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




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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 Poincaré-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 Poincaré 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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