential equations for many nonlinear systems are
often intractable (i.e., no explicit quantitative solutions are
possible), a focus of theoretical research on nonlinear systems
has been on analysis of the qualitative behavior of such systems,
in particular on analysis of the "phase space" and "trajectories"
in the phase spaces of such systems. The idea is essentially as
follows: If the state of a system depends upon N variables, the
instantaneous state of the system can be viewed as a point (phase
point) in an N-dimensional space (phase space; system
hyperspace), and as the state of the system changes, its phase
point can be viewed as describing a trajectory in its phase
space. Qualitative analysis of the possible families of solutions
of nonlinear differential equations can provide information about
such phase space trajectories, and there are certain real systems
for which qualitative analysis of the phase space trajectories of
the system has revealed significant properties of the system
otherwise difficult to delineate.
The following points are made by J.P. Gollub and M.C. Cross
(Nature 2000 404:710):
1) The techniques of nonlinear dynamics are well-developed, but
the impact of this field has been largely confined to phenomena
in which there are only a few important time-dependent
quantities. Unfortunately, this excludes a vast range of
important problems in which the behavior of one point in space
can be quite different (though statistically similar) to that at
another location. A particular example is convective behavior.
2) The traditional approach to studying nonlinear dynamical
behavior is to plot the dynamical variables of the system as a
multidimensional phase space graph indicating how the behavior
changes over time. For example, a simplified model of the Solar
System consisting of the Sun and 9 planets would require a phase
space with as many as 60 dimensions (3 position and 3 momentum
coordinates for each body). In the case of a convecting fluid, a
complete description of the flow pattern requires knowledge of
the velocity and temperature at a very large number of locations,
so the number of dimensions of the phase plot are enormous (from
thousands to millions, depending on the desired spatial
resolution). As a result, the methods of nonlinear dynamics are
cumbersome and progress has been slow, even though many
interesting examples of spatiotemporal chaos have been explored
both experimentally and numerically.
3) Recent research (D.A. Egolf et al: Nature 404:733 2000)
involving numerical studies of an accepted model of thermal
convection indicates that the origin of unpredictable motion in
chaotic thermal convective systems, at least in one particular
form of spatiotemporal chaos, lies in what occurs in small
regions of space and over short time-scales. These local changes
in the organization of the flow affect the surrounding regions in
such a way that the entire future evolution is affected. The
authors state: "This is something akin to Ed Lorenz's famous
remark [E.N. Lorenz: J. Atmos. Sci. 20:130 1963] that the
localized flapping of a butterfly's wings might change the
weather dramatically over the entire world a few weeks later."
Although such sensitivity to localized fluctuations has never
been confirmed as the source of the unpredictability of the
weather, it is apparently the origin of chaotic dynamics in
thermal convection.
4) The authors conclude: "The methods used by Egolf et al should
apply to many other forms of chaos in spatially extended systems
(physical, chemical, and biological) for which reliable model
equations are available, so that the key processes leading to the
complex dynamics can be identified. Applications to areas as
diverse as cardiology and atmospheric dynamics might be expected
eventually. Moreover, it is not unreasonable to imagine that
insight into the processes leading to unpredictability will also
lead to progress in modifying or controlling the dynamics of
these systems."
--------------------------------
ON CHAOTIC SYSTEMS
The following points are made by Andreas Albrecht (Nature 2001
412:687):
1) Chaotic behavior is well understood from a classical
perspective, and is typically discussed in the context of a
mathematical "phase space" in which there are dimensions for both
position (x) and momentum (p). A particle at a given instant can
be specified as a point in classical phase space, and the time
development of the particle describes a curve or trajectory in
phase space. In chaotic systems, particles that start out in
virtually identical states (i.e., at very close points in phase
space) rapidly evolve into completely different states (i.e.,
distant parts of phase space).
2) Because nothing is ever measured with absolute precision, one
can never realistically talk about "points" in phase space.
Instead, every point (x,p) in phase space is typically assigned a
probability P(x,p). For a well-specified particle, this
probability peaks sharply at a localized point in phase space.
For an ordinary classical object, such as a single billiard ball,
a phase-space probability distribution that starts out sharply
peaked will remain peaked over time; a small uncertainty in the
starting point results in a similar small degree of ignorance at
a later time.
3) Chaotic systems are dramatically different. A sharply peaked
initial distribution gets torn apart by the chaotic evolution, as
neighboring phase-space trajectories rapidly head off in
different directions. A small amount of ignorance at the
beginning rapidly translates into huge uncertainties later on, as
the distribution becomes highly delocalized.
--------------------------------
THERMODYNAMICS, CHAOS, AND COMPLEXITY
Ilya Prigogine (1917-2003), who received the Nobel Prize in
Chemistry in 1977 for his work in nonequilibrium thermodynamics,
was among the first theoreticians to deal with the applications
of the second law of thermodynamics to complex systems. The
second law of thermodynamics effectively holds that physical
systems tend to slide spontaneously and irreversibly toward a
state of disorder (an increase of entropy). There is no
explanation in classical thermodynamics, however, of how complex
systems can arise spontaneously from less ordered states and
maintain themselves in apparent defiance of the tendency toward
entropy. Prigogine has proposed that as long as systems receive
energy and matter from an external source, nonlinear systems
("dissipative structures") can pass through periods of
instability and then self-organization, resulting in more complex
systems whose characteristics cannot be predicted except as
statistical probabilities. The work of Prigogine has been
influential in a wide variety of fields, ranging from physical
chemistry to biology, and this work has been fundamental in the
new disciplines of chaos theory and complexity theory.
What is called "complexity theory" is a theory that proposes that
certain systems manifest behaviors that are completely
inexplicable by any conventional analysis of the constituent
parts of the system. These behaviors, commonly called "emergent
behaviors", apparently occur in many complex systems involving
living organisms. One example is the idea that human
consciousness is an emergent property of a complex network of
neurons in the brain. The major problem of complexity theory is
how to model such emergent behavior: how to devise mathematical
laws that allow emergent behavior to be explained and predicted.
This effort to establish a solid theoretical foundation for the
description of complex systems has attracted mathematicians,
physicists, biologists, economists, and social scientists.
In the research context, complexity and "chaotic behavior" are
not synonymous. If one focuses attention on the time evolution of
an emergent behavior, e.g., daily changes in temperature, that
behavior may well be completely deterministic yet
indistinguishable from a random process: the behavior is chaotic.
However, although chaos is often associated with complex systems,
not all complex systems manifest chaotic behavior. From the
standpoint of systems theory, it is the interactions of
components that create emergent patterns that are important, and
not any chaotic behavior these may patterns may display.
The following points are made by Massimo Pigliucci (Skeptic 2000
vol.8 No.3):
1) The author points out that in common non-scientific usage the
term "chaos" is a synonym for randomness, for completely non-
deterministic and irregular phenomena. In mathematical theory,
however, the term "chaos" refers to a deterministic (i.e., non-
random) phenomenon characterized by special properties that make
the predictability of outcomes very difficult: chaotic behavior
is such that although it does not occur randomly, it has the
appearance of a series of random occurrences.
2) Chaotic dynamics are usually (but not always) a property of
nonlinear systems (i.e., systems whose behavior can be described
by sets of nonlinear equations). However, the converse is not
true: not all nonlinear dynamics generate chaotic behavior.
Typically, a given system of equations can produce both non-
chaotic and chaotic outcomes, depending on the range of values
assumed by the parameters of the equations. In many systems, one
can increase the value of a key parameter and obtain a
progression of outcomes from a steady equilibrium state to
regular oscillations with two equilibria, to more complex cycles
with multiple equilibria, to finally producing the chaotic
condition.
3) Another phenomenon typically associated with chaos is the so-
called "butterfly effect": chaos is analogous to a situation in
which the flapping of a butterfly's wings in Brazil ends up
starting a cascade of events that results in a tornado in Texas.
The term for this is "sensitivity to initial conditions": a small
perturbation of a system can cause a series of effects that
eventually lead to macroscopic consequences later in the time
sequence. Had that perturbation been of a different nature, an
entirely different series of events would have occurred. a more
formal way to describe the butterfly effect is to state that the
predictability of the system decreases exponentially with time:
our predictions of where the system will be are relatively good
for the immediate future, but lose accuracy for slightly longer
intervals of time, and are soon completely useless.
4) In general, a chaotic system is one whose mathematical
function is characterized by at least one of the following: a)
The system has sensitive dependence on initial conditions on its
domain; and/or b) the system has a positive *Lyapunov exponent at
each point in its domain that is not eventually periodic. A
"Lyapunov exponent" is a convenient measure of how fast the
trajectories of the system diverge in *phase space: if the
exponent is negative, the system actually converges at an
equilibrium point; if the exponent is near zero, the system
behaves with periodic regularity; if the exponent is positive,
the system is either chaotic or (for very large positive
exponents) random.
5) Chaos theory is a component of a larger but more vague
theoretical framework called "complexity theory". Essentially,
complexity theory is an attempt to study systems that satisfy two
conditions: a) the system is made of many interacting parts; b)
the interactions result in emergent properties that are not
immediately reducible to a simple sum of the properties of the
individual components. In general, complexity theory uses
nonlinear dynamical modeling to account for the behavior of
orderly complex systems. The dynamics manifested by a given
system depend fundamentally on two parameters: the number of
parts (N) that compose the system, and the average number of
connections (K) among the parts within the system. So-called "NK"
systems then fall into 3 types, depending on the relationship
between N and K:
a) K very small compared to N: Number of connections very small
compared to the total number of parts: Each part behaves
essentially independently of other parts, and the properties of
the system are the properties of the individual parts. Such
systems tend to be static or reach simple dynamic equilibria, and
are sometimes called "sub-critical".
b) K increasing compared to N: The dynamics becomes more complex
and emergent properties appear: Local changes propagate to
distant parts of the system as a consequence of connectivity, but
this propagation usually does not cause global change, since the
ratio of K to N is still relatively small. Such systems are
called "edge of chaos" systems, or "critical systems".
c) K approaches N: Most components of the system are connected to
almost every other component: This creates the determinate but
unstable "supercritical" systems described by chaos theory.
In terms of Lyapunov exponents: a) subcritical NK systems have a
negative Lyapunov exponent; b) critical NK systems have a
Lyapunov exponent near zero; c) chaotic NK systems are
characterized by a positive Lyapunov exponent.
Most classical mathematics, physics, and biology deal with
subcritical systems; chaos theory and fractal geometry deal with
supercritical systems; complexity theory focuses on critical
systems and the transition between system types. Alleged examples
of critical systems (i.e., systems on the "edge of chaos")
include the evolution of natural populations, the developmental
biology of plants and animals, the stock market, the global
economy, and the dynamics of galaxy clusters.
Skeptic http://www.skeptic.com
ScienceWeek http://www.scienceweek.com
--
Best,
Frederick Martin McNeill
Poway, California, United States of America
mmcneill@fuzzysys.com
http://www.fuzzysys.com
http://members.cox.net/fmmcneill/
*************************
Phrase of the week :
"We are all in the gutter, but some of us are looking at the stars."
-Oscar Wilde
:-))))Snort!)
*************************
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