Arnold’s Cat Map Michael H. Dormody December 1 st, 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 Who will win? Place your bets…! 450 pixels451 pixels452 pixels

Following a single pixel across the image…

Peculiar Patterns forming in the Cat Maps: T/5 (a ghost?) 13T/75 (mini cats?) T/2 (plaid pattern?)

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 time- dependent 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: x t+1 = mod(x t + x t-1, N ), we can define the momentum as p t+1 = mod(x t + 2p t, 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, 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) “Hissssss! I don’t like non- ergodic dynamical systems! I prefer catnip…”