**Chaos notes:**

Steven Strogatz “Chaos” DVD teaching company

**Lecture 1: The
Chaos Revolution**

For a long time, science avoided chaotic systems. In the 1980s, a few
mavericks from every scientific disapline said we
need to try to understand chaotic systems. They found that chaos in different physical
systems looked the same; for example, patterns that geologists found in the
frequency of earth
quakes were similar to the patterns physicians found in heart rate variability;
and these patterns were also similar to the patterns found in internet traffic. The patterns look similar
regardless of the scale at which the patterns are viewed. (like
fractals) chaotic systems are deterministic but not predictable. Chaos theory
shifts the focus from the laws of physics to their consequences. It uses a
computer in a radical new way, to see the consequences of the laws. ; holistic; ie includes the effects
of the parts to the whole scientific
holism; Interdisciplinary; simple
systems can behave chaotically. Complex systems may exhibit simple patterns.
“Fractals are the footprints of chaos”

**Finished
lecture 1**

**Lecture 2: The
clockwork universe**

**Finished
lecture 2**

**Lecture 3:
From clockwork to chaos**

By mid 20^{th} century, three tiny cracks began to appear in our deterministic
perception of the universe: einsteins theory of
relativity, quantum mechanics , and Chaos.

**Lecture 4: Chaos
found and lost again**

Poincare found that the three body problem led to chaos. Poincare’s
method used visualization and is central to chaos theorists today. He drew
pictures of the objects traveling thru “state” space or “phase” space. The
motion of the pendulum for example, corresponds to a trajectory
moving thru state space.

State space is the collection of all possible states (all possible frames
in the movie of a pendulum. Position and velocity needed to completely define
pendulum position. A plot of phase space
shows two axes: position and velocity. Use state space diagram to show how
pendulum acts.

**Lecture 5: the
return of chaos: **

Lorentz’ original attractor had 12 variables: 12 states, so can’t
visualize

**Lecture 6: the
disorder of chaos**: the butterfly effect

**Lecture 7:
Picturing Chaos as Order: strange attractors**

strange
attractors show the order in chaos;

Water wheel: analog of convection: has 3
variables: 3 state spaces, so can visualize.

Strange attractors have 4 char: 1) deterministic: have no choice 2)
non-periodic 3) confined to the strange attractor; sensitivity to initial
conditions.

Chaos is not disorder; it is a state poised between order and disorder;

Cumulative order. The order in chaos allows it to be used in practical ways

**Lecture 8:
Animating Chaos as order: iterated maps**

Differential equations, strange attractors, and iterated maps are three
kinds of pictures; each treats time differently. Differential equations give a
frame by frame picture. Strange attractors, and
iterated maps reduce the amount of information provided in differential equations.
Strange attractor is analogous to a time lapse photograph (collect all images
together). Iterated maps are analogous to a strobe light photograph??? Leap
frog through time;
Uses feedback to recalculate.

**http://lnx-bsp.net/news/2009/01/03/
**

can the y[2]
value of the function at any peak y[2] _{n
}be used to predict the value of the function at the next peak peak y[2] _{n+1} ?_{}

if so, we
can ignore everything that happened in between the two peaks (leapfrogging)

to find
out, have to plot y[2] _{n} on x axis and y[2] _{n+1} on y axis
for all the peaks

this maping is called a Lorenz map: the picture of the map:

**http://andvari.vedur.is/~halldor/HB/Met210old/pred.html**

it shows
there is a definite relationship between y[2] _{n} and y[2] _{n+1}

_{ }

this shows
the difference between chaos and randomness. This is an astonishing pattern
arising out of chaos._{}

_{ }

Lorenz attractor: the trajectory grows until reaches a peak on one side
of the butterfly, then jumps to the alternate
trajectory on the other side of the trajectory:

**http://courses.washington.edu/phys2278/**

**order**** in chaos is always due to non-liner relationships**

**once**** again, as with Poincare, Lorenz’s discoveries of the
1970s fell on deaf ears**

**Lecture 9: How
systems turn chaotic**

Suddenly many become interested in disorder. They want to know how
systems turn chaotic. Different systems may become chaotic in the same way.
Three concepts are important:

Iterated maps, bifurcations and the transition
to chaos, linearity vs nonlinearity.

Iterated map: feed back calculation; important because provide simples
math models from order into chaos.

Two other iterated maps: logistic map: used in ecology

Bifurcation: change in qualitative long term behavior.

Robert May: biologist; urged learning about non-linear systems

**Lecture 10:
start**

**Lecture 21:
Chaos in Health and Disease**

Epileptic seizure indicated by decreasing chaos of brain:

The short term lyapunov exponents (bits/sec)
from different positions on the brain decrease and converge: the brain is
becoming synchronized.

Leon Lasemidis Chris Sackellares

Note that the brainwaves are lining up, but the chaos at the different
nodes is lining up: cannot see it in an EEG

But what about attempts to “synchronize” the
left and right hemispheres?

What can chaos theory tell us about arythmias and fibrillation
(“chaotic” behavior of heart)

can chaos
be good for your heart? Is homeostasis always best?

Ary
Goldberger Beth Israel hospital says chaos can be good for your heart

Heart rate variability: rock steady heart rate is associated with
congestive heart faliure

Totally Eratic heart beat: fibrillation

Health: variation is fractal thru time

Heart rate variability decreases as we age

**Lecture
22: Quantum Chaos**

3 conflicts between chaos and Quantum theory

Chaos: deterministic: the present determines the future

Quantum mechanics basic differential
eq is Schrodinger’s equation, but it is linear; there is no chaos in linear
systems

Heisenberg’s
uncertainty principle: destroys concept of state space on which chaos theory rests.

What is a state: the amount of info you need to predict the future of a
deterministic system.

Need initial pos & vel, but this is not
allowed by HUP.

Distribution of prime numbers:

Riemann added an improvement to the formula for prime numbers which gives
the “steps” we see in the distribution of prime numbers. The improvement was
adding frequencies

Riemann hypothesis; Riemann’s guess of frequency values needed to model prime number distribution called
the music of the primes

Riemann’s waves are the secret to the mystery of the primes.

Quantum systems have discrete energy levels, corresponding to waves
vibrating at certain frequencies.

Likewise the secret of the primes are encoded in a discrete set of wave
frequencies: the “magic frequencies” Riemann found he needed to give an
accurate formula for the prime number distribution. These are given the name
the “zeros of the Riemann Zeta function” (magic frequencies).

The frequencies of the Riemann waves look uncannily like the frequencies
of a “quantum chaotic system”.

There is some undiscovered chaotic system whose quantum counterpart would
hold the secret to the music of the primes. Chaos, atoms, prime numbers all connected. The
“atoms” of arithmetic; ie the prime numbers, are
connected to the atoms of reality, and the link between them is chaos.

**Lecture 24:
The Future of Science**

Long term order: Chaotic systems motion in state space
are not a random mess, but intricate structures called strange attractors.

Intermediate order: strobe light picture of where the system is on a
strange attractor. Called iterated map.

Strogatz
represents those who hope that chaos of complex systems and emergent properties
will yield through reverse engineering answers to the question of how to
manipulate or possibly even to create “life”: major schools funding “systems
biology” and even “synthetic biology”. Chaos theory being used to design
“biological circuits”, building living versions of things electrical engineers
have built: toggle switches, oscillators, amplifiers; not built from
transistors, but from genes and proteins. Steven Strogatz
Chaos DVD teaching company

Chaos theory also shedding light on perhaps the greatest mystery of all:
consciousness. Neuroscientists are discovering
clues. Found that when we recognize a face, or pay attention to something,
parts of the brain fire synchronized electrical signals at a specific frequency
(40 hz) This synchronized
signal is self organized and arises out of chaos. “It’s a self organizing thing
when we have consciousness... Its starting to look like all that we feel or
perceive may be just a reflection of fleeting processes of synchronization
among neurons in our brain”. Steven Strogatz Chaos
DVD teaching company