# 2.13] Can computers simulate chaos? (nonlinear science)

## Description

This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by
others.

# 2.13] Can computers simulate chaos? (nonlinear science)

Strictly speaking, chaos cannot occur on computers because they deal with

finite sets of numbers. Thus the initial condition is always precisely known,

and computer experiments are perfectly predictable, in principle. In

particular because of the finite size, every trajectory computed will

eventually have to repeat (an thus be eventually periodic). On the other hand,

computers can effectively simulate chaotic behavior for quite long times (just

so long as the discreteness is not noticeable). In particular if one uses

floating point numbers in double precision to iterate a map on the unit

square, then there are about 10^28 different points in the phase space, and

one would expect the "typical" chaotic orbit to have a period of about 10^14

(this square root of the number of points estimate is given by Rannou for

random diffeomorphisms and does not really apply to floating point operations,

but nonetheless the period should be a big number). See, e.g.,

Earn, D. J. D. and S. Tremaine, "Exact Numerical Studies of Hamiltonian

Maps: Iterating without Roundoff Error," Physica D 56, 1-22 (1992).

Binder, P. M. and R. V. Jensen, "Simulating Chaotic Behavior with Finite

State Machines," Phys. Rev. 34A, 4460-3 (1986).

Rannou, F., "Numerical Study of Discrete Plane Area-Preserving Mappings,"

Astron. and Astrophys. 31, 289-301 (1974).

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