Clas Blomberg - Physics of life-Elsevier Science (2007)

Chapter 28. Deterministic chaos 303
As mentioned above, evidence for chaotic components in sensory signals of the crayfish
nervous system has been presented by Moss and colleagues (Pei and Moss, 1996). Likewise,
certain EEG signals have been suggested to be of a chaotic nature, and there are several
attempts to characterise EEG-signals by the characteristics of chaotic processes (Babloyantz
et al., 1985; Fuchs et al., 1987). As will be discussed below, it has also been suggested that
deterministic, chaotic processes control the opening and closing of ion channels (Liebovitch
and Tóth, 1991b; see also Bassingthwaighte et al., 1994). Certain neural network models
are found to give rise to chaotic patterns, which have been compared with the EEG-patterns
(Liljenström and Wu, 1995).
Calculation of Lyapunov exponents and fractal dimensions
It is appropriate at this stage to say something about practical calculations of the fractal
dimensions and the Lyapunov exponents. Usually, the largest Lyapunov coefficient do not
provide any difficulty, while higher order ones may be more difficult to get. Still, these are
much simpler to calculate than the fractal dimension. For a 1D model, calculation of the
latter may be reasonable. In two dimensions, it may still be possible. For a strange attractor,
generated by a two-variable discrete equation, about 100,000 points with greatest possible
accuracy are needed. For a strange attractor in three dimensions of, for instance the Lorenz
equation, it is an almost hopeless task to estimate a Hausdorff dimension directly. What can
be done is to make a calculation and use some projection to a lower dimension. As we shall
see, there are other measures of a fractal dimension, which are easier to calculate, and
which can be generally used for numerical purposes.
The largest Lyapunov exponent can always be estimated by calculations of how solutions
from close initial points diverge from each other. One can get to a more systematic method, and
we can show that for a simple sequence relation such as the logistic equation. The problem is
much the same as what was previously considered for the logistic equation. We need to
describe how values differ at each step, and this is provided by taking successive derivatives.
Assume a sequence relation: x n1 f(x n ). If two initial values close to a value x 0 differ by a
small amount d, the difference between the next values is given by the derivative: f (x 0 ) d.
For the first successive steps, we can write:
x
dx1
f( x ); f( x0
)
dx
1 0
dx2
d[ f( f( x0
))]
x2 f( x1) f( f( x0))
f( x1) f( x0)
dx dx
0
0
0
(Prime as usual denotes derivation.) Thus, there is a general chain rule for these derivatives:
dx
dx
n
0
f( x ) f( x ) ..... f( x ) f( x )
n1 n2 1 0