Chaos Demonstrations Explanations
Driven Pendulum
In this driven pendulum the restoring force is proportional to the
sine of x rather than to x as in a simple harmonic oscillator:
x'' + Bx' + sin x = A sin wt
The resulting motion is complicated. With w=0.7, you see simple
periodic motion for A=0.4, a more complicated but still periodic motion
for A=0.5, and chaotic motion for A=0.6. The chaotic motion can be seen
in the real-space view 1 and in the views of sin(x) or v versus time,
but it is most easily seen in view 4 of angular velocity v (or x')
versus angular position x. Such a plot is called a phase-space plot.
Also shown is a Poincare section which shows the phase-space position at
the instant when the phase of the drive crosses zero. Periodic motion
produces distinct points the number of which is the period of the motion
divided by the drive period. Chaotic motion fills in a large portion of
the plane. The Poincare movie shows the Poincare section at successive
phases of the drive. The return map provides similar information. The
view 8 of potential energy shows a graph of 1-cos(x) versus x.
A small damping term (Bx') has been added to allow initial
conditions to damp away. The damping is caused by friction, which is
always present in real oscillators. Wait for a few dozen cycles after
each parameter change to allow the damping to occur.
Nonlinear Oscillator
In this nonlinear Ueda oscillator the restoring force is
proportional to the cube of the displacement, rather than to the first
power as in a linear oscillator that obeys Hooke's law:
3
x'' + Bx' + x = A sin wt
Such an equation might model a mass on a very hard spring or an
electrical RLC circuit with a highly nonlinear inductor. The resulting
motion is complicated. With A=2.5, you see chaotic motion for w=0.7,
periodic motion with period 1 for w=0.9, and periodic motion with period
3 for w=1.2. Vary the parameters A, B, and w and count the number of
points in the Poincare section to see how many different periodic
solutions you can find. The periodic and chaotic solutions are
intermingled in a complicated way.
A small damping term (Bx') has been added to allow the initial
conditions to damp away. Wait for a few dozen cycles after each
parameter change to allow the damping to occur.
The various views are analogous to those described in the Driven
Pendulum demonstration. The potential energy in this case is
proportional to the fourth power of x (a quartic potential well).
Duffing Oscillator
The inverted Duffing oscillator is governed by the equation:
3
x'' + Bx' - x + x = A sin wt
There are three equilibrium positions. The one at x=0 is unstable,
and the ones at x = ñ1 are stable. An example of such a system would be
a flexible rod rigidly clamped at the bottom and with a mass at the top.
The equilibrium with the rod vertical is unstable, and the rod flexes to
the right or left. When this system is driven sinusoidally, it exhibits
either periodic or chaotic motion depending on the amplitude A and
frequency w of the drive. The periodic and chaotic solutions are
intermingled in a complicated way.
A small damping term (Bx') has been added to allow the initial
conditions to damp away. Wait for a few dozen cycles after each
parameter change to allow the damping to occur.
The various views are analogous to those described in the Driven
Pendulum demonstration. The potential energy in this case is equal to
0.25 + 0.25 x^4 - 0.5 x^2.
Van der Pol Equation
The driven Van der Pol equation, given by
x'' - h(1-xý)x' + x = A sin wt
is solved here for 15 different combinations of initial velocities x'
and positions x. The results are plotted in phase space (velocity
versus position). Such an equation was first used by Van der Pol to
represent an electrical resonant (RLC) circuit in which there is both a
nonlinear resistance and an active component to overcome the resistance.
The equation has also been used to model heartbeats and pulsating stars
called Cepheids.
For h=0 and A=0, the equation is just that of an undamped harmonic
oscillator that has simple periodic orbits all of the same frequency.
For h>0, large oscillations are damped and small oscillations grow as a
result of the nonlinear damping term. All orbits are attracted to a
closed curve, which is called a limit cycle or attractor. For h<0 the
attractor becomes a repellor. With A>0, the solution may be either a
limit cycle or a two-torus (quasi-periodic) or a chaotic attractor. See
if you can find combinations of A, h, and w that give chaotic solutions.
Toggle tracking on and off and look for solutions that do not approach a
limit cycle.
Three-Body Problem
In this restricted three-body problem, N noninteracting planets
move in the inverse-square-law gravitational field of two stars. The
positions of these stars are fixed, but you can vary the ratio of their
masses (m1/m2). The stars are indicated by circles on the screen.
Note that the motion is generally chaotic, as can be shown by using
two or more planets with the same initial position but with different
initial velocities. Each time the motion is restarted, the initial
position changes. The velocities vary over a range of one percent and
cannot be changed. This sensitive dependence on initial conditions is a
general characteristic of chaotic motion.
To simplify the calculation, the stars' masses are spread uniformly
over their volumes, and the planets are allowed to penetrate the stars.
Outside the circles representing the stars, the attractive force varies
inversely with the square of the distance of the planet from the center
of the star.
Can you show Kepler's result that the orbit of a planet in an
inverse-square-law force field is an ellipse? You will have to set the
mass of one of the stars (m1) to zero and avoid penetrating the star.
Magnetic Quadrupole
A quadrupole magnetic field is produced by two infinite line-
currents flowing out of the screen at the top and bottom and two
infinite line-currents flowing into the screen at the right and left.
The resulting magnetic field vanishes on the magnetic axis at the center
of the screen. Such a field null is characterized by an X-point at
which two magnetic field lines cross. These lines form a separatrix.
A group of N, noninteracting, charged particles are started at the
same point (xi) with velocities that vary over a range of 1 percent.
The orbits of the particles are followed in time. Note that the
particles begin following the same trajectory, but after a time the
orbits separate and the chaotic motion becomes evident. Try at least
32 particles, with tracking turned off, and watch the filamentation of
their orbits.
Magnetic fields like this one are used to confine charged particles
in particle accelerators and plasma confinement devices for fusion
research. Note that though the particles tend to remain on a field
line, particles occasionally escape when they come too close to the X-
point and begin moving nearly parallel to the field.
Lorenz Attractor
The Lorenz attractor arises from a solution of three nonlinear
differential equations:
x' = ay - ax
y' = -xz + hx - y
z' = xy - bz
These equations were originally proposed by the meteorologist
Edward Lorenz in 1963 to model fluid convection and are regarded as the
first documented example of what is now called a strange attractor.
The displays show the projection of the orbit in each plane and the
variation of each variable with time. If your screen shows colors, the
colors represent different values of the third variable, giving some
sense of the three-dimensional nature of the trajectory, which never
crosses itself. In the three-dimensional view 4, the color is changed
every time x(t) changes sign.
Can you find parameters that give point attractors and limit
cycles?
Rossler Attractor
The Rossler attractor arises from a solution of three nonlinear
differential equations:
x' = - y - z
y' = x + ay
z' = b + z(x-h)
These equations were proposed by Otto Rossler in 1976 and are often
regarded as the simplest example of a chaotic flow. The equations also
have periodic solutions for various values of the parameters.
The displays show the projection of the orbit in each plane and the
variation of each variable with time. If your screen shows colors, they
represent different values of the third variable, giving some sense of
the three-dimensional nature of the trajectory, which never crosses
itself. In the three-dimensional view 4, the color is changed every
time x(t) changes sign.
Can you find parameters that give point attractors and limit
cycles?
One-Dimensional Maps
One-dimensional maps predict the next value in a series of numbers
from a function that depends only on the last value:
x(n+1) = F[x(n)]
For example, the Logistic map has F = Ax(n)[1-x(n)]. Depending on
the value of A, the solution may go to a fixed value, oscillate, or
become chaotic.
View 1 shows a graph of each of the maps available by pressing M.
In view 2, successive iterates are plotted beginning with xi, which you
may change. In view 3, there are two initial conditions that differ by
1 part in 10^8, and the difference between the two values of x is
plotted on a log scale. In view 4, many values of x are plotted versus
A after a number of iterations. In view 5, the Lyapunov exponent, given
by averaging log|dF/dx| over a large number of iterations, is plotted
versus A. A negative Lyapunov exponent means the solution is periodic,
and a positive exponent means it is chaotic. In view 6, the probability
distribution P(x) is plotted.
The logistic equation can be used to model population growth in a
biological system for a case in which the growth is governed by the
population at some previous time rather than by the present population.
Predator-Prey
The predator-prey problem is a simple ecological model described by
the following two nonlinear differential equations,
x' = ax - xy
y' = -y + xy
where x represents the number of prey (for example, rabbits), and y
represents the number of predators (foxes) at time t. The equations
might also naively model an oscillating chemical reaction. The growth
of each species depends on the populations at each instant. The problem
is discretized using the Heun method with a step size h. Such a
discretization corresponds to growth rates that depend on the
populations at a previous time rather than at the present. However,
small h corresponds to an exact solution of the above equations, and the
result is a simple periodic orbit.
The point (x=1, y=a) is an unstable fixed point. For a=1 and h<0.6
there is an invariant circle to which orbits are attracted. For a=1 and
h=0.6, a strange attractor is evident. For a=1 and h=0.7, there is a
periodic attractor of period 9. For a=1 and h>0.8, there are no bounded
orbits, and all orbits are attracted to infinity.
Chirikov Map
The Chirikov map (or standard map) is generated by starting with
various initial values p(0) and q(0) and iterating as follows with J =
1:
p(n+1) = Jp(n) - K sin q(n)
q(n+1) = q(n) + p(n+1)
This is a discretized form of the equation of motion of a pendulum in
the limit as K approaches zero. Then p represents the angular momentum,
and q the angle of the pendulum. The equation can also represent the
motion of a bouncing ball, on successive bounces, when the floor
oscillates sinusoidally.
For small K, the plot shows an interior region, in which the
pendulum swings back and forth, and an exterior region in which the
pendulum swings in a full circle. The intermediate region, in which the
pendulum nearly stalls at the top, is typically chaotic. For large K,
the motion becomes chaotic for all amplitudes, but note that even in the
extreme chaotic limit (K=6.4), there are holes in phase space that are
inaccessible to the motion. Thus even chaotic motion is subject to
rules. You can add dissipation by taking J < 1, in which case the
trajectories approach attractors.
Henon Map
The Henon map is generated by starting with random points in the
y-z plane and iterating according to the following:
y(n+1) = 1 - ay(n)ý + bz(n)
z(n+1) = y(n)
Note that after just a few iterations, the points collect in the
vicinity of a particular curve. This is an example of a strange
attractor, and all points within a given region in the y-z plane are
drawn to the attractor. Such a region is called a basin of attraction.
Points outside the basin of attraction escape to infinity. The basin of
attraction is shown in the background color in view 2. The other colors
indicate the number of iterations required for |y| to exceed an
arbitrary value of 135.
Close examination reveals that the attractor and its basin of
attraction are fractals. No matter how much a fractal is enlarged,
there is always detail at a smaller scale. The separation of the
individual lines within the attractor are proportional to b divided by
the square root of a, all raised to an integer power.
Strange Attractors
A system of equations that has chaotic solutions will often produce
an orbit that is attracted to a small region of the space whose
coordinates are the variables of the equations. Such a chaotic
attractor is called "strange" because it is neither point nor line nor
surface nor solid, but rather a fractal object, generally with non-
integer dimension.
In view 1 of this demonstration, finite difference equations of the
form
x --> a + bx + cxý + dxy + ey + fyý
y --> g + hx + ixý + jxy + ky + lyý
are solved with random choices of the coefficients a through l. Cases
that are not chaotic are discarded, and cases that are chaotic are
displayed in the x-y plane. View 2 includes a third equation with an
additional variable z, but the attractor is projected onto the x-y
plane. Views 3 through 5 display the third variable in various ways.
It is remarkable that the same simple equations produce so many
different patterns, almost none of which have ever been seen before.
Mandelbrot Set
The Mandelbrot set is created by starting with a complex number Z
(taken as zero), squaring it, and adding another complex number C:
Z(n+1) = Z(n)ý + C
The process is repeated until |Z|ý exceeds a value of 4 that
ensures that the point is outside the set. Contours of the number of
iterations required to exceed this value are plotted in the complex
C-plane.
Points in the interior region never escape, but points in the
exterior region escape rapidly. The boundary between the two regions
has visible structure at all magnifications and is a fractal. The
interior region is a basin of attraction, and all parts of this region
are connected to all other parts.
The Mandelbrot set has been described as the most complicated
mathematical object ever "seen," and yet it arises from one of the
simplest equations.
After you have a picture, press <+> or <-> to cycle among the color
palettes supported by the computer.
Julia Sets
The Julia sets are created in a manner similar to the Mandelbrot
set, by starting with a complex number Z, squaring it, and adding
another complex number C:
Z(n+1) = Z(n)ý + C
The process is repeated until |Z|ý exceeds a value of 4 that
ensures that the point is outside the set. Contours of the number of
iterations required to exceed this value are plotted in the complex
plane of the initial values of Z. Contrast this case with the
Mandelbrot set in which the points are plotted in the complex C-plane.
The various views (values of C) are chosen to correspond to values
near the fractal boundary of the Mandelbrot set. View 19 performs a
random search and produces a different pattern each time the program is
restarted. Each figure is a basin of attraction. It is remarkable that
such a diversity of shapes results from such a simple equation.
After you have a picture, press <+> or <-> to cycle among the color
palettes supported by the computer.
Diffusion
Shown here are N particles that at each iteration are moved
randomly by a unit step either up or down or right or left. The
resulting trajectory is a random walk or Brownian motion and is a
fractal. A collection of such particles undergoing random motion
exhibit diffusion--they gradually spread out from where they started.
Unlike the previous examples in which chaotic motion results from
deterministic equations, the motion here is derived using a random-
number generator.
The particles shown do not interact with one another. Imagine,
instead, that the particles are colliding with invisible background
particles and that the mean-free-path between collisions is a constant.
A characteristic of a random walk is that the mean-square
displacement of the collection of particles increases linearly in
time. At the top of the screen, the value of /t is calculated
continuously. With a sufficiently large number of particles, this
normalized diffusion rate should approach 1.0, but fluctuations equal to
the reciprocal of the square root of the number of particles are
expected, as can be seen in the simulation.
Noise
The term "noise" generally denotes a quantity that varies randomly
in time, in analogy with the random spatial motion that gives rise to
diffusion. A noisy voltage connected to a loudspeaker produces audible
noise.
The simplest kind of noise is white noise, in which the value of
the quantity at a given time is unrelated to its value at previous
times. Like white light, white noise is made up of equal amounts of all
frequencies, and thus its power spectrum is a constant independent of
frequency. Brownian motion produces a spectrum that decreases with
frequency according to 1/f^2 and appears much smoother than white noise.
It is the result of integrating white noise over time.
Another type of noise, intermediate between these two, is 1/f
noise, sometimes called pink noise. Although poorly understood
theoretically, 1/f noise is probably the most common type of noise. The
distribution of notes in music closely follows a 1/f law. The integral
of 1/f noise is 1/f^3 (sometimes called black noise) and is very smooth.
Turn on the sound in the demonstration and see which type of noise
seems most musical to you.
Deterministic Fractals
These deterministic fractals are formed by a variety of techniques.
Views 1 through 7 are formed by recursively calling a function that
replicates a pattern on successively smaller or larger scales. View 8
is a continuous but nowhere differentiable function. Views 9 through 18
are produced by the use of iterated function systems in which an initial
point is moved a prescribed distance in one of several randomly chosen
directions. The fractal tree (view 19) is formed by starting many times
at the trunk and making a turn through a given angle in one of two
randomly chosen directions after a given distance. Although random
numbers are used, the resulting patterns are deterministic.
Many of these fractals are self-similar--that is, they look the
same at any magnification. The fern is self-affine, which means it
looks the same under magnification if properly distorted. The
complexity of such patterns can be characterized by the fractal
dimension. The dimension of a fractal is defined here as twice the log
of the number of occupied sites divided by the log of the number of
available sites. A line thus has a dimension 1.0 and a surface has a
dimension 2.0.
The processes illustrated represent the way snowflakes form. You
can produce an enormous variety of such patterns by varying the rules by
which they are formed.
Random Fractals
In view 1, a point is initially illuminated at the center of the
screen. A particle begins a random walk from a random point on a circle
surrounding this point. If it contacts the point, it sticks, and is
itself illuminated; and a new particle is started from a random point on
the circle. If it intersects the circle, it is reflected so as to
remain always inside the circle. The circle expands continuously as the
pattern grows.
In view 2, a line is initially illuminated at the bottom of the
screen. Particles with random initial positions rain down from above
and stick when they touch the line or another particle that has stuck to
the line.
The process is called diffusion-limited aggregation. It leads to a
fractal whose dimension is continuously calculated and indicated at the
top of the screen. Such a simulation would model the growth of a
structure embedded in a fluid whose particles undergo random motion. It
might also model the evolution of a lightning discharge. Perhaps you've
seen patterns like this in the ice crystals that form on your windows in
the winter.
Iterated Function Systems
In an iterated function system, a point is chosen arbitrarily in
the x-y plane, and then a succession of new points is calculated from it
and plotted by randomly choosing from among a group of N linear affine
maps of the form
x --> ax + by + e
y --> cx + dy + f
Each case is generated using a different random selection of the 6N
coefficients of the system, producing a different pattern. Such a
mapping scales, rotates, shears, and reflects the image, producing a
variety of patterns with a high degree of self-similarity.
The amount of contraction is controlled by the parameter F. In
view 1 only contractions and rotations are allowed, and all N are equal,
producing a regular polygon. View 2 shows irregular polygons. View 3
shows cases with constant contraction but with arbitrary rotations.
View 4 allows all the coefficients to be arbitrary, but with a maximum
contraction of F. View 5 adds a third coordinate z and hence 6N
additional coefficients. It produces a 3-D fractal pattern that may be
viewed with special red/cyan glasses.
Coupled-Map Lattice
Nature often exhibits spatial as well as temporal chaos. An
example of spatio-temporal chaos is a turbulent fluid. One way to model
such processes is to distribute a collection of iterated maps such as
the logistic map on a regular lattice with each map coupled to a few of
its nearest neighbors with random initial conditions.
The demonstration here has a one-dimensional lattice of logistic
maps, F = AX(1-X), each coupled to N identical maps on either side of
itself according to
N
X(I) --> (1 - k) F(I) + k ä [F(I-J) + F(I+J)] / 2N
J=1
The adjustable constant k determines the strength of the coupling. You
may want to refer to the One-Dimensional Maps demonstration to review
the properties of the logistic map for various values of A.
You will observe that such coupled-map lattices have behavior very
different from single isolated maps. For example, individually chaotic
maps can lock together and produce patterns that are periodic in both
space and time. See if you can observe other types of behavior. Try
rotating the palette with the <+> and <-> keys to highlight features of
the pattern.
Mixing
A set of initial conditions, which you can consider as a collection
of individual particles, like drops of cream in a cup of coffee, is
subjected to motion governed by two clockwise-rotating vortices with
centers at Y=0 and X=ña as indicated by the small circles. The vortices
are alternately turned on with the other turned off.
With each iteration, every point is rotated through an angle h
ln(r) where r is the distance of the point from the center of the vortex
that is turned on. The strength of the rotation is controlled by the
parameter h, which is measured in radians. Negative values of h cause
the vortex on the left to rotate counter-clockwise. Such a motion might
represent approximately the stirring of a cup of coffee with a spoon
that makes a small circular motion and moves back and forth between two
places in the coffee. Note that the curve never crosses itself, but
that it becomes increasingly intertwined as it is repeatedly stretched
and folded.
You will observe the sensitivity to initial conditions that is
characteristic of chaos. However, there are regimes in which the motion
is periodic. See if you can find the chaotic and periodic regimes for
various choices of h.
Percolation
A fluid seeps through a porous medium by the process of
percolation. Perhaps the most familiar example is a coffee percolator.
Percolation theory also describes many other similar phenomena such as
the spread of forest fires and diseases, and the electrical and thermal
conductivity of amorphous materials.
This demonstration simulates the flow of a liquid through a
rectangular lattice in which gaps are randomly distributed. The liquid
starts at a cell at the top center and flows downward into all
connecting cells, like a rat exploring a maze trying to find a way out
at the bottom.
You can change the fraction of gaps across (A) and down (D) from 0
to 1 and collect statistics on the probability that the fluid percolates
through the lattice. You should observe that the probability is very
small until the fraction approaches about 0.5, whereupon the probability
rapidly rises to near 100%. This value is called the percolation
threshold. The threshold depends weakly on the anisotropy (the ratio of
A to D).
Cellular Automata
A cellular automaton is a pattern that evolves in such a way that
each point in the pattern is governed only by a few of its nearest
neighbors. A simple example is a row of positive integers all of which
are initially zero except for N which are one. At the next time step, a
new row is produced with each number dependent on its two nearest
neighbors at the previous step. One possible rule is
F(x,t+1) = [F(x-1,t) + F(x+1,t)] MOD k
which gives rise to a complicated but periodic pattern called Pascal's
triangle as shown in view 1. On a color monitor the k values of F are
indicated by different colors. Another possible rule is
F(x,t+1) = {F(x-1,t) + MAX[F(x,t),F(x+1,t)]} MOD k
which gives rise to a chaotic pattern as shown in view 2. The pattern
never repeats and exhibits extreme sensitivity to initial conditions.
The necessary nonlinearity is provided by the MAX function which returns
the larger of its two arguments. Each row constitutes a fractal with a
dimension between zero and one. View 3 is identical to view 1, and view
4 is identical to view 2 except that the N initial points are located
randomly along the first row.
Game of Life
The game of life is a classic demonstration of a two-dimensional
cellular automaton in which the evolution of points on a grid is
governed solely by their nearest neighbors. Local rules govern the
global behavior.
In the case shown here, the initial distribution of points is
chosen randomly. In the classic game of life, a point survives to the
next generation if two or three of its eight nearest neighboring points
are populated. If fewer than two points are populated, it dies of
isolation, and if more than three points are populated, it dies of
overcrowding. This rule is indicated by D = 23. An unoccupied point
with exactly three neighbors is a birth site, indicated by B = 33.
There are over 2,000 combinations of D and B for you to explore. Simple
rules allow order to emerge out of disorder, as with evolution. This
phenomenon is called "self-organization." The resulting structures go
by such descriptive names as gliders, blinkers, and traffic lights.
Although the cellular automaton has obvious application to biology,
it also models crystal growth, turbulence, ferromagnets, and galaxy
formation (with appropriately changed rules). Cellular automata are
ideal for computer simulation.
Anaglyphs
Anaglyphs are drawings that use contrasting colors to produce a
three-dimensional effect when viewed through glasses with a
corresponding colored filter over each eye. Most people require a brief
period of adjustment to experience the effect, especially when the
objects are unfamiliar. These anaglyphs of common objects supplement
those found elsewhere among the demonstrations and are included to aid
your acclimation and to amuse you.
The two-torus in view 5 illustrates a quasiperiodic orbit (a
superposition of two incommensurate frequencies) that is often
incorrectly identified as chaos since it never repeats.
Of course your computer must have a color monitor, and your glasses
must be well matched to the color of the phosphors in your monitor. If
you see a double image with either eye closed, the colors are not well
matched. You might try altering any adjustments provided on your
monitor or using different glasses. Sometimes the ghost image can be
eliminated by looking through two pairs of glasses.
The figures should appear to the correct scale when viewed from
arm's length with a 12- to 15-inch monitor. If you still have trouble
seeing the effect, try viewing from closer to the screen or use the
magnify (M) command to make the object smaller.