Oscillations, Waves, and Interactions - GWDG

Complex dynamics of nonlinear systems 419
To investigate the scaling in the limit k/N → 0 one can decrease the number of
neighbours k or increase the number of data points N.
All methods for computing fractal dimensions from (large) data sets can be considerably
accelerated by using fast search algorithms [30] for the nearest neighbours
of the data points. Algorithms for these tasks and for estimating many other useful
characteristics of nonlinear systems were implemented at the DPI and are publicly
available in the Matlab T M toolbox TSTOOL 5 .
More details about dimension estimation methods, their possible pitfalls, extensions,
and further references are given in many review articles and textbooks [31–40].
3.2 Lyapunov exponents
Lyapunov exponents describe the mean exponential increase or decrease of small
perturbations on an attractor and are invariant with respect to diffeomorphic changes
of the coordinate system. The full set of Lyapunov exponents of a d-dimensional
system constitutes the Lyapunov spectrum which is an ordered set of real numbers
{λ1, λ2, . . . , λd} with λi ≥ λi+1.
When the largest Lyapunov exponent λ1 is positive, the system is said to be chaotic
and it shows sensitive dependence on initial conditions.
The meaning of the Lyapunov exponents is illustrated in Fig. 10. An infinitesimally
small ball of initial conditions forming neighbouring points of some reference state is
transformed into an ellipsoid due to the temporal evolution of the system (linearized
around the trajectory of the reference state). The principal axes of this ellipsoid grow
or shrink proportial to exp(λit).
For an exact definition of the Lyapunov exponents and computational details see
Geist et al. [41] or Abarbanel [38].
Figure 10. Illustration of the local
temporal evolution of a ball of
neighbouring states evolving into
an ellipsoid with principle axes
whose lengths are proportional to
exp(λit).
4 Time series analysis
In mathematical models of dynamical systems the dynamics is described in their
state space, whose (integer) dimension is given by the number of the dependent variables
of the model. In experiments, however, often just one variable is measured as
a function of time, and the state space is usually not known. How, then, to arrive
at the attractor that may characterise the system? This gap between the theoretical
notions and observable quantities was filled in 1980 when Packard, Crutchfield,
Farmer, and Shaw [42] published their fundamental paper “Geometry from a time
5 TSTOOL URL: //http:www.physik3.gwdg.de/tstool/