(Click on the figure to see a bigger
version.) The bifurcation diagram of the logistic map shows an
extraordinary complexity as its parameter r is
varied. Some details of this diagram are still not understood.
(Click on the figure to see a bigger version.) Although the
global behavior of this chaotic orbit (blue thread) of the Lorenz
equations is difficult to understand, the outward spiraling
motion along a nearly planar surface plus the approach to such a
surface from an opposing fixed point can be largely understood in
terms of the linearized dynamics near the unstable fixed points
at the center of each "butterfly wing". These hyperbolic fixed
points have one stable direction (here indicated in red) and two
unstable directions (indicated as green arrows around the fixed
points), with the latter corresponding to a pair of complex
eigenvalues with positive real part. This image comes from
this site, which has other nice visualizations of the
behavior of orbits near unstable fixed points.
(Click on the figure to see a bigger version.) Power spectral
density P(f,p) of Saturn's dynamical motion as a function
of frequency f and fractional change in Saturn's
initial distance p from the Sun, based on a numerical
simulation of all the planets and Sun in the Solar System
interacting via their gravitational forces. For some values of
the parameter p, e.g., between -0.002 and 0.002, the
power spectra have well defined peaks and so the dynamics is
periodic and predictable. For other parameter values, e.g.,
between .003 and 0.007, the spectra are broad band and so the
dynamics are chaotic and unpredictable, implying that the Solar
System itself is chaotic. From this figure, can you deduce
whether the transition to chaos is supercritical as the
parameter p is varied? Figure was taken from
this web site, based on the research of Ferenc
Varadi.
The essence of how
chaos is generated as revealed in a taffy making machine in a
candy store in Savannah, Georgia. (The scale of the figure is
about one meter.) The taffy is repeatedly stretched and folded as
the two metal arms counterrotate periodically. This leads to an
exponentially rapid separation on average of any two nearby
points within the taffy. In the phase space of some dynamical
system, the stretching would arise from instability that is more
rapid in some directions than others, the folding from the
boundedness of the dynamics. (Picture by H. Greenside.)
A challenge for nonlinear dynamics
researchers is understanding how biological brains process
sensory information, store memory, recognize objects in the
environment, learn new behaviors, and orchestrate muscle
movements. Each of the many neurons in a brain is itself a
complicated nonlinear processing unit that receives electrical
impulses (spikes) from other neurons (from as many as 100,000
inputs for a Purkinje cell in the mammalian cerebellum!) and
then sends out spikes of its own at a rate that depends on its
input. In this image taken from
Richard Axel's lab at Columbia University, neurons are
shown in the second (yellow-orange) and third (green) stages of
the olfactory processing pathway of a fly brain. (Not shown is
the first stage of olfactory receptor neurons that are directly
activated by odorants that come in contact with the fly's
antennae.) How the resulting neuronal dynamics enables a fly to
identify different odors and to navigate toward or away some
odor---often in the presence of fluctuating turbulent
air---remains a mystery.
Fractals are sets of points that have a self-similar geometric or
statistical structure upon magnification of any part of the
set. An especially intriguing and modern set of points that is
partly fractal is
the positions of 200,000 galaxies up to two billion light
years away from Earth (about 1/6 the diameter of the known
universe!) as measured recently by the Sloan Digital Sky Survey. Each
dot represents a separate galaxy and the color of the dot
represents that galaxy's luminosity. An analysis of these data
indicate that the baryonic matter that Earth is made of
constitutes only 5 percent (!) of the mass of the
universe. The rest of the mass consists 25 percent of "dark
matter" and 70 percent of "dark energy". What these dark
matter and energy consist of are two of the most interesting
unsolved questions of current science. And how did the expansion
of the universe and the gravitational coupling of matter to dark
matter and energy produce this unusual clustered geometric
structure?
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