ChaoticMarket 43Take a look at Figures 3-5 and 3-6. Figure 3-5 depicts the time
series of a simulated weather system, suggesting behavior that is
completely random and resembles typical graphs of stoc kmar ket
prices. Figure 3-6 depicts a phase portrait of the same system, re-
vealing a strange attractor. Again, loo kat Figure 3-6 as a side shot
of the time-series plot in Figure 3-5, condensing that Cartesian fig-
ure into a phase space portrait.
Limit cycle attractors and point cycle attractors do not exhibit
any sensitive dependence on initial conditions: A pendulum without
permanent mechanical force will always end up at the point of origin
(its point attractor) no matter where it started, and a pendulum with
permanent mechanical force will always orbit in its loop (its limit
cycle attractor) no matter where it started.
Systems containing strange attractors do exhibit sensitive depen-
dence on initial conditions: Where the system is at some future mo-
ment will be determined by where the system started (or by where
it was at any time before).
Stretching
Phase portraits depict all possible states of a system by plotting a
variable’s value against the possible values of all other variables. The
dimension of the phase space is equal to the number of variables
that describe the system. Whether a system exhibits sensitive depen-
dence on initial conditions can be determined by numbers called
Lyapunov exponents (LEs), named for the Russian mathematician
Aleksandr Lyapunov, who discovered them.^8
LEs measure the speed of a variable’s movements in phase space
versus another variable. Positive LEs measure stretching in phase
space—the speed of divergence of one variable with respect to an-
other variable. Negative LEs measure contracting in phase space—
the speed of system restoration after being perturbed. Thus, LEs for
point attractors and limit cycles never are positive because such sys-
tems are always contracting.
In the case of a point attractor, the dimensions always converge
to a fixed point, the origin; in the case of a limit cycle attractor, all
the dimensions converge into one another except one, whose relative
position creates the loop by not changing (and whose LE is therefore
zero). For a strange attractor—involving a system that does exhibit
sensitive dependence on initial conditions—at least one LE must be
positive such that there is divergence in the nearby orbits.