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[1.1] What's New?
Fixed lots of broken and outdated links. A few sites seem to be gone,
and some new sites appeared.
To some extent this FAQ is now been superseded by the Dynamical Systems site
run by SIAM. See http://www.dynamicalsystems.org There you will find a
glossary that contains most of the answers in this FAQ plus new ones. There is
also a growing software list. You are encouraged to contribute to this list,
and can do so interactively.
[1] About Sci.nonlinear FAQ
[1.1] What's New?
[2] Basic Theory
[2.1] What is nonlinear?
[2.2] What is nonlinear science?
[2.3] What is a dynamical system?
[2.4] What is phase space?
[2.5] What is a degree of freedom?
[2.6] What is a map?
[2.7] How are maps related to flows (differential equations)?
[2.8] What is an attractor?
[2.9] What is chaos?
[2.10] What is sensitive dependence on initial conditions?
[2.11] What are Lyapunov exponents?
[2.12] What is a Strange Attractor?
[2.13] Can computers simulate chaos?
[2.14] What is generic?
[2.15] What is the minimum phase space dimension for chaos?
[3] Applications and Advanced Theory
[3.1] What are complex systems?
[3.2] What are fractals?
[3.3] What do fractals have to do with chaos?
[3.4] What are topological and fractal dimension?
[3.5] What is a Cantor set?
[3.6] What is quantum chaos?
[3.7] How do I know if my data are deterministic?
[3.8] What is the control of chaos?
[3.9] How can I build a chaotic circuit?
[3.10] What are simple experiments to demonstrate chaos?
[3.11] What is targeting?
[3.12] What is time series analysis?
[3.13] Is there chaos in the stock market?
[3.14] What are solitons?
[3.15] What is spatio-temporal chaos?
[3.16] What are cellular automata?
[3.17] What is a Bifurcation?
[3.18] What is a Hamiltonian Chaos?
[4] To Learn More
[4.1] What should I read to learn more?
[4.2] What technical journals have nonlinear science articles?
[4.3] What are net sites for nonlinear science materials?
[5] Computational Resources
[5.1] What are general computational resources?
[5.2] Where can I find specialized programs for nonlinear science?
[6] Acknowledgments
[2] Basic Theory
[2.1] What is nonlinear?
In geometry, linearity refers to Euclidean objects: lines, planes, (flat)
three-dimensional space, etc.--these objects appear the same no matter how we
examine them. A nonlinear object, a sphere for example, looks different on
different scales--when looked at closely enough it looks like a plane, and
from a far enough distance it looks like a point.
In algebra, we define linearity in terms of functions that have the property
f(x+y) = f(x)+f(y) and f(ax) = af(x). Nonlinear is defined as the negation of
linear. This means that the result f may be out of proportion to the input x
or y. The result may be more than linear, as when a diode begins to pass
current; or less than linear, as when finite resources limit Malthusian
population growth. Thus the fundamental simplifying tools of linear analysis
are no longer available: for example, for a linear system, if we have two
zeros, f(x) = 0 and f(y) = 0, then we automatically have a third zero f(x+y) =
0 (in fact there are infinitely many zeros as well, since linearity implies
that f(ax+by) = 0 for any a and b). This is called the principle of
superposition--it gives many solutions from a few. For nonlinear systems, each
solution must be fought for (generally) with unvarying ardor!
[2.2] What is nonlinear science?
Stanislaw Ulam reportedly said (something like) "Calling a science 'nonlinear'
is like calling zoology 'the study of non-human animals'. So why do we have a
name that appears to be merely a negative?
Firstly, linearity is rather special, and no model of a real system is truly
linear. Some things are profitably studied as linear approximations to the
real models--for example the fact that Hooke's law, the linear law of
elasticity (strain is proportional to stress) is approximately valid for a
pendulum of small amplitude implies that its period is approximately
independent of amplitude. However, as the amplitude gets large the period gets
longer, a fundamental effect of nonlinearity in the pendulum equations (see
http://monet.physik.unibas.ch/~elmer/pendulum/upend.htm and [3.10]).
(You might protest that quantum mechanics is the fundamental theory and that
it is linear! However this is at the expense of infinite dimensionality which
is just as bad or worse--and 'any' finite dimensional nonlinear model can be
turned into an infinite dimensional linear one--e.g. a map x' = f(x) is
equivalent to the linear integral equation often called the Perron-Frobenius
equation
p'(x) = integral [ p(y) \delta(x-f(y)) dy ])
Here p(x) is a density, which could be interpreted as the probability of
finding oneself at the point x, and the Dirac-delta function effectively moves
the points according to the map f to give the new density. So even a nonlinear
map is equivalent to a linear operator.)
Secondly, nonlinear systems have been shown to exhibit surprising and complex
effects that would never be anticipated by a scientist trained only in linear
techniques. Prominent examples of these include bifurcation, chaos, and
solitons. Nonlinearity has its most profound effects on dynamical systems (see
[2.3]).
Further, while we can enumerate the linear objects, nonlinear ones are
nondenumerable, and as of yet mostly unclassified. We currently have no
general techniques (and very few special ones) for telling whether a
particular nonlinear system will exhibit the complexity of chaos, or the
simplicity of order. Thus since we cannot yet subdivide nonlinear science into
proper subfields, it exists as a whole.
Nonlinear science has applications to a wide variety of fields, from
mathematics, physics, biology, and chemistry, to engineering, economics, and
medicine. This is one of its most exciting aspects--that it brings researchers
from many disciplines together with a common language.
[2.3] What is a dynamical system?
A dynamical system consists of an abstract phase space or state space, whose
coordinates describe the dynamical state at any instant; and a dynamical rule
which specifies the immediate future trend of all state variables, given only
the present values of those same state variables. Mathematically, a dynamical
system is described by an initial value problem.
Dynamical systems are "deterministic" if there is a unique consequent to every
state, and "stochastic" or "random" if there is more than one consequent
chosen from some probability distribution (the "perfect" coin toss has two
consequents with equal probability for each initial state). Most of nonlinear
science--and everything in this FAQ--deals with deterministic systems.
A dynamical system can have discrete or continuous time. The discrete case is
defined by a map, z_1 = f(z_0), that gives the state z_1 resulting from the
initial state z_0 at the next time value. The continuous case is defined by a
"flow", z(t) = \phi_t(z_0), which gives the state at time t, given that the
state was z_0 at time 0. A smooth flow can be differentiated w.r.t. time to
give a differential equation, dz/dt = F(z). In this case we call F(z) a
"vector field," it gives a vector pointing in the direction of the velocity at
every point in phase space.
[2.4] What is phase space?
Phase space is the collection of possible states of a dynamical system. A
phase space can be finite (e.g. for the ideal coin toss, we have two states
heads and tails), countably infinite (e.g. state variables are integers), or
uncountably infinite (e.g. state variables are real numbers). Implicit in the
notion is that a particular state in phase space specifies the system
completely; it is all we need to know about the system to have complete
knowledge of the immediate future. Thus the phase space of the planar pendulum
is two-dimensional, consisting of the position (angle) and velocity. According
to Newton, specification of these two variables uniquely determines the
subsequent motion of the pendulum.
Note that if we have a non-autonomous system, where the map or vector field
depends explicitly on time (e.g. a model for plant growth depending on solar
flux), then according to our definition of phase space, we must include time
as a phase space coordinate--since one must specify a specific time (e.g. 3PM
on Tuesday) to know the subsequent motion. Thus dz/dt = F(z,t) is a dynamical
system on the phase space consisting of (z,t), with the addition of the new
dynamics dt/dt = 1.
The path in phase space traced out by a solution of an initial value problem
is called an orbit or trajectory of the dynamical system. If the state
variables take real values in a continuum, the orbit of a continuous-time
system is a curve, while the orbit of a discrete-time system is a sequence of
points.
[2.5] What is a degree of freedom?
The notion of "degrees of freedom" as it is used for
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Hamiltonian systems means
one canonical conjugate pair, a configuration, q, and its conjugate momentum
p. Hamiltonian systems (sometimes mistakenly identified with the notion of
conservative systems) always have such pairs of variables, and so the phase
space is even dimensional.
In the study of dissipative systems the term "degree of freedom" is often used
differently, to mean a single coordinate dimension of the phase space. This
can lead to confusion, and it is advisable to check which meaning of the term
is intended in a particular context.
Those with a physics background generally prefer to stick with the Hamiltonian
definition of the term "degree of freedom." For a more general system the
proper term is "order" which is equal to the dimension of the phase space.
Note that a dynamical system with N d.o.f. Hamiltonian nominally moves in a
2N dimensional phase space. However, if H(q,p) is time independent, then
energy is conserved, and therefore the motion is really on a 2N-1 dimensional
energy surface, H(q,p) = E. Thus e.g. the planar, circular restricted 3 body
problem is 2 d.o.f., and motion is on the 3D energy surface of constant
"Jacobi constant." It can be reduced to a 2D area preserving map by Poincaré
section (see [2.6]).
If the Hamiltonian is time dependent, then we generally say it has an
additional 1/2 degree of freedom, since this adds one dimension to the phase
space. (i.e. 1 1/2 d.o.f. means three variables, q, p and t, and energy is no
longer conserved).
[2.6] What is a map?
A map is simply a function, f, on the phase space that gives the next state,
f(z) (the image), of the system given its current state, z. (Often you will
find the notation z' = f(z), where the prime means the next point, not the
derivative.)
Now a function must have a single value for each state, but there could be
several different states that give rise to the same image. Maps that allow
every state in the phase space to be accessed (onto) and which have precisely
one pre-image for each state (one-to-one) are invertible. If in addition the
map and its inverse are continuous (with respect to the phase space coordinate
z), then it is called a homeomorphism. A homeomorphism that has at least one
continuous derivative (w.r.t. z) and a continuously differentiable inverse is
a diffeomorphism.
Iteration of a map means repeatedly applying the map to the consequents of the
previous application. Thus we get a sequence
n
z = f(z ) = f(f(z )...) = f (z )
n n-1 n-2 0
This sequence is the orbit or trajectory of the dynamical system with initial
condition z_0.
[2.7] How are maps related to flows (differential equations)?
Every differential equation gives rise to a map, the time one map, defined by
advancing the flow one unit of time. This map may or may not be useful. If the
differential equation contains a term or terms periodic in time, then the time
T map (where T is the period) is very useful--it is an example of a Poincaré
section. The time T map in a system with periodic terms is also called a
stroboscopic map, since we are effectively looking at the location in phase
space with a stroboscope tuned to the period T. This map is useful because it
permits us to dispense with time as a phase space coordinate: the remaining
coordinates describe the state completely so long as we agree to consider the
same instant within every period.
In autonomous systems (no time-dependent terms in the equations), it may also
be possible to define a Poincaré section and again reduce the phase space
dimension by one. Here the Poincaré section is defined not by a fixed time
interval, but by successive times when an orbit crosses a fixed surface in
phase space. (Surface here means a manifold of dimension one less than the
phase space dimension).
However, not every flow has a global Poincaré section (e.g. any flow with an
equilibrium point), which would need to be transverse to every possible orbit.
Maps arising from stroboscopic sampling or Poincaré section of a flow are
necessarily invertible, because the flow has a unique solution through any
point in phase space--the solution is unique both forward and backward in
time. However, noninvertible maps can be relevant to differential equations:
Poincaré maps are sometimes very well approximated by noninvertible maps. For
example, the Henon map (x,y) -> (-y-a+x^2,bx) with small |b| is close to the
logistic map, x -> -a+x^2.
It is often (though not always) possible to go backwards, from an invertible
map to a differential equation having the map as its Poincaré map. This is
called a suspension of the map. One can also do this procedure approximately
for maps that are close to the identity, giving a flow that approximates the
map to some order. This is extremely useful in bifurcation theory.
Note that any numerical solution procedure for a differential initial value
problem which uses discrete time steps in the approximation is effectively a
map. This is not a trivial observation; it helps explain for example why a
continuous-time system which should not exhibit chaos may have numerical
solutions which do--see [2.15].
[2.8] What is an attractor?
Informally an attractor is simply a state into which a system settles (thus
dissipation is needed). Thus in the long term, a dissipative dynamical system
may settle into an attractor.
Interestingly enough, there is still some controversy in the mathematics
community as to an appropriate definition of this term. Most people adopt the
definition
Attractor: A set in the phase space that has a neighborhood in which every
point stays nearby and approaches the attractor as time goes to infinity.
Thus imagine a ball rolling inside of a bowl. If we start the ball at a point
in the bowl with a velocity too small to reach the edge of the bowl, then
eventually the ball will settle down to the bottom of the bowl with zero
velocity: thus this equilibrium point is an attractor. The neighborhood of
points that eventually approach the attractor is the basin of attraction for
the attractor. In our example the basin is the set of all configurations
corresponding to the ball in the bowl, and for each such point all small
enough velocities (it is a set in the four dimensional phase space [2.4]).
Attractors can be simple, as the previous example. Another example of an
attractor is a limit cycle, which is a periodic orbit that is attracting
(limit cycles can also be repelling). More surprisingly, attractors can be
chaotic (see [2.9]) and/or strange (see [2.12]).
The boundary of a basin of attraction is often a very interesting object
since it distinguishes between different types of motion. Typically a basin
boundary is a saddle orbit, or such an orbit and its stable manifold. A crisis
is the change in an attractor when its basin boundary is destroyed.
An alternative definition of attractor is sometimes used because there
are systems that have sets that attract most, but not all, initial conditions
in their neighborhood (such phenomena is sometimes called riddling of the
basin). Thus, Milnor defines an attractor as a set for which a positive
measure (probability, if you like) of initial conditions in a neighborhood are
asymptotic to the set.
[2.9] What is chaos?
It has been said that "Chaos is a name for any order that produces confusion
in our minds." (George Santayana, thanks to Fred Klingener for finding this).
However, the mathematical definition is, roughly speaking,
Chaos: effectively unpredictable long time behavior arising in a deterministic
dynamical system because of sensitivity to initial conditions.
It must be emphasized that a deterministic dynamical system is perfectly
predictable given perfect knowledge of the initial condition, and is in
practice always predictable in the short term. The key to long-term
unpredictability is a property known as sensitivity to (or sensitive
dependence on) initial conditions.
For a dynamical system to be chaotic it must have a 'large' set of initial
conditions which are highly unstable. No matter how precisely you measure the
initial condition in these systems, your prediction of its subsequent motion
goes radically wrong after a short time. Typically (see [2.14] for one
definition of 'typical'), the predictability horizon grows only
logarithmically with the precision of measurement (for positive Lyapunov
exponents, see [2.11]). Thus for each increase in precision by a factor of 10,
say, you may only be able to predict two more time units (measured in units of
the Lyapunov time, i.e. the inverse of the Lyapunov exponent).
More precisely: A map f is chaotic on a compact invariant set S if
(i) f is transitive on S (there is a point x whose orbit is dense in S), and
(ii) f exhibits sensitive dependence on S (see [2.10]).
To these two requirements #DevaneyDevaney adds the requirement that periodic
points are dense in S, but this doesn't seem to be really in the spirit of the
notion, and is probably better treated as a theorem (very difficult and very
important), and not part of the definition.
Usually we would like the set S to be a large set. It is too much to hope for
except in special examples that S be the entire phase space. If the dynamical
system is dissipative then we hope that S is an attractor (see [2.8]) with a
large basin. However, this need not be the case--we can have a chaotic saddle,
an orbit that has some unstable directions as well as stable directions.
As a consequence of long-term unpredictability, time series from chaotic
systems may appear irregular and disorderly. However, chaos is definitely not
(as the name might suggest) complete disorder; it is disorder in a
deterministic dynamical system, which is always predictable for short times.
The notion of chaos seems to conflict with that attributed to Laplace: given
precise knowledge of the initial conditions, it should be possible to predict
the future of the universe. However, Laplace's dictum is certainly true for
any deterministic system, recall [2.3]. The main consequence of chaotic motion
is that given imperfect knowledge, the predictability horizon in a
deterministic system is much shorter than one might expect, due to the
exponential growth of errors. The belief that small errors should have small
consequences was perhaps engendered by the success of Newton's mechanics
applied to planetary motions. Though these happen to be regular on human
historic time scales, they are chaotic on the 5 million year time scale (see
e.g. "Newton's Clock", by Ivars Peterson (1993 W.H. Freeman).
[2.10] What is sensitive dependence on initial conditions?
Consider a boulder precariously perched on the top of an ideal hill. The
slightest push will cause the boulder to roll down one side of the hill or the
other: the subsequent behavior depends sensitively on the direction of the
push--and the push can be arbitrarily small. Of course, it is of great
importance to you which direction the boulder will go if you are standing at
the bottom of the hill on one side or the other!
Sensitive dependence is the equivalent behavior for every initial condition--
every point in the phase space is effectively perched on the top of a hill.
More precisely a set S exhibits sensitive dependence if there is an r such
that for any epsilon > 0 and for each x in S, there is a y such that |x - y| <
epsilon, and |x_n - y_n| > r for some n > 0. Then there is a fixed distance r
(say 1), such that no matter how precisely one specifies an initial state
there are nearby states that eventually get a distance r away.
Note: sensitive dependence does not require exponential growth of
perturbations (positive Lyapunov exponent), but this is typical (see [2.14])
for chaotic systems. Note also that we most definitely do not require ALL
nearby initial points diverge--generically [2.14] this does not happen--some
nearby points may converge. (We may modify our hilltop analogy slightly and
say that every point in phase space acts like a high mountain pass.) Finally,
the words "initial conditions" are a bit misleading: a typical small
disturbance introduced at any time will grow similarly. Think of "initial" as
meaning "a time when a disturbance or error is introduced," not necessarily
time zero.
[2.11] What are Lyapunov exponents?
(Thanks to Ronnie Mainieri & Fred Klingener for contributing to this answer)
The hardest thing to get right about Lyapunov exponents is the spelling of
Lyapunov, which you will variously find as Liapunov, Lyapunof and even
Liapunoff. Of course Lyapunov is really spelled in the Cyrillic alphabet:
(Lambda)(backwards r)(pi)(Y)(H)(0)(B). Now that there is an ANSI standard of
transliteration for Cyrillic, we expect all references to converge on the
version Lyapunov.
Lyapunov was born in Russia in 6 June 1857. He was greatly influenced by
Chebyshev and was a student with Markov. He was also a passionate man:
Lyapunov shot himself the day his wife died. He died 3 Nov. 1918, three days
later. According to the request on a note he left, Lyapunov was buried with
his wife. [biographical data from a biography by A. T. Grigorian].
Lyapunov left us with more than just a simple note. He left a collection of
papers on the equilibrium shape of rotating liquids, on probability, and on
the stability of low-dimensional dynamical systems. It was from his
dissertation that the notion of Lyapunov exponent emerged. Lyapunov was
interested in showing how to discover if a solution to a dynamical system is
stable or not for all times. The usual method of studying stability, i.e.
linear stability, was not good enough, because if you waited long enough the
small errors due to linearization would pile up and make the approximation
invalid. Lyapunov developed concepts (now called Lyapunov Stability) to
overcome these difficulties.
Lyapunov exponents measure the rate at which nearby orbits converge or
diverge. There are as many Lyapunov exponents as there are dimensions in the
state space of the system, but the largest is usually the most important.
Roughly speaking the (maximal) Lyapunov exponent is the time constant, lambda,
in the expression for the distance between two nearby orbits, exp(lambda *
t). If lambda is negative, then the orbits converge in time, and the
dynamical system is insensitive to initial conditions. However, if lambda is
positive, then the distance between nearby orbits grows exponentially in time,
and the system exhibits sensitive dependence on initial conditions.
There are basically two ways to compute Lyapunov exponents. In one way one
chooses two nearby points, evolves them in time, measuring the growth rate of
the distance between them. This is useful when one has a time series, but has
the disadvantage that the growth rate is really not a local effect as the
points separate. A better way is to measure the growth rate of tangent vectors
to a given orbit.
More precisely, consider a map f in an m dimensional phase space, and its
derivative matrix Df(x). Let v be a tangent vector at the point x. Then we
define a function
1 n
L(x,v) = lim --- ln |( Df (x)v )|
n -> oo n
Now the Multiplicative Ergodic Theorem of Oseledec states that this limit
exists for almost all points x and all tangent vectors v. There are at most m
distinct values of L as we let v range over the tangent space. These are the
Lyapunov exponents at x.
For more information on computing the exponents see
Wolf, A., J. B. Swift, et al. (1985). "Determining Lyapunov Exponents from a
Time Series." Physica D 16: 285-317.
Eckmann, J.-P., S. O. Kamphorst, et al. (1986). "Liapunov exponents from
time series." Phys. Rev. A 34: 4971-4979.
[2.12] What is a Strange Attractor?
Before Chaos (BC?), the only known attractors (see [2.8]) were fixed
points, periodic orbits (limit cycles), and invariant tori (quasiperiodic
orbits). In fact the famous Poincaré-Bendixson theorem states that for a pair
of first order differential equations, only fixed points and limit cycles can
occur (there is no chaos in 2D flows).
In a famous paper in 1963, Ed Lorenz discovered that simple systems of
three differential equations can have complicated attractors. The Lorenz
attractor (with its butterfly wings reminding us of sensitive dependence (see
[2.10])) is the "icon" of chaos
http://kong.apmaths.uwo.ca/~bfraser/version1/lorenzintro.html. Lorenz showed
that his attractor was chaotic, since it exhibited sensitive dependence.
Moreover, his attractor is also "strange," which means that it is a fractal
(see [3.2]).
The term strange attractor was introduced by Ruelle and Takens in 1970
in their discussion of a scenario for the onset of turbulence in fluid flow.
They noted that when periodic motion goes unstable (with three or more modes),
the typical (see [2.14]) result will be a geometrically strange object.
Unfortunately, the term strange attractor is often used for any chaotic
attractor. However, the term should be reserved for attractors that are
"geometrically" strange, e.g. fractal. One can have chaotic attractors that
are not strange (a trivial example would be to take a system like the cat map,
which has the whole plane as a chaotic set, and add a third dimension which is
simply contracting onto the plane). There are also strange, nonchaotic
attractors (see Grebogi, C., et al. (1984). "Strange Attractors that are not
Chaotic." Physica D 13: 261-268).
[2.13] Can computers simulate chaos?
Strictly speaking, chaos cannot occur on computers because they deal with
finite sets of numbers. Thus the initial condition is always precisely known,
and computer experiments are perfectly predictable, in principle. In
particular because of the finite size, every trajectory computed will
eventually have to repeat (an thus be eventually periodic). On the other hand,
computers can effectively simulate chaotic behavior for quite long times (just
so long as the discreteness is not noticeable). In particular if one uses
floating point numbers in double precision to iterate a map on the unit
square, then there are about 10^28 different points in the phase space, and
one would expect the "typical" chaotic orbit to have a period of about 10^14
(this square root of the number of points estimate is given by Rannou for
random diffeomorphisms and does not really apply to floating point operations,
but nonetheless the period should be a big number). See, e.g.,
Earn, D. J. D. and S. Tremaine, "Exact Numerical Studies of Hamiltonian
Maps: Iterating without Roundoff Error," Physica D 56, 1-22 (1992).
Binder, P. M. and R. V. Jensen, "Simulating Chaotic Behavior with Finite
State Machines," Phys. Rev. 34A, 4460-3 (1986).
Rannou, F., "Numerical Study of Discrete Plane Area-Preserving Mappings,"
Astron. and Astrophys. 31, 289-301 (1974).
[2.14] What is generic?
(Thanks to Hawley Rising for contributing to this answer)
Generic in dynamical systems is intended to convey "usual" or, more properly,
"observable". Roughly speaking, a property is generic over a class if any
system in the class can be modified ever so slightly (perturbed), into one
with that property.
The formal definition is done in the language of topology: Consider the class
to be a space of systems, and suppose it has a topology (some notion of a
neighborhood, or an open set). A subset of this space is dense if its closure
(the subset plus the limits of all sequences in the subset) is the whole
space. It is open and dense if it is also an open set (union of
neighborhoods). A set is countable if it can be put into 1-1 correspondence
with the counting numbers. A countable intersection of open dense sets is the
intersection of a countable number of open dense sets. If all such
intersections in a space are also dense, then the space is called a Baire
space, which basically means it is big enough. If we have such a Baire space
of dynamical systems, and there is a property which is true on a countable
intersection of open dense sets, then that property is generic.
If all this sounds too complicated, think of it as a precise way of defining a
set which is near every system in the collection (dense), which isn't too big
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