# 2.15] What is the minimum phase space dimension for 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.

# 2.15] What is the minimum phase space dimension for chaos? (nonlinear science)

This is a slightly confusing topic, since the answer depends on the type of

system considered. First consider a flow (or system of differential

equations). In this case the PoincarĂ©-Bendixson theorem tells us that there is

no chaos in one or two-dimensional phase spaces. Chaos is possible in three-

dimensional flows--standard examples such as the Lorenz equations are indeed

three-dimensional, and there are mathematical 3D flows that are provably

chaotic (e.g. the 'solenoid').

Note: if the flow is non-autonomous then time is a phase space coordinate, so

a system with two physical variables + time becomes three-dimensional, and

chaos is possible (i.e. Forced second-order oscillators do exhibit chaos.)

For maps, it is possible to have chaos in one dimension, but only if the map

is not invertible. A prominent example is the Logistic map

x' = f(x) = rx(1-x).

This is provably chaotic for r = 4, and many other values of r as well (see

e.g. #DevaneyDevaney). Note that every point x < f(1/2) has two preimages, so

this map is not invertible.

For homeomorphisms, we must have at least two-dimensional phase space for

chaos. This is equivalent to the flow result, since a three-dimensional flow

gives rise to a two-dimensional homeomorphism by PoincarĂ© section (see [2.7]).

Note that a numerical algorithm for a differential equation is a map, because

time on the computer is necessarily discrete. Thus numerical solutions of two

and even one dimensional systems of ordinary differential equations may

exhibit chaos. Usually this results from choosing the size of the time step

too large. For example Euler discretization of the Logistic differential

equation, dx/dt = rx(1-x), is equivalent to the logistic map. See e.g. S.

Ushiki, "Central difference scheme and chaos," Physica 4D (1982) 407-424.

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