Lyapunov fractals

Q17a: Where are the popular periodically-forced Lyapunov fractals described?
A17a: See:

1.  A. K. Dewdney, Leaping into Lyapunov Space, _Scientific American_, Sept.
1991, pp. 178-180.

2.  M. Markus and B. Hess, Lyapunov Exponents of the Logistic Map with
Periodic Forcing, _Computers and Graphics_ 13, 4 (1989), pp. 553-558.

3.  M. Markus, Chaos in Maps with Continuous and Discontinuous Maxima,
_Computers in Physics_, Sep/Oct 1990, pp. 481-493.

Q17b: What are Lyapunov exponents?
A17b:

Lyapunov exponents quantify the amount of linear stability or instability of
an attractor, or an asymptotically long orbit of a dynamical system.  There
are as many lyapunov exponents as there are dimensions in the state space of
the system, but the largest is usually the most important.

Given two initial conditions for a chaotic system, a and b, which are close
together, the average values obtained in successive iterations for a and b
will differ by an exponentially increasing amount.  In other words, the two
sets of numbers drift apart exponentially.  If this is written e^(n*(lambda))
for n iterations, then e^(lambda) is the factor by which the distance between
closely related points becomes stretched or contracted in one iteration.
Lambda is the Lyapunov exponent.  At least one Lyapunov exponent must be
positive in a chaotic system.  A simple derivation is available in:

1.  H. G. Schuster, _Deterministic Chaos: An Introduction_, Physics Verlag,
1984.

Q17c: How can Lyapunov exponents be calculated?
A17c: For the common periodic forcing pictures, the lyapunov exponent is:

lambda = limit as N->infinity of 1/N times sum from n=1 to N of log2(abs(dx
sub n+1 over dx sub n))

In other words, at each point in the sequence, the derivative of the iterated
equation is evaluated.  The Lyapunov exponent is the average value of the log
of the derivative.  If the value is negative, the iteration is stable.  Note
that summing the logs corresponds to multiplying the derivatives; if the
product of the derivatives has magnitude < 1, points will get pulled closer
together as they go through the iteration.

MS-DOS and Unix programs for estimating Lyapunov exponents from short time
series are available by ftp: lyapunov.ucsd.edu:/pub/ncsu .

Computing Lyapunov exponents in general is more difficult.  Some references
are:

1.  H. D. I. Abarbanel, R. Brown and M. B. Kennel, Lyapunov Exponents in
Chaotic Systems: Their importance and their evaluation using observed data,
_International Journal of Modern Physics B_ 56, 9 (1991), pp. 1347-1375.

2.  A. K. Dewdney, Leaping into Lyapunov Space, _Scientific American_, Sept.
1991, pp. 178-180.

3.  M. Frank and T. Stenges, _Journal of Economic Surveys_ 2 (1988), pp. 103-
133.

4.  T. S. Parker and L. O. Chua, _Practical Numerical Algorithms for Chaotic
Systems_, Springer Verlag, 1989.