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: ftp://lyapunov.ucsd.edu/pub/ .  
  
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.