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| | 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://inls.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. |
| |