Transcript Arnold’s Cat Map - Physics Department

Arnold’s Cat Map
Michael H. Dormody
December 1st, 2006
Classical Mechanics 210
UC Santa Cruz
A Primer on Chaotic
Systems
• Time evolution of the universe: deterministic chaos v.
random behavior.
• Sensitive dependence on initial conditions: Initially similar
systems will move together, but will diverge as time
evolves.
• Ergodic: a system will return to its initial state after a
given time. Underlying order to chaos?
• Applications to physics: nonlinear dynamical systems
including quantum orbits using classical models, Poincare
recurrence.
• We can use chaotic systems to study quantum physics on
a classical level.
An Introduction
To Anosov Maps
• Anosov map: A mapping on a manifold (space) from itself
to itself with explicit instructions to make the manifold
expand and contract.
• Anosov diffeomorphism: an invertible mapping.
• Arnold Cat Map: specific Anosov map with hyperbolic
behavior (one side expands, the other contracts) and is a
diffeomorphism. For a digital image, a pixel shifts from its
location to another location in the image according to a
specific rule.
• In physics, it can describe the motion of a bead hopping
sites on a circular ring with N sites. Perfect for periodic
boundary conditions!
• Can help to bridge the gap between classical chaos and
quantum chaos.
An Example of a Cat Map
• An Arnold Cat Map is a
mapping of the form: (x,y)
 modulus(x+y,x+2y, N)
• The mapping shears the
original image, and the
modulus function folds the
image into the original
image area.
• We can illustrate these
mappings using a digital
image. To stick with
tradition, I’ve chosen my cat
Missy.
A Race To The Finish:
Ergodic Motion
• Using a digital image of Missy the cat, we will transform
the image under the instruction of the Arnold cat map.
• We use three similar images with one pixel difference
between them: 450 x 450, 451 x 451, and 452 x 452.
• Recurrence Period: increase with pixels?
• If so, by how much?
The Running of the
Cat Maps
450 pixels
451 pixels
452 pixels
Who will win? Place your bets…!
Following a single pixel
across the image…
Motion Of The Origin Pixel
N = 450 pixels
Motion Of The Origin Pixel
N = 452 pixels
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Motion Of The Origin Pixel
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Peculiar Patterns
forming in the Cat Maps:
T/5
(a ghost?)
T/2
(plaid pattern?)
13T/75
(mini cats?)
Functional Relation of
Recurrence Period and Number
of Pixels
We can see a a few functional relationships, but no solid fit. A
“universal function” fails.
We can see special cases:
T = N, T = 2N, T=1/2 N…
Relation to Properties
of Dynamical Systems
• Mixing Theorem: If we take a solution of 20%
rum and 80% Coke, and stir vigorously enough,
every part of the solution will consist of 20%
rum and 80% Coke. Because each Coke and
rum particle are being randomly moved, unlike
Arnold’s Cat Map, the system will never return
to its original state (unless you froze the solution
and drained out the alcohol).
• Poincare Recurrence: if a dynamical system is
“shuffled” for a long enough time, it will return
to its initial state.
• In the Arnold Cat Map, the image returns to its
initial shape after a certain amount of time
because it has a definite assignment.
Concrete Examples
• By quantizing a classically chaotic system, we can
observe a quantum distribution with a continuous
energy distribution: A charged particle constrained
to the plane under the influence of a timedependent field experiences little “kicks” that push
the particle into various states. The time evolution
operator applied to this system yields a continuous
energy spectrum.
• Time evolution operator (propagators): operators
that measures how a system evolves.
• Also, If we are simulating a linear chain of N sites,
then the bead hopping around is modified by a
modulus function. If we say that its position is
defined by: xt+1 = mod(xt + xt-1, N), we can define
the momentum as pt+1 = mod(xt + 2pt, N). The
phase space for this system represents chaotic
motion, and is a perfect example of the Arnold Cat
Map in Motion.
Summary / Further Readings
• Arnold, V. I. & Avez, A. Ergodic Problems of
Classical Mechanics. W.A. Benjamin, Inc.: New
York, 1968.
• Weigert, S. The Configurational Quantum Cat
Map. Condensed Matter 80, 34 (1990).
• Knabe, S. On the Quantization of Arnold’s Cat.
J. Phys. A: Math. Gen. 23 (1990) 2013-2025.
“Hissssss! I don’t like nonergodic dynamical systems! I
prefer catnip…”